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in  @xcite , @xcite and @xcite we carried out a program to generalize l.c . siebenmann s famous manifold collaring theorem @xcite in ways applicable to manifolds with non - stable fundamental group at infinity . motivated by some important examples of finite - dimensional manifolds and a seminal paper by t.a . chapman and siebenmann @xcite on hilbert cube manifolds , we chose the following definitions . a manifold @xmath2 with compact boundary is called a _ homotopy collar _ if @xmath3 is a homotopy equivalence . if @xmath2 contains arbitrarily small homotopy collar neighborhoods of infinity , we call @xmath2 a _ pseudo - collar . _    clearly , an actual open collar @xmath2 , i.e. , @xmath4 , is a special case of a pseudo - collar . fundamental to @xcite , @xcite , and our earlier work , is the notion of ` inward tameness ' . a manifold @xmath5 is _ inward tame _ if each of its clean neighborhoods of infinity is finitely dominated ; it is _ absolutely inward tame _ if those neighborhoods all have finite homotopy type . an alternative formulation of this definition ( see  [ subsection : topology of ends of manifolds ] ) justifies the adjective ` inward'a term that helps distinguish this version of tameness from a similar , but inequivalent , version found elsewhere in the literature .    in @xcite a classification of pseudo - collarable @xmath0-manifolds for @xmath6 was obtained . in simplified form , it says :    [ pseudo - collarability characterization simple version][pct ]  a 1-ended @xmath0-manifold @xmath5 ( @xmath1 ) with compact boundary is pseudo - collarable iff the following conditions hold :    1 . @xmath5 is absolutely inward tame , and 2 . the fundamental group at infinity is @xmath7-semistable .    a ` @xmath7-semistable ( or perfectly semistable ) fundamental group at infinity ' indicates that an inverse sequence of fundamental groups of neighborhoods of infinity can be arranged so that bonding homomorphisms are surjective with perfect kernels .    by way of comparison , the simple version of siebenmann s collaring theorem is obtained by replacing b ) with the stronger condition of @xmath8-stability , while chapman and siebenmann s pseudo - collarability characterization for hilbert cube manifolds is obtained by omitting b ) entirely . thus , the differences among these three results lie entirely in the fundamental group at infinity .    in this paper we take a close look at @xmath0-manifolds satisfying only the inward tameness hypothesis . by necessity , our attention turns to the group theory at the ends of those spaces . unlike the case of infinite - dimensional manifolds , cw complexes , or even @xmath0-manifolds with noncompact boundary , inward tameness has major implications for the fundamental group at the ends of @xmath0-manifolds with compact boundary . unfortunately , inward tameness ( ordinary or absolute ) does not imply @xmath7-semistability  an example from @xcite attests to that  but it comes remarkably close . one of the main results of this paper is the following . [ theorem : inward tame implies ap - semistability]let @xmath5 be an inward tame @xmath0-manifold with compact boundary . then @xmath5 has an @xmath9-semistable ( almost perfectly semistable ) fundamental group at each of its finitely many ends .    developing the appropriate group theory ( including the definition of @xmath9-semistable ) and proving the above theorem are the initial goals of this paper . after that is accomplished , we apply those investigations by proving a structure theorem for manifolds that are inward tame , but not necessarily pseudo - collarable .    [ near pseudo - collarability characterization simple version][theorem : npc characterization simple version]a 1-ended @xmath0-manifold @xmath5 ( @xmath1 ) with compact boundary is nearly pseudo - collarable iff the following conditions hold :    1 . @xmath5 is absolutely inward tame , and 2 . the fundamental group at infinity is @xmath10-semistable . the notion of ` near pseudo - collarability ' will be defined and explored in  [ section : generalizing 1-sided cobordisms , homotopy collars , and pseudocollars ] . for now , we note that nearly pseudo - collarable manifolds admit arbitrarily small clean neighborhoods of infinity @xmath11 , containing codimension 0 submanifolds @xmath12 for which @xmath13 is a homotopy equivalence . obtaining a near pseudo - collar structure requires a slight strengthening of @xmath9-semistability to @xmath10-semistability ( strong almost perfect semistability ) . the essential nature of this stronger condition is verified by a final result , in which our group - theoretic explorations come together in a concrete set of examples .    [ theorem : existence of counterexamples]for all @xmath1 , there exist 1-ended open @xmath0-manifolds that are absolutely inward tame but do not have @xmath10-semistable fundamental group at infinity , and thus , are not nearly pseudo - collarable .    in  [ section : remaining questions ] , we close with a discussion of some open questions . _ throughout this paper attention is restricted to noncompact manifolds with compact boundaries . when a boundary is noncompact , its end topology gets entangled with that of the ambient manifold , leading to very different issues . in the study of noncompact manifolds , a focus on those with compact boundaries is analogous to a focus on closed manifolds in the study of compact manifolds . an investigation of manifolds with noncompact boundaries is planned for @xcite . _ in this subsection we review the definition of ` perfect group ' and discuss some variations .    given elements @xmath14 and @xmath15 of a group @xmath16 , the _ commutator _ @xmath17 will be denoted @xmath18   $ ] . the _ commutator subgroup _ of @xmath16 , denoted @xmath19   $ ] is the subgroup generated by all commutators . it is a standard fact that @xmath19   $ ] is normal in @xmath16 and is the smallest such subgroup with an abelian quotient . we call @xmath16 _ perfect _ if @xmath20   $ ] .    now suppose @xmath16 and @xmath21 are normal subgroups of @xmath22 . define @xmath23   $ ] to be the subgroup of @xmath22 generated by the set of commutators @xmath24   = \left\ {   k^{-1}j^{-1}kj\mid k\in k\text { and } j\in j\right\ }   .\ ] ] the following is standard and easy to verify . [ lemma : generalized commutator subgroups]for normal subgroups @xmath16 and @xmath21 of a group @xmath22 ,    1 . @xmath25   \trianglelefteq g$ ] , 2 . @xmath25   \trianglelefteq k$ ] and @xmath25 \trianglelefteq j$ ] , and 3 . @xmath25   = \left [   j , k\right ]   $ ] . given the above setup , we say that @xmath16 is @xmath21-_perfect _ if @xmath26   $ ] , and that @xmath16 is _ strongly _ @xmath21-_perfect _ if @xmath27   $ ] _ _ . _ _ by lemma [ lemma : generalized commutator subgroups ] , both of these conditions imply that @xmath28 ; so we customarily begin with that as an assumption . the following two lemmas are immediate . we state them explicitly for the purpose of comparison . let @xmath28 be normal subgroups of @xmath22 .    1 . @xmath16 is perfect if and only if each element of @xmath16 can be expressed as @xmath29   $ ] where @xmath30 for all @xmath31 . @xmath16 is @xmath21-perfect if and only if each element of @xmath16 can be expressed as @xmath29   $ ] where @xmath32 for all @xmath31 . @xmath16 is strongly @xmath21-perfect if and only if each element of @xmath16 can be expressed as @xmath29   $ ] where @xmath33 and @xmath34 for all @xmath31 . [ basic implications]let @xmath35 be normal subgroups of @xmath22 .    1 . if @xmath16 is [ strongly ] @xmath21-perfect , then @xmath16 is [ strongly ] @xmath36-perfect for every normal subgroup @xmath36 containing @xmath21 . @xmath16 is [ strongly ] @xmath16-perfect if and only if @xmath16 is a perfect group . _ lemma  [ basic implications ] suggests a key theme : the smaller the group _ _ for which _ @xmath16 _ is [ strongly ] _ @xmath36-_perfect , the closer _ @xmath16 _ is to being a genuine perfect group . _    the various levels of perfectness can be nicely characterized using _ group homology_. the @xmath37-homology of a group @xmath22 may be defined as the @xmath37-homology of a @xmath38 space @xmath39 . if @xmath40 is a homomorphism , there is a map @xmath41 , unique up to basepoint preserving homotopy , inducing @xmath42 on fundamental groups . define @xmath43 to be the homomorphisms induced by @xmath44 . [ lemma : group homology]let @xmath28 , @xmath45 be inclusion , and @xmath46 be projection .    1 . @xmath16 is perfect if and only if @xmath47 . @xmath16 is @xmath21-perfect if and only if @xmath48 if and only if @xmath49 3 . @xmath16 is strongly @xmath21-perfect if and only if @xmath16 is @xmath21-perfect and @xmath50 is surjective . claim 1 ) is clear from the standard fact that @xmath51   $ ] . claim 2 ) can be verified with elementary group theory . claim 3 ) follows from a well - known 5-term exact sequence  due to stallings @xcite and stammbach @xcite . due to its importance in this paper , we state it as a separate lemma . [ 5-term exact sequence for group homology][lemma : 5 term exact ] given a normal subgroup @xmath16 of a group @xmath21 , there is a natural exact sequence:@xmath52   \rightarrow h_{1}\left (   j;\mathbb{z } \right )   \rightarrow h_{1}\left (   j / k;\mathbb{z } \right )   \rightarrow0\text{.}\ ] ]    the following elementary facts about group homology will be useful . [ lemma : induced surjections on h_2]let @xmath53 be a map between connected cw complexes and @xmath54 the induced homomorphism . then    1 . @xmath55 , 2 . @xmath56 realizes @xmath57 , and 3 .   if @xmath58 is surjective , then @xmath59 is also surjective . build a @xmath60 complex @xmath61 by attaching cells of dimension @xmath62 to @xmath63 and a @xmath64 complex @xmath65 by attaching cells of dimension @xmath62 to @xmath66 . both @xmath67 and @xmath68 induce isomorphisms on @xmath8 and @xmath69 , so ( 1 ) follows . use the asphericity of @xmath65 to extend @xmath70 to @xmath71 , also inducing @xmath42 on @xmath8 . clearly @xmath72 and @xmath73 are surjective . this gives a commutative diagram@xmath74{ccc}h_{2}\left (   x;\mathbb{z } \right )   & \overset{f_{\ast}}{\twoheadrightarrow } & h_{2}\left (   y;\mathbb{z } \right ) \\ i_{\ast}\downarrow &   & \downarrow j_{\ast}\\ h_{2}\left (   \pi_{1}\left (   x\right )   ; \mathbb{z } \right )   & \overset{f_{\ast}^{\prime}}{\longrightarrow } & h_{2}\left (   \pi _ { 1}\left (   y\right )   ; \mathbb{z } \right ) \end{array}\ ] ] since the other maps are all surjective , so is @xmath75 . lastly we offer a topological characterization of the various levels of perfectness . for the purposes of this paper , these are possibly the most useful . let @xmath76 denote a compact orientable surface of genus @xmath77 with a single boundary component . a collection of oriented simple closed curves @xmath78 on @xmath76 with the property that each @xmath79 intersects @xmath80 transversely at a single point , and each of @xmath81 , @xmath82 , and @xmath83 is empty when @xmath84 , is called a _ complete set of handle curves _ for @xmath76 . a complete set of handle curves on @xmath76 is not unique ; however given any such set , there exists a homeomorphism of @xmath76 to the ` disk with @xmath77 handles ' pictured in figure [ fig1-handle curves ] taking each @xmath79 and @xmath80 to the corresponding curves in the diagram . [ ptb ]    figure1-ends4.eps    given a ( not necessarily embedded ) loop @xmath85 in a topological space @xmath63 , we say that @xmath85 _ bounds a compact orientable surface in _ @xmath63 if , for some @xmath77 , there exists a map @xmath86 such that @xmath87 . notice that we do not require that @xmath70 be an embedding . we often abuse terminology slightly by saying that @xmath85 bounds the surface @xmath76 in @xmath63 . similarly , we often do not distinguish between a set of handle curves on @xmath76 and their images in @xmath63 . [ lemma : geometric perfect ] let @xmath63 be a space with @xmath88 and let @xmath28 be normal subgroups of @xmath22 . then    1 . @xmath16 is perfect if and only if each loop @xmath85 in @xmath63 representing an element of @xmath16 bounds a surface @xmath76 in @xmath63 containing a complete set of handle curves @xmath89 with each @xmath79 and @xmath80 belonging to @xmath16 . @xmath16 is @xmath21-perfect if and only if each loop @xmath85 in @xmath63 representing an element of @xmath16 bounds a surface @xmath76 in @xmath63 containing a complete set of handle curves @xmath89 with each @xmath79 and @xmath80 belonging to @xmath21 . @xmath16 is strongly @xmath21-perfect if and only if each loop @xmath85 in @xmath63 representing an element of @xmath16 bounds a surface @xmath76 in @xmath63 containing a complete set of handle curves @xmath90 with each @xmath79 belonging to @xmath16 and each @xmath80 belonging to @xmath21 . _ we are being informal in the statement of lemma [ lemma : geometric perfect ] . since the handle curves are not based , we should also choose , for each pair _ @xmath91 _ _ , an arc _ _ @xmath92 _ in _ @xmath76 _ from _ @xmath93 _ to _ @xmath94__. the element of _ _ @xmath95 _ represented by _ @xmath79 _ is then _ @xmath96 _ _ , and similarly for _ _ @xmath80__. notice that , by normality , the question of whether one of these loops belongs to _ _ @xmath16 _ or _ @xmath21 _ is independent of the choice of _ @xmath97__. _ _      understanding the ` fundamental group at infinity ' requires the language of inverse sequences . we briefly review the necessary definitions and terminology .    