I asked recently on MO about algebraic structures admitted by topologically homogenous continua like the Hilbert cube $\ I^{\mathbb N}\ $ or the Knaster pseudo-arc. There is a relation between the algebraic and geometric structures in this context. And this is the topic of this post.


Let $\ \*\ $ stand for a single-point space. Let us consider the following 3 properties of a compact space $\ X$:


* **(C0)** -- $X\ $ is acyclic, meaning that $\ H^\*(X;\mathbb Z) = H^\*(\*;\mathbb Z)$;
* **(C1)** -- $X\ $ has a shape of a single-point space, i.e. every continuous map of $\ X\ $ into any (compact finite) polyhedron is homotopic to a constant map;
* **(C2)** -- $X\ $ is contractible.


Of course $\ (C2)\Rightarrow (C1)\Rightarrow (C0).\ $ Now the following three conjectures are open (or someone may provide a reference):




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**(Hk)** If a continuum $\ X\ $ ($|X|>1$)  has property (Ck) then there does not exist a homogenous metric in $\ X\ $ which induces the topology of $\ X$;




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for $\ k=0\ 1\ 2.\ $


**MOTIVATION**: if a metric compact space $\ X\ $ admits a structure of a left or right quotient $\ G/H\ $ where $\ G\ $ is a compact metric group, and $\ H\ $ is a closed (not necessarily normal) subgroup of $\ G\ $ then $\ X\ $ admits a homegenous metric which induces the given topology. Indeed, one can apply the Haar measure on $\ G$; first to obtain a homogenous metric on $\ G$, and then again on $\ X=G/H\ $ itself (I hope this is right).



Can an acyclic continuum be metrically homogenous? (I'd say: no way! :-)
It seems that the conjecture (H2) can be confirmed with help of the recent result of Hofmann and Kramer (<http://arxiv.org/pdf/1301.5114.pdf>) who proved that for a compact topological group $G$ and a closed subgroup $H\subset G$ the homogeneous space $X=G/H$ is a manifold if and only if $X$ contains a non-empty open set contractible in $X$. This result of Hofmann and Kramer implies that a contractible continuum $X$ admitting a homogeneous metric is a contractible compact manifold, which is a singleton (see the discussion <https://math.stackexchange.com/questions/418509/is-there-a-compact-contractible-manifold>).


Moreover, a recent result of Antonyan and Dobrowolski (Locally contractible coset spaces, Forum Mathematicum. 27:4 (2015), 2157–2175) implies that each finite-dimensional locally connected compact homogeneous space $X$ is a manifold. This implies that an acyclic finite-dimensional Peano continuum admits no homogeneous metric. 


