Here are three vague theorems rolled up in one.



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> Let $X$ and $Y$ be sufficiently nice topological spaces and $f:X \to Y$ a sufficiently nice surjection. If for each $y \in Y$, the fiber $f^{-1}(y) \subset X$ is (topologically, homotopically, homologously) equivalent to a point, then $f$ induces a (topological, homotopic, homological) equivalence between $X$ and $Y$.
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There are many such theorems (Vietoris-Begle for homology, Smale for homotopy), and generalizations abound. For instance, instead of requiring fibers to be homotopy-equivalent to points, you can ask that they be $n$-connected for some $n$; under this weaker hypotheses one can still extract the fact that the homotopy groups of $X$ and $Y$ agree up to dimension $n-1$.


This suggests a fairly general template: take a nice map $f:X \to Y$, and assume that each fiber $f^{-1}(y)$ is equivalent to a point under [insert topological invariant $I$ here]. Then, $X$ and $Y$ are $I$-equivalent.



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> **Question:** Is there a general theorem like the one suggested above?
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For instance, one might ask: if each fiber has the Euler characteristic of a point, must we obtain $\chi(X) = \chi(Y)$? The reason I thought to ask is because I would like to know if such a theorem holds for simple homotopy equivalence (see [here](http://ncatlab.org/nlab/show/simple+homotopy+theory) for instance). Even if nothing like the super-general theorem above can hold, I'd still be interested in knowing if one has the following precise, restricted version:



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> Let $X$ and $Y$ be connected CW complexes and $f:X \to Y$ a cellular surjection. If for each $y \in Y$, the fiber $f^{-1}(y) \subset X$ is simple homotopy equivalent to a point, then $X$ and $Y$ are simple homotopy equivalent.
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If this is true, where can one find this theorem in the literature? I scoured Marshall Cohen's book on Simple Homotopy Theory but could not find it.



Is there a general theory of fiber theorems?
**Edit:** I have added some definitions and details to my answer.


In the most general form I can find, your third question is a consequence of two results regarding cell-like maps and fine homotopy equivalences. It is closely related to and makes use of Chapman's celebrated theorem on topological invariance of Whitehead torsion.


Firstly, a *cell-like map* is a map which is proper, and whose fibres are all cell-like spaces. A compact metric space is called *cell-like* if it is "shape contractible" in the sense of having trivial (Borsuk) shape. Secondly, a map $f:X\to Y$ is a *fine homotopy equivalence* if, for any open cover $\mathcal{U}$ of $Y$, $f$ has a homotopy inverse $g$, and corresponding homotopies to the two identity maps which both follow paths never leaving some open in $\mathcal{U}$. Finally, recall that ANR abbreviates *absolute neighbourhood retract*. Importantly, and to avoid piling up qualifiers, all ANRs in this answer will be implicitly assumed metric and separable. A good example of a (metric separable) ANR is a locally finite, countable CW-complex.


The main theorem in the article "[Mappings between ANRs that are fine homotopy equivalences](http://projecteuclid.org/euclid.pjm/1102905677)" by William Haver implies that any cell-like map between locally compact ANRs is a (proper) fine homotopy equivalence. In that article, a cell-like map would be called a proper $UV^\infty$-map instead.


On the other hand, corollary 3.2 in the article "[The homeomorphism group of a compact Hilbert cube manifold is an ANR](http://www.maths.ed.ac.uk/~aar/papers/haver.pdf)" by Steve Ferry implies that any proper fine homotopy equivalence between locally compact ANRs is a simple homotopy equivalence. The proof of this result by Ferry uses the theory of Hilbert cube manifolds, via the following facts: (1) a proper fine homotopy equivalence between Hilbert cube manifolds is approximable by homeomorphisms (as proved by Ferry in the article cited above), and (2) the product of a locally compact ANR with the Hilbert cube is a Hilbert cube manifold (by a result of Robert Edwards). Finally, it uses Chapman's results on the topological invariance of Whitehead torsion.


In any case, putting together the two results stated above, we conclude the following.



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> Any cell-like map between locally finite, countable CW-complexes is a simple homotopy equivalence.
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To relate this to the actual question, first observe that simple homotopy type is not defined for every space. In the greatest generality I am aware of, it is definable for locally compact ANRs. Then being simple homotopy equivalent to a point is equivalent to being a compact contractible ANR. Since such a space is necessarily cell-like, we obtain the following answer to your last question.



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> Let $f:X\to Y$ be a proper map between locally finite, countable CW-complexes. Assume each fibre of $f$ is a contractible ANR, i.e. has trivial simple homotopy type (since the fibres are compact). Then $f$ is a cell-like map. By the above result, $f$ is a simple homotopy equivalence.
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That is the most general statement I can find. If you only care about the case of finite CW-complexes, then the result admits a formulation with fewer classifiers, and was also proved a few years earlier. For example, it is stated explicitly as a theorem in page 17 of the article by Lacher cited below.



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> If $f:X\to Y$ is a cell-like map between compact ANRs, then $f$ is a simple homotopy equivalence. In particular, if $f:X\to Y$ is a map between finite CW-complexes whose fibres are all contractible ANRs, then $f$ is a simple homotopy equivalence.
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Finally, two nice overviews of the theory of cell-like maps and related topics are:


* the [1978 ICM address of Robert Edwards](http://www.mathunion.org/ICM/ICM1978.1/Main/icm1978.1.0111.0128.ocr.pdf) which describes many interesting connections with the topology of manifolds;
* the article "[Cell-like mappings and their generalizations](http://dx.doi.org/10.1090/S0002-9904-1977-14321-8)" by Lacher, which gives a thorough exposition of the theory of cell-like maps.


Moreover, the first part of the article by Lacher contains an interesting exposition of theorems of Vietoris-Begle type, much like the ones mentioned in the question. It might be a good place to start looking for a general theory of such results.