throughout this subsection all arrows denote homomorphisms , while those of type @xmath98 or @xmath99 specify surjections . the symbol @xmath100 denotes isomorphisms . let @xmath101 be an inverse sequence of groups and homomorphisms . subsequence _ is an inverse sequence of the form @xmath102 in the future we denote a composition @xmath103 ( @xmath104 ) by @xmath105 . sequences @xmath106 and @xmath107 are _ pro - isomorphic _ if , after passing to subsequences , there exists a commuting diagram : @xmath74{ccccccc}g_{i_{0 } } & \overset{\lambda_{i_{0}+1,i_{1}}}{\longleftarrow } & g_{i_{1 } } & \overset{\lambda_{i_{1}+1,i_{2}}}{\longleftarrow } & g_{i_{2 } } & \overset { \lambda_{i_{2}+1,i_{3}}}{\longleftarrow } & \cdots\\ & \nwarrow\quad\swarrow &   & \nwarrow\quad\swarrow &   & \nwarrow\quad\swarrow & \\ & h_{j_{0 } } & \overset{\mu_{j_{0}+1,j_{1}}}{\longleftarrow } & h_{j_{1 } } & \overset{\mu_{j_{1}+1,j_{2}}}{\longleftarrow } & h_{j_{2 } } & \cdots \end{array } .\ ] ] clearly an inverse sequence is pro - isomorphic to each of its subsequences . to avoid tedious notation , we often do not distinguish @xmath106 from its subsequences . instead we assume @xmath108 has the properties of a preferred subsequence prefaced by the words ` after passing to a subsequence and relabeling ' . an inverse sequence @xmath106 is _ stable _ if it is pro - isomorphic to an @xmath109 for which each @xmath110 is an isomorphism . a more usable formulation is that @xmath108 is stable if , after passing to a subsequence and relabeling , there is a commutative diagram of the form @xmath74{ccccccccc}g_{0 } & \overset{\lambda_{1}}{\longleftarrow } & g_{1 } & \overset{\lambda_{2}}{\longleftarrow } & g_{2 } & \overset{\lambda_{3}}{\longleftarrow } & g_{3 } & \overset{\lambda_{4}}{\longleftarrow } & \cdots\\ & \nwarrow\quad\swarrow &   & \nwarrow\quad\swarrow &   & \nwarrow\quad\swarrow &   &   & \\ & im(\lambda_{1 } ) & \overset{\cong}{\longleftarrow } & im(\lambda_{2 } ) & \overset{\cong}{\longleftarrow } & im(\lambda_{3 } ) & \overset{\cong}{\longleftarrow } & \cdots & \end{array } \tag{$\ast$}\ ] ] where all unlabeled maps are obtained by restriction . if @xmath109 can be chosen so that each @xmath110 is an epimorphism , we call our sequence _ semistable _ ( or _ mittag - leffler , _ or _ _  pro - epimorphic _ _ ) . in that case , it can be arranged that the maps in the bottom row of ( @xmath111 ) are epimorphisms . similarly , if @xmath109 can be chosen so that each @xmath110 is a monomorphism , we call our sequence _ pro - monomorphic _ ; it can then be arranged that the restriction maps in the bottom row of ( @xmath111 ) are monomorphisms . it is easy to show that an inverse sequence that is semistable and pro - monomorphic is stable . an inverse sequence is _ perfectly semistable _ if it is pro - isomorphic to an inverse sequence @xmath112 of finitely presentable groups and surjections where each @xmath113 is perfect . a straightforward argument ( * ? ? ? 1 ) shows that sequences of this type behave well under passage to subsequences . an _ augmentation _ of an inverse sequence @xmath114 is a sequence @xmath115 , where @xmath116 and @xmath117 for each @xmath31 . the corresponding _ augmentation sequence _ is the sequence @xmath118 . the _ minimal augmentation _ ( or the _ unaugmented case _ ) occurs when @xmath119 ; the _ maximal augmentation _ is the case where @xmath120 ; and the _ standard augmentation _ occurs when @xmath121 for each @xmath31 . any augmentation where @xmath122 for each @xmath31 is called a _ small augmentation . _ for each subsequence @xmath123 of a sequence @xmath108 augmented by @xmath115 , there is a corresponding augmentation @xmath124 . we say that @xmath106 satisfies the @xmath125__-perfectness property _ _ if , for each @xmath31 , @xmath126 is @xmath127-perfect ; it satisfies the _ strong _ @xmath115__-perfectness property _ _ if each @xmath128 is strongly @xmath129-perfect . more concisely , if @xmath130 and @xmath131 , these conditions require that each @xmath132 be [ strongly ] @xmath133-perfect .    employing the above terminology , we can restate the definition perfect semistability ( abbreviated @xmath7__-semistable _ _ ) by requiring that the sequence be pro - isomorphic an inverse sequence of finitely presented groups and surjections satisfying the @xmath115-perfectness property for the minimal augmentation @xmath134 . more generally , we call an inverse sequence of groups :    * @xmath9__-semistable _ _ ( for almost perfectly semistable ) if it is pro - isomorphic to an inverse sequence @xmath135 of finitely presentable groups and surjections , satisfying the @xmath125-perfectness property for some small augmentation @xmath125 , and * @xmath10__-semistable _ _ ( for strongly almost perfectly semistable ) if it is pro - isomorphic to an inverse sequence @xmath108 of finitely presentable groups and surjections satisfying the strong @xmath115-perfectness property for some small augmentation @xmath115 .    _ note that an inverse sequence satisfies the [ strong ] _ @xmath125__-perfectness property for _ _ some _ small augmentation _ @xmath115 _ if and only if it satisfies that property for the standard augmentation . _    when applying sequences of the above types to geometric constructions , it is frequently desirable to pass to subsequences without losing the defining property of the sequence . for that reason , the following observation is crucial . [ prop : nearly sub]if an inverse sequence @xmath114 of surjections augmented by @xmath115 satisfies the [ strong ] @xmath115-perfectness property , then any subsequence @xmath123 satisfies the corresponding [ strong ] @xmath124-perfectness property . since the proofs for perfectness and strong perfectness are similar , we prove only the latter . assume @xmath106 augmented by @xmath115 satisfies strong @xmath136-perfectness . simplifying notation , a portion of the given subsequence becomes@xmath137 where @xmath138 . we must show that @xmath139   $ ] . suppose the proposition holds for @xmath140 . if @xmath141 , then @xmath142 , and the result follows by hypothesis . now , assume @xmath143 and write @xmath144 let @xmath145 ; then @xmath146 . by induction , @xmath147@xmath148   $ ] ; so , @xmath149 is a product of commutators @xmath150 $ ] where @xmath151 and @xmath152 . since @xmath153 is surjective over @xmath154 we identify for each @xmath155 a pair of elements @xmath156 that map to @xmath157 and @xmath158 , respectively . thus , @xmath159 , @xmath160 , and @xmath161\in\lbrack\ker\left ( \lambda_{b+1,c}\right )   , \lambda_{b+1,c}^{-1}\left (   l_{b}\right )   ] $ ] .    now , let @xmath162 be the product of the commutators with @xmath163   $ ] replacing @xmath164   $ ] . by construction , @xmath165 and @xmath166 $ ] . thus,@xmath167 \subseteq\left [   \ker\left (   \lambda_{b+1,c}\right )   , \lambda_{b+1,c}^{-1}\left (   l_{b}\right )   \right ]   \text{.}\ ] ] consequently , @xmath168 $ ] as well , completing the proof of the proposition . next we supply some topological definitions and background . throughout the paper , @xmath169 represents homeomorphism and @xmath170 indicates homotopic maps or homotopy equivalent spaces . the word _ manifold _ means _ manifold with ( possibly empty ) boundary_. a manifold is _ open _ if it is non - compact and has no boundary . as noted earlier , we restrict our attention to manifolds with compact boundaries .    for convenience , all manifolds are assumed to be pl ; analogous results may be obtained for smooth or topological manifolds in the usual ways . our standard resource for pl topology is @xcite . some of the results presented here are valid in all dimensions . others are valid in dimensions @xmath171 or @xmath172 , but require the purely topological @xmath173-dimensional techniques found in @xcite for the @xmath173 and/or @xmath174 dimensional cases ; there the conclusions are only topological . the main focus of this paper is on dimensions @xmath175 . let @xmath5 be a manifold with compact ( possibly empty ) boundary . a set @xmath176 is a _ neighborhood of infinity _ if @xmath177 is compact . a neighborhood of infinity @xmath11 is _ clean _ if    * @xmath11 is a closed subset of @xmath5 , * @xmath178 , and * @xmath11 is a codimension 0 submanifold of @xmath5 with bicollared boundary . it is easy to see that each neighborhood of infinity contains a clean neighborhood of infinity . we say that @xmath5 _ has _ _ ends _ if it contains a compactum @xmath180 such that , for every compactum @xmath181 with @xmath182 , @xmath183 has exactly @xmath179 unbounded components , i.e. , @xmath179 components with noncompact closures . when @xmath179 exists , it is uniquely determined ; if @xmath179 does not exist , we say @xmath5 _ has infinitely many ends_. if @xmath5 is @xmath179-ended , then it contains a clean neighborhood of infinity @xmath11 consisting of @xmath179 connected components , each of which is a 1-ended manifold with compact boundary . thus , when studying manifolds with finitely many ends , it suffices to understand the @xmath184__-ended _ _ situation . that is the case in this paper , where our standard hypotheses ensure finitely many ends . ( see theorem  [ semistable ] . ) a connected clean neighborhood of infinity with connected boundary is called a _ 0-neighborhood of infinity_. a @xmath185-neighborhood of infinity @xmath11 for which @xmath186 induces a @xmath8-isomorphism is called a _ generalized 1-neighborhood of infinity_. if , in addition , @xmath187 for @xmath188 , then @xmath11 is a _ generalized k - neighborhood of infinity_.    a nested sequence @xmath189 of neighborhoods of infinity is _ cofinal _ if @xmath190 . we will refer to any cofinal sequence @xmath191 of closed neighborhoods of infinity with @xmath192 , for all @xmath31 , as an _ end structure _ for @xmath5 . descriptors will be added to indicate end structures with additional properties . for example , if each @xmath193 is clean we call @xmath191 a _ clean end structure _ ; if each @xmath193 is clean and connected we call @xmath194 a _ clean connected end structure _ ; and if each @xmath193 is a generalized @xmath179-neighborhood of infinity , we call @xmath194 a _ generalized k - neighborhood end structure_.    _ the word ` generalized ' in the above definitions is in deference to siebenmann s terminology in @xcite where the ambient manifold _ @xmath5 _ _ is assumed to have stable fundamental group at infinity . there a ( non - generalized ) k - neighborhood of infinity  _ _ @xmath11 _ is also required to satisfy _ @xmath195__. _ _    building upon the above terminology , the primary goal of this paper can be described as : identify , construct , and detect the existence of various end structures for manifolds . a central example  the _ pseudo - collar _ can be described as an end structure @xmath194 where each @xmath193 is a homotopy collar . we say @xmath196is _ inward tame _ if , for arbitrarily small neighborhoods of infinity @xmath11 , there exist homotopies @xmath197   \rightarrow n$ ] such that @xmath198 and @xmath199 is compact . thus inward tameness means each neighborhood of infinity can be pulled into a compact subset of itself . in this case we refer to @xmath200 as a _ taming homotopy_.    in @xcite , the existence of generalized @xmath201-neighborhood end structures is shown for all inward tame @xmath5 ( @xmath202 ) . recall that a space @xmath63 is _ finitely dominated _ if there exists a finite complex @xmath16 and maps @xmath203 and @xmath204 such that @xmath205 . the following lemma uses this notion to offer equivalent formulations of inward tameness . [ lemma : inward tame ] ( see ( * ? ? ? * lemma 2.4 ) ) for a manifold @xmath5 , the following are equivalent .    1 . @xmath5 is inward tame . each clean neighborhood of infinity in @xmath5 is finitely dominated . 3 .   for each clean end structure @xmath194 , the inverse sequence @xmath206 is pro - homotopy equivalent to an inverse sequence of finite polyhedra .    given a clean connected end structure @xmath207 , base points @xmath208 , and paths @xmath209 connecting @xmath210 to @xmath211 , we obtain an inverse sequence : @xmath212 here , each @xmath213 is the homomorphism induced by inclusion followed by the change of base point isomorphism determined by @xmath79 . the singular ray obtained by piecing together the @xmath79 s is called the _ base ray _ for the inverse sequence . provided the sequence is semistable , its pro - isomorphism class does not depend on any of the choices made above ( see @xcite or @xcite ) . in the absence of semistability , the pro - isomorphism class of the inverse sequence depends on the base ray ; hence , the ray becomes part of the data . the same procedure may be used to define @xmath214 for all @xmath215 . similarly , but without need for a base ray or connectedness , we may define @xmath216 . in @xcite , wall showed that each finitely dominated connected space @xmath63 determines a well - defined @xmath217   \right )   $ ] ( the reduced projective class group ) that vanishes if and only if @xmath63 has the homotopy type of a finite complex . given a clean connected end structure @xmath218 for an inward tame @xmath5 , we have a wall finiteness obstruction @xmath219 for each @xmath31 . these may be combined into a single obstruction @xmath220   \right)\ ] ] that is well - defined and which vanishes if and only if each clean neighborhood of infinity in @xmath5 has finite homotopy type . see @xcite or @xcite for details . we may now state the full version of the main theorem of @xcite . [ pseudo - collarability characterization complete version][pct technical version]a 1-ended @xmath0-manifold @xmath5 ( @xmath1 ) with compact boundary is pseudo - collarable if and only if the following conditions hold :    1 . @xmath5 is inward tame , 2 . @xmath221 is @xmath7-semistable , and 3 . in this section we show that , for manifolds with compact boundary , the inward tameness condition , by itself , has significant implications . the main goal is a proof of theorem [ theorem : inward tame implies ap - semistability]that every inward tame manifold with compact boundary has @xmath9-semistable fundamental group at each of its finitely many ends . results in this section are valid in all ( finite ) dimensions .    begin by recalling a theorem from @xcite .    [ semistable]if an @xmath0-manifold with compact ( possibly empty ) boundary is inward tame , then it has finitely many ends , each of which has semistable fundamental group and stable homology in all dimensions . _ note that _ none _ of the above conclusions is valid for hilbert cube manifolds , polyhedra , or manifolds with noncompact boundary . see , for example , @xcite . _    as preparation for the proof of theorem [ theorem : inward tame implies ap - semistability ] , we look at an easier result that follows directly from theorem [ semistable ] . let @xmath5 be an inward tame @xmath0-manifold with compact boundary . since @xmath5 is finite - ended , there is no loss of generality in assuming that @xmath5 is 1-ended . by taking a product with @xmath223 ( @xmath224 ) if necessary , we may arrange that @xmath1 , without changing the fundamental group at infinity . so , by the semistability conclusion of theorem  [ semistable ] combined with the generalized @xmath225-neighborhood theorem ( * ? ? ? * th.4 ) , we may choose a generalized @xmath225-neighborhood end structure @xmath194 for which each bonding map in the inverse sequence @xmath226 is surjective . abelianization gives an inverse sequence@xmath227 which , by theorem  [ semistable ] , is stable . it follows that all but finitely many of the epimorphisms in ( [ sequence : h_1 sequence ] ) are isomorphisms , so by omitting finitely many terms ( then relabeling ) , we may assume all bonds in ( [ sequence : h_1 sequence ] ) are isomorphisms . a term - by - term application of lemma  [ lemma : group homology ] gives the following . [ prop : near perfect 1]every 1-ended inward tame manifold @xmath5 with compact boundary admits a generalized @xmath225-neighborhood end structure @xmath191 for which all bonding maps in the sequence @xmath228 are surjective and each @xmath128 is @xmath229-perfect ; in other words , if @xmath230 is the maximal augmentation , then @xmath231 satisfies the @xmath115-perfectness property . theorem [ theorem : inward tame implies ap - semistability ] is a stronger version of proposition [ prop : near perfect 1 ] . for clarity , we restate it in a similar form . [ prop : near perfect 2]every 1-ended inward tame manifold @xmath5 with compact boundary admits a generalized @xmath225-neighborhood end structure @xmath191 for which all bonding maps in the sequence @xmath228 are surjective and , if we let @xmath130 for each @xmath232 ( the standard augmentation ) , then @xmath132 is @xmath233-perfect for all @xmath234 . in other words , @xmath231 satisfies the @xmath235-perfectness property ; so @xmath5 has @xmath9-semistable fundamental group at infinity . assume the sequence @xmath194 was chosen so that , for each @xmath31 , @xmath236 is sufficiently small that a taming homotopy @xmath237 pulls @xmath193 into @xmath238 , i.e. , @xmath239 , and @xmath240 is sufficiently small that @xmath241   \right )   \cap n_{i+3}=\varnothing$ ] . by compactness of @xmath242 and @xmath243   \right )   $ ] those choices can be made . now let @xmath234 be fixed and @xmath244 be the universal covering projection . let @xmath245 and for @xmath246 , @xmath247 and @xmath248 then @xmath249 ; and @xmath250 lifts to a proper homotopy @xmath251 that pulls @xmath252 into @xmath253 and for which @xmath254   \right )   $ ] misses @xmath255 . we may associate @xmath233 with @xmath256 and @xmath132 with @xmath257 . thus , an arbitrary element of @xmath132 may be viewed as a loop @xmath258 in @xmath259 that bounds a disk @xmath181 in @xmath260 . to prove the proposition , it suffices to show that @xmath258 bounds an orientable surface in @xmath261 . by @xmath8-surjectivity and the fact that the @xmath262 s are generalized @xmath225-neighborhoods , @xmath258 may be homotoped within @xmath263 to a loop @xmath264 in @xmath265 . let @xmath266 be the cylinder in @xmath263 between @xmath258 and @xmath264 traced out by that homotopy . then the disk @xmath267 may be viewed as an element @xmath268\in h_{2}\left (   \widehat{a}_{i}\cup\widehat{a}_{i-1},\partial\widehat{n}_{i+1}\right )   $ ] . let @xmath269\cup_{\partial\widehat { n}_{i}\times\{0\}}\widehat{a}_{i}\rightarrow\widetilde{a}_{i-2}\cup \widehat{a}_{i-1}\cup\widehat{a}_{i}\ ] ] be the identity on @xmath263 and @xmath270 on @xmath271 $ ] . by pl transversality theory ( see @xcite or @xcite ) , we may  after a small proper adjustment that does not alter @xmath272 on @xmath273assume that @xmath274 is a manifold with boundary that is a homeomorphism over a collar neighborhood of @xmath265 . let @xmath275 be the component of @xmath276 containing that neighborhood . then , by local characterization of degree , @xmath277 is a proper degree @xmath225 map , and @xmath278 . thus we have a surjection@xmath279 let @xmath280   $ ] be a preimage of @xmath281 $ ] . we may assume that @xmath282 is an orientable surface with boundary in @xmath275 . since @xmath272 is the identity on @xmath283 , @xmath284 is homologous in @xmath283 to @xmath285 . so , without loss of generality , we man assume that @xmath286 . since @xmath275 lies in @xmath271\cup_{\partial\widehat{n}_{i}\times\{0\}}\widehat{a}_{i}$ ] , we may push @xmath282 , rel boundary , into @xmath263 . this provides an orientable surface in @xmath287 with boundary @xmath264 . gluing the cylinder @xmath266 to that surface along @xmath264 produces the bounding surface for @xmath258 that we desire . early attempts to prove @xmath7-semistability ( hence pseudo - collarability ) with only an assumption of inward tameness , were brought to a halt by the discovery of a key example presented in @xcite . ideas contained in that example play an important role here , so we provide a quick review .    an easy way to denote normal subgroups will be helpful . let @xmath22 be a group and @xmath288 . the _ normal closure of s in g _ is the smallest normal subgroup of @xmath22 containing @xmath289 . we denote it by @xmath290 . [ main example from @xcite][example : 1 ] for all @xmath1 , there exist 1-ended absolutely inward tame open @xmath0-manifolds with fundamental group system@xmath112 where @xmath291,a_{2}=\left [   a_{2},a_{1}\right ]   , \cdots , a_{i}=\left [   a_{i},a_{i-1}\right ]   \right\rangle\ ] ] and @xmath292 sends @xmath293 to @xmath293 for @xmath294 and @xmath295 to @xmath225 .    by a largely algebraic argument , it was shown that these examples do not have @xmath7-semistable fundamental group at infinity , and thus , are not pseudo - collarable . notice , however , that each @xmath130 is the normal closure of @xmath295 and @xmath296   $ ] in @xmath297 ; so @xmath298   $ ] . in other words , @xmath114 satisfies the strong @xmath235-perfectness property , and is therefore @xmath10-semistable . in addition to the above algebra , these examples have nice topological properties . although they do not contain small homotopy collar neighborhoods of infinity , they do contain arbitrarily small generalized @xmath225-neighborhoods of infinity @xmath11 for which @xmath186 is @xmath37-homology equivalence . in fact , they contain a sequence @xmath191 of generalized @xmath225-neighborhoods of infinity with @xmath299 and @xmath300 a @xmath301$]-homology equivalence . these observations provide much of the motivation for the remainder of this paper . we begin developing ideas for placing example [ example : 1 ] into a general context . we will see that end structures like those found in that example are possible only when kernels satisfy a strong relative perfectness condition . conversely , we will show that whenever such a group theoretic condition is present , a corresponding ` near pseudo - collar ' structure is attainable . we have already defined pseudo - collar structure on the end of a manifold @xmath5 to be an end structure @xmath194 for which each @xmath193 is a homotopy collar , i.e. , each @xmath302 is a homotopy equivalence . the existence of such a structure allows us to express each @xmath193 as a union@xmath303 where @xmath304 , and each triple @xmath305 is a compact _ one - sided h - cobordism _ in the sense that @xmath306 is a homotopy equivalence ( and @xmath307 is probably not ) . one - sided cobordisms play an important role in manifold topology in general , and the topology of ends in particular . see @xcite for details . for later use , we review a few key properties of one - sided h - cobordisms . see , for example , ( * ? ? ? * theorem 2.5 )    [ one - sided]let @xmath308 be a compact cobordism between closed manifolds with @xmath309 a homotopy equivalence . then    1 . @xmath309 and @xmath310 are @xmath311   $ ] -homology equivalences , i.e. , @xmath312   \right )   \allowbreak=\allowbreak 0\allowbreak=\allowbreak h_{\ast}\left (   w , q;\mathbb{z } \left [   \pi_{1}\left (   w\right )   \right ]   \right )   $ ] 2 . @xmath313 is surjective , and @xmath314 3 . @xmath315 is perfect    moving forward , we require generalizations of the fundamental concepts : homotopy equivalence , homotopy collar , one - sided h - cobordism and pseudo - collar . they are as follows :    * let @xmath316 be a cw - pair for which @xmath317 induces a @xmath8-isomorphism , and let @xmath318 . call @xmath31 a @xmath319__-homotopy equivalence _ _ if @xmath320\right )   = 0 $ ] for all @xmath111 . extension to arbitrary maps is accomplished by use of mapping cylinders . * a manifold @xmath11 with compact boundary is a @xmath319__-homotopy collar _ _ if @xmath321 and @xmath186 is a @xmath319-homotopy equivalence . * let @xmath322 be a compact cobordism between closed manifolds and @xmath323 . we call @xmath324 a @xmath319__-one - sided h - cobordism _ _ if @xmath325 is a @xmath319-homotopy equivalence and @xmath326 induces a surjection on fundamental groups . * let @xmath194 be a generalized @xmath225-neighborhood end structure on a manifold @xmath5 , chosen so that the bonding maps in @xmath327 are surjective , and let @xmath115 be an augmentation of this sequence . call @xmath194 a @xmath328 _ pseudo - collar structure _ if each @xmath302 is a @xmath329-homotopy equivalence .    _ i ) _ * * _ each of the above definitions reduces to its traditional counterpart when the subgroup(s ) involved are trivial . _    _ ii ) _ _ in the generalization of one - sided h - cobordism , we _ require @xmath330 _ to be surjective  a condition that is automatic when _ @xmath331 _ _ , but not in general . analogs of the other two assertions of theorem [ one - sided ] will be shown to follow . _ _    _ iii ) _ * * _ for the maximal augmentation , the generalization of pseudo - collar requires only that _ _ each _ @xmath300 _ be a _ @xmath37__-homology equivalence ; whereas , for the trivial augmentation , we have a genuine pseudo - collar . the key dividing line between those extremes occurs when _ _ @xmath115 _ _ is a small augmentation ( _ _ @xmath332 _ for all _ @xmath31 _ _ ) . in those cases , we call _ _ @xmath194 _ a _ near pseudo - collar _ structure , and say that a 1-ended _ @xmath5 _ with compact boundary is _ nearly pseudo - collarable _ if it admits such a structure . the geometric significance of the ` small augmentation ' requirement will become clear in the proof of theorem [ npct technical version ] . further discussion of that topic is contained in [ section : remaining questions ] . _    the following lemma adds topological meaning to the definition of @xmath333-homotopy equivalence . [ lemma : ( modl ) h.e.]let @xmath316 be a cw - pair for which @xmath317 induces a @xmath8-isomorphism , @xmath334 , and @xmath335 for which @xmath336 . obtain @xmath337 from @xmath12 by attaching a @xmath338-disk @xmath339 along each @xmath340 ; let @xmath341 , and @xmath342 . then @xmath31 is a @xmath319-homotopy equivalence if and only if @xmath343 is a homotopy equivalence . let @xmath344 be the covering projection corresponding to @xmath36 . then @xmath345 is the cover of @xmath12 corresponding to @xmath36 . viewing @xmath289 as a collection of loops in @xmath12 and @xmath346 the set of all lifts of those loops , then attaching @xmath338-disks to @xmath347 ( and , simultaneously @xmath348 ) along @xmath346 produces universal covers @xmath349 and @xmath350 . assume now that @xmath317 is a @xmath351-homotopy equivalence . then by shapiro s lemma @xcite , @xmath352 , so by excision @xmath353 . since both spaces are simply connected , the relative hurewicz theorem implies that @xmath354 ; therefore @xmath355 . by whitehead s theorem @xmath343 is a homotopy equivalence . conversely , if @xmath343 is a homotopy equivalence , then its lift @xmath356 is a homotopy equivalence . therefore @xmath357 , so by excision @xmath358 , and by shapiro s lemma @xmath320\right )   = 0 $ ] . the following is a useful corollary . [ lemma : ( modq ) h.e . ] let @xmath316 be a cw - pair for which @xmath317 induces a @xmath8-isomorphism . suppose @xmath359 . if @xmath360\right )   = 0 $ ] , then @xmath361\right )   = 0 $ ] for any @xmath21 with @xmath362 . in particular , @xmath363    the next observation is a direct analog of theorem [ one - sided ] . [ thm : one - sided general ] let @xmath322 be a compact @xmath351-one - sided h - cobordism between closed manifolds with @xmath323 . let @xmath326 and @xmath364 . then    1 .   both @xmath309 and @xmath310 are @xmath365$]-homology equivalences , i.e. , @xmath366\right )   = \allowbreak0=\allowbreak h_{\ast}\left (   w , q;\mathbb{z } \mathbb{[\pi}_{1}\left (   w\right )   /l]\right )   $ ] , and 2 . @xmath367 @xmath368 is strongly @xmath369-perfect .    first note that by the surjectivity of @xmath370 , there is a canonical isomorphism @xmath371 that is assumed throughout . let @xmath372 be the covering projection corresponding to @xmath36 , @xmath373 and @xmath374 . then both @xmath375 and @xmath376 are connected , and their projections onto @xmath377 and @xmath378 are the coverings corresponding to @xmath36 and @xmath369 .    the assertion that @xmath366\right )   = 0 $ ] is part of the hypothesis , and ( by shapiro s lemma @xcite ) equivalent to the assumption that @xmath379 . to show that @xmath380\right )   $ ] vanishes in all dimensions , it suffices to show that @xmath381 . this will follow from poincar duality for noncompact manifolds if we can verify :    _ claim . _ @xmath382 _ _ , where the  _ _ @xmath70__ indicates cellular cohomology based on finite cochains . ( see ( * ? ? ? _    applying lemma [ lemma : ( modl ) h.e . ] , attach @xmath338-cells to @xmath383 along a collection @xmath289 of loops in @xmath377 to kill @xmath36 , obtaining spaces @xmath384 and @xmath385 , and a homotopy equivalence @xmath386 . since @xmath383 is compact , any strong deformation retraction of @xmath385 onto @xmath384 is proper , and hence , lifts to a proper strong deformation retraction of universal covers @xmath387 onto @xmath388 @xcite . it follows that @xmath389 . both universal covers are obtained by attaching disks along the collection @xmath346 of lifts to @xmath375 and @xmath390 of the loops in @xmath289 . by excising the interiors of those disks , we conclude that @xmath391 .    to verify assertion ( 2 ) , consider the short exact sequence @xmath392 where @xmath393 may be identified with @xmath36 . lemma [ lemma : 5 term exact ] provides the 5-term exact sequence @xmath394   \rightarrow h_{1}\left ( l^{\prime};\mathbb{z } \right )   \overset{q_{\ast1}}{\longrightarrow}h_{1}\left (   l^{\prime}/k;\mathbb{z } \right )   \rightarrow0\text{.}\ ] ] from which the @xmath369-perfectness of @xmath16 can be deduced by showing that @xmath395 is an epimorphism and @xmath396 an isomorphism . since @xmath397 induces @xmath398 and since @xmath399 , the the long exact sequence for that pair ensures that @xmath400 . in addition , the surjectivity of @xmath401 combines with lemma [ lemma : induced surjections on h_2 ] to imply surjectivity of @xmath402 . with all necessary definitions in place , we are ready to prove the second main theorem described in the introduction . we begin by stating a strong form of the theorem , written in the style of earlier characterization theorems from @xcite and @xcite . [ near pseudo - collarability characterization][npct technical version]a 1-ended @xmath0-manifold @xmath5 ( @xmath1 ) with compact boundary is nearly pseudo - collarable iff each of the following conditions holds :    1 . @xmath5 is inward tame , 2 . the fundamental group at infinity is @xmath10-semistable , and 3 . @xmath222 . recall that condition ( 2 ) presumes the existence of a representation of @xmath403 of the form@xmath404 with a small augmentation @xmath115 ( @xmath405 , for all @xmath31 ) so that each @xmath132 is strongly @xmath133-perfect , where @xmath131 . first we verify that a nearly pseudo - collarable 1-ended manifold with compact boundary must satisfy conditions ( 1)-(3 ) . the hypothesis provides a generalized @xmath225-neighborhood end structure @xmath191 on @xmath5 with group data @xmath406 ( @xmath407 ) and a small augmentation @xmath125 ( @xmath408 ) such that each @xmath193 is a @xmath409-homotopy collar .    to simultaneously verify ( 1 ) and ( 3 ) , it suffices to exhibit a cofinal sequence of clean neighborhoods of infinity , each having finite homotopy type . lemma [ lemma : ( modq ) h.e . ] ensures that each @xmath193 is a @xmath410-homotopy collar , and since each @xmath292 is a surjection between finitely presented groups , each @xmath132 is finitely generated as a normal subgroup of @xmath297 . let @xmath31 be fixed , and @xmath411 a finite collection of loops in @xmath412 that normally generates @xmath132 in @xmath297 . by lemma [ lemma : ( modl ) h.e . ] , if we abstractly attach a @xmath338-disk @xmath413 along each @xmath414 , we obtain a homotopy equivalence @xmath415 in particular , @xmath416 has the homotopy type of a finite complex . but , since each @xmath414 represents an element of @xmath128 , we may assume that each @xmath413 is properly embedded in @xmath417 . by thickening these @xmath338-disks to @xmath338-handles , we obtain a clean neighborhood of infinity @xmath418 with finite homotopy type , lying in @xmath419 . this leaves only @xmath10-semistability to be checked . we will show that ( [ inverse sequence in structure theorem ] ) satisfies the strong @xmath115-perfectness property ; in other words , each @xmath132 is strongly @xmath133-perfect , where @xmath420 .    for each @xmath421 , let @xmath422 .    _ claim . _ @xmath423 _ is a _ @xmath424__-one - sided h - cobordism . _ _    fix @xmath31 and let @xmath425 be the covering corresponding to @xmath426 ; let @xmath427 denote @xmath428 ; and let @xmath261 denote @xmath429 . then @xmath427 is the cover of @xmath430 corresponding to @xmath431 , and @xmath261 is the cover of @xmath193 corresponding to @xmath432 . by lemma [ lemma : ( modq ) h.e . ] and shapiro s lemma@xmath433\right )   \cong h_{\ast}\left (   \widehat{n}_{i},\partial\widehat{n}_{i}\partial n_{i};\mathbb{z } \right )   \text{,}\ ] ] and from the long exact homology sequence for the triple @xmath434 , excision , and shapiro s lemma @xmath435\right )   = 0.\ ] ] the claim follows . finally , since the bonding map @xmath436 is represented by the inclusion @xmath437 , @xmath132 is strongly @xmath133-perfect by theorem [ thm : one - sided general ] .    for the converse , we must show that conditions ( 1)-(3 ) imply the existence of a near pseudo - collar structure on @xmath5 . though the proof is rather complicated , it follows the same outline as @xcite , which followed the original proof in @xcite . for a full understanding , the reader should be familiar with @xcite . the new argument presented here generalizes the the final portions of that proof . a concise review of @xcite can be found in @xcite .    in @xcite and @xcite the goal was to improve arbitrarily small neighborhoods of infinity to homotopy collars . that is impossible with our weaker hypotheses ; instead , the goal is to improve neighborhoods of infinity to homotopy collars modulo certain subgroups of their fundamental groups .    by condition ( 2 ) the pro - isomorphism class of @xmath438 may be represented by a sequence@xmath112 of finitely presented groups , along with a small augmentation @xmath125 ( @xmath408 , for all @xmath31 ) so that each @xmath132 is strongly @xmath133-perfect , where @xmath439 .    by ( * ? ? ? * lemma 8) there is a sequence @xmath194 of generalized @xmath225-neighborhoods of infinity whose inverse sequence of fundamental groups is isomorphic to a subsequence of @xmath440 . @xmath74{ccccccccc}g_{i_{0 } } & \overset{\lambda_{i_{0}+1,i_{1}}}{\twoheadleftarrow } & g_{i_{1 } } & \overset{\lambda_{i_{1}+1,i_{2}}}{\twoheadleftarrow } & g_{i_{2 } } & \overset{\lambda_{i_{2}+1,i_{3}}}{\twoheadleftarrow } & g_{i_{3 } } & \overset{\lambda_{i_{3}+1,i_{4}}}{\twoheadleftarrow } & \cdots\\ \updownarrow\cong &   & \updownarrow\cong &   & \updownarrow\cong &   & \updownarrow\cong &   & \\ \pi_{1}\left (   n_{0},p_{0}\right )   & \overset{inc_{\#}}{\twoheadleftarrow } & \pi_{1}\left (   n_{1},p_{1}\right )   & \overset{inc_{\#}}{\twoheadleftarrow } & \pi_{1}\left (   n_{2},p_{2}\right )   & \overset{inc_{\#}}{\twoheadleftarrow } & \pi_{1}\left (   n_{3},p_{3}\right )   & \overset{inc_{\#}}{\twoheadleftarrow } & \cdots \end{array}\ ] ] this diagram and proposition [ prop : nearly sub ] ensure that , for each @xmath441 , @xmath442 is strongly@xmath443-perfect . so by passing to this subsequence and relabeling , we may assume that sequence ( [ sequence : pi1-setup for main theorem ] ) and the corresponding subgroup data matches the fundamental group data of @xmath194 . note here that the @xmath21-groups ( which are not viewed as part of the original data ) are not the same as the previous @xmath21-groups ; they are now preimages of _ compositions _ of the original bonding maps . next we inductively improve the sequence @xmath444 to generalized @xmath179-neighborhoods of infinity for increasing values of @xmath179 , up to @xmath445 . we must frequently pass to subsequences , however , each improvement of a given @xmath262 leaves its fundamental group and that of @xmath446 intact ; so at each stage , the ` new ' fundamental group data will be a subsequence of the original ( [ sequence : pi1-setup for main theorem ] ) , along with the subsequence augmentation . the @xmath21-groups will change as per their definition , but , by proposition [ prop : nearly sub ] , we always maintain the appropriate strong relative perfectness condition . this neighborhood improvement process uses only the hypothesis that @xmath5 is inward tame ; it is identical that used in ( * ? ? ? * th . 5 ) and outlined in ( * ? ? ? * theorem 3.2 ) . to save on notation we relabel the neighborhood sequences and their corresponding groups at each stage , designating the resulting cofinal sequence of generalized @xmath447-neighborhoods of infinity by @xmath191 , with @xmath407 , @xmath448 the corresponding homomorphism , @xmath408 , and @xmath449 .    for each @xmath31 , let @xmath450 and consider the collection of cobordisms @xmath451 . the following summary comprises the contents of lemmas 11 and 12 of @xcite , along with new hypotheses regarding kernels .    1 . each @xmath193 is a generalized @xmath452-neighborhood of infinity . each induced bonding map @xmath453 is surjective . 3 .   each inclusion @xmath454 induces a @xmath8-isomorphism . 4 .   each @xmath455 induces a @xmath8-epimorphism with kernel strongly @xmath133-perfect . @xmath456 for all @xmath457 and all @xmath31 each @xmath458 admits a handle decomposition based on @xmath446 containing handles only of index @xmath447 and @xmath459 . each @xmath193 admits an infinite handle decomposition with handles only of index @xmath447 and @xmath459 . each @xmath460 has the homotopy type of a relative cw pair @xmath461 with@xmath462 . the obvious next goal is attempting to improve the @xmath193 to generalized @xmath459-neighborhoods of infinity , which by item viii ) would necessarily be homotopy collars . in previous work @xcite , @xcite and @xcite , that is the final ( also the most difficult and interesting ) step . the same is true here , where the weakened hypotheses create greater difficulties and the strategy and end goal must eventually be altered . for now , we continue with the earlier strategies by turning attention to @xmath463 , which may be viewed as a @xmath464$]-module @xmath465)$ ] . the content of ( * ? ? ? * lemma 13 ) is given by the next two items .    1 . @xmath466 is a finitely generated projective @xmath464$]-module . 2 .   as an element of @xmath467\right ) $ ] , @xmath468   = \left (   -1\right )   ^{n}\sigma\left (   n_{i}\right )   $ ] , where @xmath469 is the wall finiteness obstruction for @xmath193 .    taken together , these elements of @xmath467\right )   $ ] determine the obstruction @xmath470 found in condition ( 3 ) . from now on we assume that @xmath471 vanishes . this is equivalent to assuming that each @xmath469 is the trivial element of @xmath467\right )   $ ] , in other words , each @xmath466 is a stably free @xmath464$]-module . therefore we have :    1 .   by carving out finitely many trivial @xmath447-handles from each @xmath193 , we can arrange that @xmath466 is a finitely generated free @xmath464$]-module . item ( xi ) can be done so that these sets remain a generalized @xmath472-neighborhood of infinity , and so that their fundamental groups and those of their boundaries are unchanged . again , to save on notation , we denote the improved collection by @xmath194 . see ( * ? ? ? * lemma 14 ) for details . the finite generation of @xmath466 allows us to , after again passing to a subsequence and relabeling , assume that    1 . @xmath473 is surjective for each @xmath31 . the long exact sequence for the triple @xmath474 from there shows that    1 . @xmath475 is an isomorphism for each @xmath31 ( hence , @xmath476 is a finitely generated free @xmath477$]-module ) . as above , we may choose handle decompositions for the @xmath478 based on @xmath446 having handles only of index @xmath479 and @xmath480 .    from now on , let @xmath31 be fixed . after introducing some trivial @xmath481-handle pairs , an algebraic lemma and some handle slides allows us to obtain a handle decomposition of @xmath478 based on @xmath412 with @xmath459-handles @xmath482 and an integer @xmath483 , such that the subcollection @xmath484 is a free @xmath485   $ ] -basis for @xmath486 . so we have :    1 . @xmath485   $ ] -cellular chain complex for @xmath487 may be expressed as@xmath488    where    * @xmath489 and @xmath490 represent free @xmath485   $ ] -submodules of @xmath491 generated by the corresponding handles ; * @xmath492  is the free @xmath485   $ ] -module generated by the @xmath447-handles in @xmath478 ; * @xmath493 ; and * @xmath494 takes @xmath495 injectively into @xmath496 . item xiv ) and the preceding paragraph are the content ( * ? ? ? * lemma 15 ) .    to this point , we have only used the hypotheses of inward tameness and triviality of the wall obstruction to build the structure described by items ( i)-(xiv ) . all arguments used thus far appear in @xcite and @xcite , with simpler analogs in @xcite .    under the @xmath8-stability hypothesis of @xcite , @xmath497 can now be killed by sliding the offending @xmath459-handles @xmath498 off the @xmath447-handles and carving out their interiors . under the weaker @xmath7-semistability hypothesis of @xcite , a similar strategy works , but only after a significant preparatory step , made possible by perfect kernels . in @xcite an alternate strategy was employed . instead of killing @xmath499 by removing its generating handles @xmath500 , the task was accomplished by introducing new @xmath447-handles , which became images of the @xmath498 under the resulting boundary map , thereby trivializing the kernel . complete discussions of these approaches can be found in @xcite and @xcite ; the strategy employed here is based on the latter . it is helpful to change our perspective by switching to the dual handle decomposition of @xmath478 . let @xmath501 be a closed collar neighborhood of @xmath502 in @xmath478 , and for each @xmath459-handle @xmath503 identified earlier , let @xmath504 be its dual , attached to @xmath501 . similarly , for each @xmath447-handle @xmath505 , let @xmath506 be its dual . as is standard , the attaching and belt spheres of a given handle switch roles in its dual . let @xmath507 , @xmath508 , and @xmath509 be a closed collar on @xmath510 in @xmath511 . observe that @xmath512 . see figure [ fig2-ri ] . [ ptb ]    figure2-ends4.eps    a simplified view of the next step is that we will find a collection of @xmath513-handles @xmath514 attached to the left hand boundary of @xmath478 and lying in @xmath515 so that the collection @xmath516 of attaching spheres of those @xmath513-handles is algebraically dual to the belt spheres of @xmath517 and has trivial algebraic intersection with the belt spheres of @xmath518 . adding those @xmath513-handles to the mix , then inverting the handle decomposition again , results in a cobordism with chain complex @xmath519 in which @xmath520 as desired  but with a caveat . although addition of the @xmath513-handles does not change the fundamental group of the cobordism , the arranged algebraic intersections between the attaching spheres of @xmath521 and the belt spheres of the existing @xmath338-handles are @xmath522   $ ] -intersection numbers ; this is the best the hypotheses will allow . then , to arrive at the desired conclusion  that we have effectively killed the relative second homology , it is necessary to switch the coefficient ring to @xmath522   $ ] ( in other words , mod out by @xmath523 ) , and reinterpret ( [ chain complex - enhanced ] ) as a @xmath522   $ ] -complex . then , letting @xmath524 , it follows that : @xmath525 ; @xmath526 induces a @xmath8-isomorphism ; and @xmath527   ) = 0 $ ] . in other words , @xmath528 is a @xmath529-homotopy collar .    in order to carry out the above program , we first identify a collection @xmath516 of pairwise disjoint @xmath338-spheres in @xmath510 algebraically dual over @xmath522   $ ] to the collection @xmath530 of belt spheres of the @xmath338-handles @xmath531 and having trivial @xmath522   $ ] -intersections with the belt spheres @xmath532of the remaining @xmath338-handles @xmath533 . keeping in mind that @xmath534 is canonically isomorphic to @xmath535 , and using the hypothesis that @xmath536 is strongly @xmath537-perfect , such a collection @xmath516 exists , as is shown in @xcite . by general position , the collection can be made disjoint from the attaching tubes of the @xmath513-handles @xmath538 , so they may be viewed as lying in @xmath446 . if the collection @xmath539 bounds a pairwise disjoint collection of embedded @xmath513-disks in @xmath515 , regular neighborhoods of those disks would provide the desired @xmath513-handles , and the proof is complete . ( the argument from @xcite provides details . )    for @xmath540 , the issue is just whether the @xmath338-spheres @xmath541 contract in @xmath515 . ( in dimension @xmath542 , a special argument is needed to get pairwise disjoint embeddings . ) contractibility is not guaranteed ; but with additional work it can be arranged . the additional work  involves the _ spherical alteration _  of @xmath338-handles developed in @xcite . the idea is to alter the @xmath338-handles @xmath543 in a preplanned manner so that the correspondingly altered @xmath516 contract in the new @xmath515 . along the way it will be necessary to reconstruct the @xmath513-handles @xmath538 as well ; for later use , let @xmath544 denote the attaching spheres of those handles . all details were carefully laid out in @xcite , with this application in mind . the tailor - made lemma , stated in the final section of that paper , is repeated here . [ ( * ? ? ? * lemma 6.1)][lemma from spherical alterations paper]let @xmath545 be a pair of n - manifolds @xmath546 with a common boundary component @xmath547 , and suppose there is a subgroup @xmath369 of @xmath548 for which @xmath549 is strongly @xmath369-perfect . suppose further that there is a clean submanifold @xmath550 consisting of a finite collection @xmath551 of 2-handles in @xmath552 attached to a collar neighborhood @xmath289 of @xmath547 with @xmath553 inducing a @xmath554-isomorphism ( the 2-handles precisely kill the group @xmath16 ) and a finite collection @xmath555 of pairwise disjoint embedded 2-spheres in @xmath556 , each of which contracts in @xmath552 . then on any subcollection @xmath557 , one may perform spherical alterations to obtain 2-handles @xmath558 in @xmath552 so that in @xmath559 ( where @xmath560 is the correspondingly altered version of @xmath561 ) there is a collection of 2-spheres @xmath562 algebraically dual over @xmath563   $ ] to the belt spheres @xmath564 common to @xmath565 and @xmath558 with the property that each @xmath566 contracts in @xmath567 . furthermore , each correspondingly altered 2-sphere @xmath568 ( now lying in @xmath559 ) has the same @xmath569   $ ] -intersection number with those belt spheres and with any other oriented @xmath452-manifold lying in both @xmath556 and @xmath570 as did @xmath571 . whereas the 2-spheres @xmath572 each contracted in @xmath552 , the @xmath568 each contract in @xmath567 . apply lemma [ lemma from spherical alterations paper ] to the current setup , with the following substitutions : @xmath74{llll}\underline{\mathbf{lemma}\text { \ref{lemma from spherical alterations paper } } } &   & \hspace*{0.05 in } & \underline{\mathbf{current\ situation}\text { } } \\ r^{\prime } & \leftrightarrow &   & r_{i}\\ r & \leftrightarrow &   & r_{i}\bigcup r_{i-1}\\ b & \leftrightarrow &   & \partial n_{i+1}\\ \mathcal{h}^{2 } & \leftrightarrow &   & \left\ {   \overline{h}_{1}^{2},\cdots,\overline{h}_{s}^{2},\overline{h}_{s+1}^{2},\cdots,\overline{h}_{r}^{2}\right\ } \\ l^{\prime } & \leftrightarrow &   & j_{i+1}=\lambda_{i+1}^{-1}\left ( l_{i}\right ) \\ t & \leftrightarrow &   & t_{i}=s_{i}\bigcup\left (   \overline{h}_{1}^{2}\cup\cdots\cup\overline{h}_{s}^{2}\cup\overline{h}_{s+1}^{2}\cup\cdots \cup\overline{h}_{r}^{2}\right ) \\ k\in\mathbb{z } & \leftrightarrow &   & s\in\mathbb{z}\\ \left\ {   h_{j}^{2}\right\ }   _ { j=1}^{k } & \leftrightarrow &   & \left\ { \overline{h}_{j}^{2}\right\ }   _ { j=1}^{s}\\ \left\ {   \gamma_{j}^{2}\right\ }   _ { j=1}^{k } & \leftrightarrow &   & \left\ { \gamma_{j}^{2}\right\ }   _ { j=1}^{s}\\ \left\ {   \theta_{t}^{2}\right\ }   & \leftrightarrow &   & \left\ {   \theta _ { j}^{2}\right\ }   _ { j=1}^{t}\end{array}\ ] ]    after applying this lemma , the collection @xmath573 is replaced by altered versions @xmath574 and the original collection @xmath575 is retained . let @xmath576 and @xmath577 . the collections @xmath516 and @xmath578 are replaced by altered versions @xmath579 and @xmath580 which lie in @xmath581 and contract in @xmath582 . the original @xmath513-handles @xmath583 must be discarded since their attaching tubes have been disrupted ; replacements will be constructed shortly . when @xmath540 , use general position to choose pairwise disjoint collection of properly embedded @xmath513-disks in @xmath582 with boundaries corresponding to the @xmath338-spheres @xmath584 . those @xmath513-disks may be thickened to @xmath513-handles by taking regular neighborhoods . with all of these handles finally in place , the argument described earlier completes the proof . when @xmath585 , the same is true , but the @xmath586-@xmath586 argument used in ( * ? ? 4.2 & 5.3 ) is needed in order to find pairwise disjoint embedded @xmath513-disks .    _ in reality , we have shown a stronger result than what is stated in theorem [ npct technical version ] . specifically , the near pseudo - collar structures obtained are as close to actual pseudo - collars as the augmentation is to the trivial augmentation . for example , if _ @xmath136 _ is the trivial augmentation , the above argument contains an alternative proof of the main result of @xcite ( stated here as theorem [ pct technical version ] ) . more generally , if _ @xmath136 _ lies somewhere between the trivial augmentation and the standard augmentation , then a near pseudo - collar structure on _ @xmath5 _ can be chosen to reflect that augmentation . _ the main examples of @xcite , described here in example [ example : 1 ] , proved the existence of ( absolutely ) inward tame open manifolds that are not pseudo - collarable . in this section we construct open manifolds that are absolutely inward tame but not nearly pseudo - collarable . since the examples from @xcite are nearly pseudo - collarable , the new examples fill a gap in the spectrum of known end structures . the examples of @xcite began with algebra . the main theorems of that paper showed that all inward tame open manifolds have pro - finitely generated , semistable fundamental group , and stable @xmath37-homology , at infinity . the missing ingredient for detecting a pseudo - collar structure was @xmath7-semistability . with that knowledge , an inverse sequence of groups satisfying the necessary properties , but failing @xmath7-semistability , became the blueprint for an example . a nontrivial handle - theoretic strategy was needed to realize the examples , but the heart of the matter was the group theory . a similar story plays out here . we will begin with an inverse sequence of finitely presented groups with surjective bonding maps that become isomorphisms upon abelianization ; but this time , in light of theorems [ theorem : inward tame implies ap - semistability ] and [ theorem : npc characterization  simple version ] , we want an @xmath9-semistable sequence that is not @xmath10-semistable . the first step is to identify such a sequence . let @xmath587 , the free group on the three generators ; @xmath588   $ ] , @xmath589   $ ] , and @xmath590   $ ] ; @xmath591 ; and @xmath592 . notice that @xmath593 is precisely the commutator subgroup @xmath594   $ ] , so @xmath595 . let @xmath596   $ ] , @xmath597   $ ] , and @xmath598   $ ] ; @xmath599 ; and @xmath600 . since @xmath601 , there is an induced epimorphism @xmath602 which abelianizes to the identity map on @xmath603 . continue inductively , letting @xmath604   $ ] , @xmath605   $ ] , and @xmath606   $ ] ; @xmath607 ; and @xmath608 . the result is a nested sequence of normal subgroups of @xmath609 , @xmath610 , and a corresponding inverse sequence of quotient groups @xmath611 which abelianizes to the constant inverse sequence @xmath612 a more delicate motivation for our choices is the following : for each @xmath613 , @xmath614 ; similarly , for each @xmath615 , @xmath616 moreover , since the elements of @xmath617 are precisely the commutators of the elements of @xmath618 , @xmath619   .\ ] ] so , for the standard augmentation , @xmath121 , ( [ chosen inverse sequence ] ) is @xmath115-perfect , hence , @xmath9-semistable . two tasks remain :    * prove that ( [ chosen inverse sequence ] ) is not @xmath10-semistable , and * construct 1-ended absolutely inward tame open manifolds with fundamental groups at infinity representable by ( [ chosen inverse sequence ] ) . since these tasks are independent , the ordering of the following two  subsections is arbitrary . let @xmath620 , the free group on @xmath0 generators . we will exploit two standard constructions from group theory . the _ derived series of _ @xmath621 is defined by @xmath622 and @xmath623   $ ] for @xmath624 . the _ lower central series of _ @xmath625 is given by @xmath626 and then @xmath627   $ ] for @xmath624 . by inspection @xmath628 , @xmath629 , and @xmath630 for all @xmath179 . a well - known fact , similar in spirit to our goal in this subsection , is that @xmath631 . the following representation of @xmath621 was discovered by magnus ; our general reference is @xcite . [ prop : representation ]  ( * ? ? ? * proposition 10.1 ) let @xmath632 be the non - commuting power series ring in indeterminates @xmath633 with @xmath634 for @xmath635 . then the function @xmath636 @xmath637 induces a faithful representation of @xmath621 into @xmath638 , the multiplicative group of units of @xmath632 .    in @xmath632 , the _ fundamental ideal _ @xmath639 is the kernel of the homomorphism @xmath640 that takes each @xmath641 to 0 . the elements of @xmath639 are all sums of the form @xmath642 where each @xmath643 is a homogeneous polynomial of degree at least one . consequently , for any positive integer @xmath179 the ideal @xmath644 is made of all sums of the form @xmath645 where each @xmath643 is a homogeneous polynomial of degree at least @xmath179 . the next proposition and lemma are useful for monitoring the location of commutators in a group .    [ prop : commutator ]  ( * ? ? ? * proposition 10.2 ) let @xmath646 be the representation given above . if @xmath647 such that @xmath648 and @xmath649 , then @xmath650   \right )   -1\in\delta^{r+s}$ ] .    by applying proposition [ prop : commutator ] inductively , we obtain the following useful facts . [ lemma : commutator]for all integers @xmath651 ,    1 . @xmath652 2 . @xmath653 3 . @xmath654 , and 4 . @xmath655 . we now focus our attention on @xmath609 and its subgroups @xmath656 , as defined earlier . [ lemma : topology prelim]for each @xmath215 and @xmath657 ,    1 . @xmath658 is a member of at least one free basis for @xmath659 , and   2 . @xmath660 .    assertion ( 1 ) can be obtained from an inductive argument using schreier systems . a model argument can be found in ( * ? ? ? * example 8.1 ) . assertion ( 2 ) follows from ( 1 ) , since the quotient map @xmath661 is the abelianization of @xmath662 .    since @xmath663 , the following is an easy consequence of lemmas [ lemma : commutator ] and [ lemma : topology prelim ] . [ lemma : specialized commutators]for each @xmath232 and @xmath664 ,    1 . @xmath665 , and 2 . @xmath666 . the definitions of derived and lower central series are clearly applicable to arbitrary groups . to expand those notions further , the following definition is useful . for @xmath667 , let @xmath668 and @xmath669   $ ] for @xmath670 . by normality , @xmath671 . when @xmath200 is strongly @xmath22-perfect , @xmath672 for all @xmath179 . [ prop : omega groups get small]for each @xmath232 , there exists @xmath673 and @xmath674 such that    1 .   for each @xmath675 , @xmath676 , and 2 .   @xmath677 . let @xmath31 be fixed . existence of @xmath210 follows from item ( 3 ) of lemma [ lemma : commutator ] . existence of @xmath678 may be obtained from an inductive application of [ prop : commutator ] . we shift focus one more time ; from @xmath609 and its subgroups to the quotient groups @xmath679 and their subgroups . in doing so , we will allow a word in the generators of @xmath609 to represent both an element of @xmath609 and the corresponding element of a @xmath297 . for example , recalling that @xmath680 , we say @xmath681 . the following is simple but useful . [ lemma : images of omega]suppose @xmath682 is a surjective homomorphism , @xmath667 , and @xmath683 . then @xmath684 . lemma [ lemma : images of omega ] ensures that , for each @xmath685 and all @xmath683 , the quotient maps @xmath686 restrict to epimorphisms @xmath687 [ equation : key epimorphism ]    [ prop : no omega ] for @xmath210 and @xmath678 as chosen in proposition [ prop : omega groups get small ] , and each @xmath675 , @xmath688 whenever @xmath689 . suppose @xmath690 . surjection ( [ equation : key epimorphism ] ) provides a @xmath691 with cosets @xmath692 . consequently , there is an @xmath693 with @xmath694 in @xmath609 . then@xmath74{rcl}\beta(r_{i , j})-1 & = & \beta(h)\beta\left (   w\right )   -1\\ &   & \beta(h)\beta\left (   w\right )   -\beta(h)+\beta(h)-1\\ &   & \beta(h)\left (   \beta\left (   w\right )   -1\right )   + \left (   \beta ( h)-1\right ) \end{array}\ ] ] since @xmath695 and @xmath696 , then @xmath697 , violating the choice of @xmath210 . we are now ready for the main result of this subsection ..    [ theorem : example ] the inverse sequence @xmath698 is not @xmath10-semistable . in fact , @xmath699 is not pro - isomorphic to any inverse sequence @xmath109 of surjections that satisfies the strong @xmath700-perfectness property . we proceed directly to the stronger assertion . suppose @xmath106 is pro - isomorphic to an inverse sequence @xmath701 of surjections , that is strongly @xmath702-perfect ; in other words , @xmath703 $ ] for all @xmath31 .    by proposition [ prop : nearly sub ] , each subsequence of @xmath109 satisfies the same essential property , so by our assumption , @xmath106 contains a subsequence that fits into a commutative diagram of the following form : @xmath704    passing to a further subsequence if necessary , we may assume that @xmath705 for all @xmath0 .    by lemma [ lemma : topology prelim ] , @xmath706 . choose @xmath707 with @xmath708 . then , @xmath709 , and consequently @xmath710   $ ] , since @xmath711 is strongly @xmath712-perfect ( again using proposition [ prop : nearly sub ] ) . therefore @xmath713 for all @xmath714 . moreover , since @xmath715 , @xmath716 for all @xmath714 , thereby contradicting proposition [ prop : no omega ] . the goal of this subsection is to construct , for each @xmath1 , a 1-ended open manifold @xmath5 that is absolutely inward tame and has fundamental group at infinity represented by the inverse sequence ( [ chosen inverse sequence ] ) . by theorem [ theorem : npc characterization simple version ] or [ npct technical version ] , such an example fails to be nearly pseudo - collarable , thus completing the proof of theorem [ theorem : existence of counterexamples ] . we will construct @xmath5 as a countable union of codimension 0 submanifolds @xmath717 where @xmath718 is a compact core  and @xmath719 is a sequence of compact cobordisms between closed connected @xmath720-manifolds where @xmath721 for each @xmath232 , and @xmath722 . letting @xmath723 gives a preferred end structure @xmath194 with @xmath724 for each @xmath31 . see figure [ fig3-union of cobordisms ] . [ ptb ]    figure3-ends4.eps    so that @xmath725-@xmath726 is represented by ( [ chosen inverse sequence ] ) , the @xmath727 will be constructed to satisfy :    1 .   for all @xmath232 , @xmath728 and @xmath729 induces a @xmath8-isomorphism , and 2 . the isomorphism between @xmath730 and @xmath297 may be chosen so that the following diagram commutes : @xmath74{ccccc}g_{i } &   & \overset{\lambda_{i+1}}{\longleftarrow } &   & g_{i+1}\\ \downarrow\cong &   &   &   & \downarrow\cong\\ \pi_{1}(\gamma_{i},p_{i } ) & \overset{\cong}{\longleftarrow } & \pi_{1}\left ( a_{i},p_{i}\right )   & \overset{\psi_{i+1}}{\longleftarrow } & \pi_{1}(\gamma_{i+1},p_{i+1})\medskip \end{array}\ ] ] here @xmath731 is the composition@xmath732 where @xmath733 is induced by inclusion and @xmath734 is a change - of - basepoint isomorphism  with respect to a path @xmath735 in @xmath727 between @xmath210 and @xmath211 . from there it follows from van kampen s theorem that each @xmath736 induces a  @xmath8-isomorphism , so by repeated application of ( a ) and ( b ) , the inverse sequence@xmath737 is isomorphic to ( [ chosen inverse sequence ] ) .    it will be also be shown that each @xmath193 has finite homotopy type ; so @xmath5 is absolutely inward tame . that argument requires specific details of the construction ; it will be presented later . recall that a @xmath738-handle @xmath739 attached to an @xmath0-manifold @xmath740 and a @xmath741-handle @xmath742 attached to @xmath743 form a _ complementary pair _ if the attaching sphere of @xmath742 intersects the belt sphere of @xmath739 transversely in a single point . in that case @xmath744 ; moreover , we may arrange ( by an isotopy of the attaching sphere of @xmath742 ) that @xmath745 is an @xmath746-ball in @xmath747 . conversely , for any ball @xmath748 , one may introduce a pair of complementary handles @xmath749 so that @xmath750 . we call @xmath751 a _ trivial handle pair . _ note that the difference between a complementary pair and trivial pair is just a matter of perspective . in general , we say that @xmath739 is _ attached trivially _ to @xmath740 if it is possible to attach an @xmath742 so that @xmath751 is a complementary pair . after a preliminary step where we construct the core manifold @xmath718 , our proof proceeds inductively . at the @xmath752 stage we construct the cobordism @xmath753 , along with a compact manifold @xmath754 with @xmath755 , to be used in the following stage . throughout the construction , we abuse notation slightly by letting @xmath756   $ ] denote a small regular neighborhood of @xmath757 in @xmath758 and @xmath759   $ ] to denote a small regular neighborhood of @xmath760 in @xmath727    * step 0 . * ( preliminaries )    let @xmath761 be the @xmath0-manifold obtained by attaching three orientable 1-handles @xmath762 to the @xmath0-ball @xmath763 . choose a basepoint @xmath764 and let @xmath765 , and @xmath766 be be embedded loops in @xmath767 intersecting only at @xmath768 . abuse notation slightly by writing @xmath769 a convenient way to arrange that the 1-handles are orientable is by attaching three trivial @xmath770-handle pairs @xmath771 , then discarding the @xmath338-handles . recall that @xmath772 where @xmath588   $ ] , @xmath589   $ ] , and @xmath590   $ ] . attach a trio of @xmath338-handles @xmath773 to @xmath761 , where @xmath774 has attaching circle @xmath775 . choose the framings of these handles so that , if the @xmath338-handles @xmath776 were added back in , then @xmath777 would be trivially attached ( to an @xmath0-ball ) . let @xmath778 and note that @xmath779 . * step 1 . * ( constructing @xmath780 and @xmath781 )    attach three trivial @xmath782-handle pairs to @xmath718 , disjoint from the existing handles , then perform handle slides on each of the trivial 2-handles ( over the handles @xmath783 ) so that the resulting 2-handles @xmath784 , @xmath785 and @xmath786 have attaching circles spelling out the words @xmath787 , @xmath788 and @xmath789 , respectively . this is possible since each @xmath790 can be viewed as a product of the loops @xmath791 and their inverses , which are the attaching circles of @xmath783 . sliding a 2-handle over @xmath774 inserts the loop @xmath792 into the new attaching circle of that 2-handle ( with @xmath793 depending on the orientation chosen ) .    by keeping track of the attaching 2-spheres of the trivial 3-handles after the handle slides , it is possible to attach 3-handles @xmath794 , @xmath795 , and @xmath796 to @xmath797 that are complementary to @xmath784 , @xmath785 , and @xmath786 , respectively . then @xmath798 for later purposes , it is useful to have a schematic image of the attaching circles of @xmath783 and the attaching @xmath338-spheres of the complementary handles @xmath783 . figure [ fig4-attachingcircle ] provides such an image for one complementary pair . [ ptb ]    figure4-ends4.eps    the outer loop represents the attaching circle for an @xmath799 and the shaded region represents the ` lower hemisphere ' of the attaching 2-sphere of @xmath800 ; the ` upper hemisphere ' , which is not shown , is a parallel copy of the core of @xmath799 . within the lower hemisphere , the small central disk represents the lower hemisphere of the 2-sphere before handle slides . the arms are narrow strips whose centerlines are the paths along which the handle slides were performed ; diametrically opposite paths lead to the same 2-handle , and are chosen to be parallel to a fixed path . we have indicated this by labeling one pair of centerlines @xmath42 and the other @xmath801 . the four outer disks are parallel to the cores of the 2-handles over which the slides were made . a twist in the strip leading to an outer disk is used to reverse the orientation of the boundary of that disk . thus , diametrically opposite outer disks are parallel to each other , but with opposite orientations . center points of the outer disks represent transverse intersections with belt spheres of those handles ; thus , @xmath802 and @xmath803 are nearby intersections with the same belt sphere , and similarly for @xmath804 and @xmath805 .    by rewriting @xmath806 as @xmath807 , we may reorder the handles so that @xmath784 , @xmath785 , and @xmath786 are attached first . define @xmath808 and note that @xmath809 . furthermore , @xmath810 so , if we let @xmath811   ) \cup\left ( \cup_{j=1}^{3}h_{1,j}^{2}\right )   \cup\left (   \cup_{j=1}^{3}h_{2,j}^{3}\right )   \text{,}\ ] ] ( the result of excising the interior of a slightly shrunken copy of @xmath781 ) , then @xmath812 . by letting @xmath813 and @xmath814 we obtain the first cobordism of the construction @xmath815 . by avoiding the base point @xmath764 in all of the above handle additions , we may let the arc @xmath816 be the product line @xmath817   $ ] , with @xmath818 and @xmath819 its end points . conditions ( a ) and ( b ) are then clear .    * inductive step . * ( constructing @xmath727 and @xmath754 )    assume the existence of a cobordism @xmath820 satisfying ( a ) and ( b ) along with a compact manifold @xmath821 , with the attaching circle of each @xmath822 representing the relator @xmath823 in the presentation of @xmath297 , and @xmath824 . attach three trivial @xmath782-handle pairs to @xmath758 , then perform handle slides on each of the trivial 2-handles ( over the handles @xmath825 ) so that the resulting 2-handles @xmath826 , @xmath827 and @xmath828 have attaching circles spelling out the words @xmath829 , @xmath830 and @xmath831 , respectively . this is possible since each @xmath832 can be viewed as a product of the loops @xmath833 and their inverses , which are the attaching circles of @xmath825 .    by keeping track of the attaching 2-spheres of the trivial 3-handles under the above handle slides , it is possible to attach 3-handles @xmath834 , @xmath835 , and @xmath836 to @xmath837 that are complementary to @xmath826 , @xmath827 , and @xmath828 , respectively . then @xmath838 a picture like figure [ fig4-attachingcircle ] , but with different indices , describes the current situation . rewrite @xmath839 as @xmath840 , then reorder the handles so that @xmath826 , @xmath827 , and @xmath828 are attached first . define @xmath841 and note that @xmath842 . furthermore , @xmath843 excising the interior of a slightly shrunken copy of @xmath754 gives@xmath844   ) \cup\left ( \cup_{j=1}^{3}h_{i , j}^{2}\right )   \cup\left (   \cup_{j=1}^{3}h_{i+1,j}^{3}\right )   \text{,}\ ] ] then @xmath845 . noting that @xmath846 and letting @xmath847 , we obtain @xmath753 . by avoiding @xmath848 in all of the handle additions , letting @xmath849 be the product line @xmath850   $ ] , and @xmath211 the new end point , conditions ( a ) and ( b ) are clear .    assembling the pieces in the manner described in figure [ fig3-union of cobordisms ] completes the construction . in particular , we obtain a 1-ended open manifold@xmath851 whose fundamental group at infinity is represented by the inverse sequence ( [ chosen inverse sequence ] ) .    [ remark : dual handle decompositions]_in the construction of _ @xmath753 _ _ , we have written _ _ @xmath760 _ on the left and _ @xmath852 _ on the right to match the blueprint laid out in figure [ fig3-union of cobordisms ] . in that case , the handle decomposition of _ _ implicit in the construction goes from right to left , with handles being attached to a collar neighborhood _ $ ] _ of _ @xmath852__. later , when our perspective becomes reversed , we will pass to the dual decomposition _ _ @xmath854   ) \cup\left (   \cup _ { j=1}^{3}\overline{h}_{1,j}^{n-3}\right )   \cup\left (   \cup_{j=1}^{3}\overline{h}_{2,j}^{n-2}\right)\ ] ] _ where each _ @xmath855 _ is the dual of an original _ @xmath739 _ and _ @xmath759   $ ] _ is a thin collar neighborhood of _ @xmath760__. _ _      the following proposition will complete the proof of theorem [ theorem : existence of counterexamples ] . [ prop : absolute inward tameness of mn]for the manifolds @xmath5 constructed above , each clean neighborhood of infinity @xmath856 has finite homotopy type . thus , @xmath5 is absolutely inward tame . this will be accomplished by examining @xmath857 ( equivalently , @xmath858 viewed as a @xmath859-module ) where @xmath860 . in particular , we will prove :    [ claim : homology of n_i]for each @xmath31 , @xmath857 is trivial in all dimensions except for @xmath861 , where it is isomorphic to the free module @xmath862 . once this claim is established , proposition [ prop : absolute inward tameness of mn ] follows from ( * ? ? ? * lemma 6.2 ) . in remark [ siebenmann s lemma ] at the conclusion of this section , we explain why this final observation is elementary , requiring no discussion of finite dominations or finiteness obstructions .    for proving the claim , it is useful to consider compact subsets of the form@xmath863 by repeated application of remark [ remark : dual handle decompositions ] , there is a handle decomposition of @xmath864 based on @xmath865   $ ] with handles only of indices @xmath479 and @xmath480 . by reordering the handles , @xmath866 is seen to be homotopy equivalent to a finite relative cw complex @xmath867 where @xmath868 consists of @xmath760 with an @xmath447-cell attached for each @xmath447-handle of @xmath864 followed by an @xmath459-cell for each @xmath869-handle . in the usual way , the @xmath859-incidence number of an @xmath459-cell with an @xmath472-cell is equal to the @xmath859-intersection number between the belt sphere of the corresponding @xmath447-handle and the attaching sphere of the corresponding @xmath459-handle . this process produces a sequence@xmath870 of relative cw complexes with direct limit a relative cw pair @xmath871 homotopy equivalent to @xmath872 . so we can determine @xmath857 by calculating @xmath873 and taking the direct limit as @xmath874    the @xmath859-handle chain complex for @xmath866 ( equivalently , the @xmath859-cellular chain complex for @xmath867 ) looks like@xmath875 where @xmath876 and @xmath877 are finitely generated free @xmath859-modules generated by the handles of @xmath864 , and the boundary map is determined by @xmath859-intersection numbers between the belt spheres of @xmath201-handles and attaching spheres of the @xmath459-handles . these intersection numbers will be determined by returning to the construction . beginning with the compact manifold @xmath821 , attach three trivial @xmath782-handle pairs , then perform handle slides on the 2-handles ( over the handles @xmath825 ) to obtain @xmath826 , @xmath827 and @xmath828 with attaching circles @xmath829 , @xmath830 and @xmath878 , respectively . having kept track of the attaching 2-spheres of the trivial 3-handles under the handle slides , attach 3-handles @xmath834 , @xmath835 , and @xmath836 to @xmath837 that are complementary to @xmath826 , @xmath827 , and @xmath828 , respectively ( all as described ` inductive step ' above ) . this can all be done so that @xmath834 , @xmath835 , and @xmath836 do not touch the earlier @xmath338-handles @xmath879 , @xmath880 and @xmath881 . next attach a second trio of trivial @xmath782-handle pairs , taking care that they are disjoint from the existing handles , and slide the trivial @xmath338-handles over the @xmath338-handles @xmath882 so that the resulting 2-handles @xmath883 have attaching circles @xmath884 , @xmath885 and @xmath886 . again , having kept track of the attaching 2-spheres of the trivial 3-handles under the handle slides , attach 3-handles @xmath887 , @xmath888 , and @xmath889 to@xmath890 that are complementary to @xmath891 , @xmath892 , and @xmath893 , respectively , while taking care that these new 3-handles are completely disjoint from all 2- and 3-handles of lower index . continue this process @xmath894 times , at each stage : attaching three trivial @xmath782-handle pairs disjoint from the existing handles ; sliding the trivial @xmath338-handles over the @xmath338-handles created in the previous step , in the manner prescribed above ; then attaching 3-handles complementary to these new 2-handles ( and disjoint from earlier 2- and 3-handles ) along the images of the attaching 2-spheres of the trivial 3-handles after the handle slides . since all of the 2- and 3-handles mentioned above , except for the original 2-handles @xmath879 , @xmath880 and @xmath881 , occur in complementary pairs , the manifold we just created is just a thickened copy of @xmath758 ; let us call it @xmath895 . by the standard reordering lemma , we may arrange that the 2-handles are pairwise disjoint , and all are attached before any of the 3-handles  which are also are attached in a pairwise disjoint manner . then @xmath896 where , going from the first to the second line , we apply the definition of @xmath758 ; going from the second to the third , we bring the last triple of 2-handles forward to the beginning ; and in going from the third to the fourth , we apply the definition of @xmath897 . excising a slightly shrunken copy of the interior of @xmath897 from @xmath895 results in a cobordism between @xmath898 and @xmath899 , which has a handle decomposition@xmath900   ) \cup\left (   \cup_{j=1}^{3}h_{i , j}^{2}\right )   \cup\left (   \cup_{s=1}^{k-1}\left (   \cup_{j=1}^{3}h_{i+s , j}^{2}\right )   \right )   \cup\left (   \cup_{s=1}^{k}\left ( \cup_{j=1}^{3}h_{i+s , j}^{3}\right )   \right )   \text{.}\ ] ] comparing this handle decomposition to our earlier construction , reveals that this cobordism is precisely @xmath901 . in order to match the orientation of figure [ fig3-union of cobordisms ] , view @xmath902 as the right - hand boundary and @xmath760 as the left - hand boundary , with 2- and 3-handles being attached from right to left . before switching to the dual handle decomposition , we analyze the @xmath859-intersection numbers between the attaching spheres of the 3-handles and the belt spheres of the 2-handles . all should be viewed as submanifolds of the left - hand boundary of @xmath903 ) \cup\left (   \cup_{j=1}^{3}h_{i , j}^{2}\right )   \cup\left (   \cup_{s=1}^{k-1}\left (   \cup_{j=1}^{3}h_{i+s , j}^{2}\right )   \right )   $ ] , which has fundamental group @xmath297 .    for each @xmath904 and @xmath675 let @xmath905 denote the attaching 2-sphere of @xmath906 ; and for each @xmath907 and @xmath908 let @xmath909 denote the belt @xmath201-sphere of @xmath910 there are three cases to consider . * case 1 . * @xmath911 . then for each @xmath441 , the pair @xmath912 is complementary ; in other words @xmath913 intersects @xmath914 transversely in a single point . adjusting base paths , if necessary , and being indifferent to orientation ( since it will not affect our computations ) , we have @xmath915 . if @xmath916 , then @xmath906 does not intersect @xmath917 , so @xmath918 .    * case 2 . * @xmath919 .    for each @xmath441 , @xmath913 can be split into a pair of disks . the ` upper hemisphere ' lies in the the 2-handle @xmath920 and intersects @xmath921 transversely in a single point ; that point of intersection was accounted for in case 1 . the ` lower hemisphere ' is analogous to the one pictured in figure [ fig4-attachingcircle ] . if @xmath922 , then one pair of the diametrically opposite disks has boundaries labelled @xmath923 and @xmath924 and the disks are parallel to the core of @xmath925 , so each intersects @xmath926 transversely in points @xmath927 and @xmath928 . due to the flipped orientation of one of the disks , these points of intersection , between @xmath913 and @xmath926 , have opposite sign . connecting @xmath927 and @xmath928 by a path homotopic to @xmath929 in @xmath905 and a short path @xmath930 connecting @xmath927 and @xmath928 in @xmath926 yields a loop that is contractible in the left - hand boundary of @xmath903   ) \cup\left ( \cup_{j=1}^{3}h_{i , j}^{2}\right )   \cup\left (   \cup_{s=1}^{k-1}\left ( \cup_{j=1}^{3}h_{i+s , j}^{2}\right )   \right )   $ ] . so together @xmath927 and @xmath928 contribute @xmath185 to to the @xmath859-intersection number of @xmath913 and @xmath926 ; hence , @xmath931 . similarly @xmath932 . finally , @xmath913 and @xmath933 do not intersect , so @xmath934 , as well . * @xmath935 .    in this case , the handles @xmath906 and @xmath936 are disjoint , so @xmath937 . now invert the above handle decomposition , to obtain a handle decomposition of the cobordism @xmath938 , based on @xmath939 containing only @xmath447- and @xmath940-handles . specifically , we have@xmath941   ) \cup\left (   \cup_{s=1}^{k}\left (   \cup_{j=1}^{3}\overline{h}_{i+s , j}^{n-3}\right )   \right ) \cup\left (   \cup_{j=1}^{3}\overline{h}_{i , j}^{n-2}\right )   \cup\left ( \cup_{s=1}^{k-1}\left (   \cup_{j=1}^{3}\overline{h}_{i+s , j}^{n-2}\right ) \right )   \text{.}\ ] ] since the belt sphere of each @xmath942 is the attaching sphere of its dual @xmath943 and the attaching sphere of of each @xmath944 is the belt sphere of its dual @xmath945 , the incidence numbers between these handles of this handle decomposition are determined ( up to sign ) by the earlier calculations . so the cellular @xmath859-chain complex for the @xmath866 is isomorphic to @xmath946 where , the @xmath947 summands on the left are generated by the handles @xmath948 and those on the right by @xmath949 . since @xmath915 for all @xmath950 and all other intersection numbers are @xmath185 , the boundary map is trivial on the @xmath951 copy of @xmath947 ; misses the @xmath952 copy of @xmath947 in the range ; and restricts to an isomorphism @xmath953 elsewhere . thus @xmath954 where @xmath955 is generated by the @xmath956 summand , and @xmath957 is generated by the @xmath958 summand . now consider the inclusion @xmath959 and the corresponding inclusion of @xmath859-chain complexes . the chain complex of @xmath960 will contain an extra @xmath947 summand in each dimension , generated by @xmath961 and @xmath962 , respectively . the boundary map takes the new summand in the domain onto the previous cokernel , thereby killing @xmath963 , and replacing it with a cokernel generated by @xmath964 . said differently , the inclusion induced map@xmath965 is trivial . on the other hand , the expansion from @xmath868 to @xmath966 does not change @xmath967 , which is still generated by the handles @xmath968 . in other words , the inclusion induced map @xmath969 is an isomorphism .    taking direct limits , we have @xmath970{ccc}\left ( \mathbb{z } g_{i}\right )   ^{3 } &   & \text{if } \ast\ = n-2\\ 0 &   & \text{otherwise}\end{array } \right .   \text{.}\ ] ] so the claim is proved .    _ [ siebenmann s lemma]the appeal to ( * ? ? ? * lemma 6.2 ) may give the impression that obtaining proposition [ prop : absolute inward tameness of mn ] from claim [ claim : homology of n_i ] is complicated  that is not the case . the conclusion can be obtained directly as follows : if _ @xmath971 _ represents the cores of the _ @xmath869__-handles _ _ @xmath972 _ _ , which generate _ _ @xmath857 _ _ , abstractly attach _ _ @xmath459__-disks _ _ @xmath973 _ to _ @xmath760 _ along their boundaries . this does not affect fundamental groups , so by excision , the pair _ @xmath974 _ has the same _ @xmath859__-homology as _ _ @xmath872 _ _ , with the same generating set . now attach an _ _ @xmath746__-cell _ _ @xmath975 _ along each sphere _ @xmath976 _ to obtain a pair _ @xmath977 _ with trivial _ @xmath859__-homology in all dimensions . it follows that _ _ @xmath978 _ is a homotopy equivalence . but notice that each _ @xmath979 _ has a free face _ _ _ @xmath981 _ collapses onto _ @xmath193__. therefore , _ _ @xmath193 _ is homotopy equivalent to _ @xmath982__. _ _ in the introduction we commented that nearly pseudo - collarable manifolds admit arbitrarily small clean neighborhoods of infinity @xmath11 , containing codimension 0 submanifolds @xmath12 for which @xmath13 is a homotopy equivalence . call such a pair @xmath983 a _ wide homotopy collar_. the difference , of course , between a wide homotopy collar and a homotopy collar is that , in the latter , the subspace is required to be a codimension 0 submanifold . the fact that nearly pseudo - collarable manifolds contain arbitrarily small wide homotopy collars is immediate from the following easy lemma . suppose @xmath984 is a ( @xmath985)-homotopy collar neighborhood of infinity in a manifold @xmath5 ( @xmath986 , where @xmath21 is a normally finitely generated subgroup of @xmath987 . then @xmath5 contains a wide homotopy collar neighborhood of infinity @xmath988 , where @xmath989 . choose a finite collection of pairwise disjoint properly embedded 2-disks @xmath990 in @xmath991 with boundaries comprising a normal generating set for @xmath992 . let @xmath983 be a regular neighborhood pair for @xmath993 and apply lemma [ lemma : ( modl ) h.e . ] . examples constructed in this paper and in @xcite show that the existence of arbitrarily small wide homotopy collars in a manifold @xmath5 does not imply the existence of a pseudo - collar structure . the following seems likely but , so far , we have been unable to find a proof .
we continue our study of ends of non - compact manifolds , with a focus on the inward tameness condition . for manifolds with compact boundary , inward tameness , by itself , 
 has significant implications . 
 for example , such manifolds have stable homology at infinity in all dimensions . 
 we show that these manifolds have ` almost perfectly semistable ' fundamental group at each of their ends . 
 that observation leads to further analysis of the group theoretic conditions at infinity , and to the notion of a ` near pseudo - collar ' structure . 
 we obtain a complete characterization of @xmath0-manifolds ( @xmath1 ) admitting such a structure , thereby generalizing @xcite . 
 we also construct examples illustrating the necessity and usefulness of the new conditions introduced here . 
 variations on the notion of a perfect group , with corresponding versions of the quillen plus construction , form an underlying theme of this work .