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Given a compact manifold and a Morse function, we obtain a convenient cellular decomposition. But suppose I have a (finite dimensional, possibly singular) space $X$ that is not a manifold and I wish to find a simplicial set $S$ whose geometric realization is homeomorphic to $X$. Surely no tool will be as convenient as Morse theory, but there ought to be techniques that are similarly explicit.

Here's a test question that should be easy if the tool is good enough.

Let $G$ be a finite dimensional connected Lie group, and define $T_G$ to be the subspace of $G \times G$ consisting of commuting pairs $T_G = \left\{ (g,h) \in G \times G \; \; | \; \; gh=hg \right\}$. Supposing that $T_G$ is connected, does there exist a reduced simplicial set $S$ with simplicial subset $S_0 \subseteq S$ so that the geometric realization of $S$ is homeomorphic to $G \times G$ and the geometric realization of $S_0$ is the subspace $T_G$?

Note that the connectivity of $G$ makes it plausible for $S$ to be reduced. More generally, if a space is $n$-connected, it would be nice if we could assume the simplicial set to have a trivial $(n-1)$ skeleton.

John Wiltshire-Gordon
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  • You need more hypotheses on $X$ for such a thing to exist; your condition implies that $X$ is homeomorphic to a CW complex and it's an open problem whether this is always true even for compact topological $4$-manifolds (http://mathoverflow.net/questions/73428/when-is-a-compact-topological-4-manifold-a-cw-complex). – Qiaochu Yuan Jul 25 '14 at 18:55
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    If you weaken "homeomorphic" to "weak homotopy equivalent to" then the singular simplicial set always works, and if you weaken "homeomorphic" to "homotopy equivalent to" then the singular simplicial set works for spaces homotopy equivalent to CW complexes. – Qiaochu Yuan Jul 25 '14 at 18:59
  • If your space admits a Whitney stratification then apparently there is a notion of stratified Morse theory which can be used in this case. – Qiaochu Yuan Jul 25 '14 at 19:01
  • Just to supplement Qiaochu's above reference: http://math.stackexchange.com/questions/593041/cw-complexes-and-manifolds – Dan Rust Jul 25 '14 at 19:32
  • @QiaochuYuan Thank you for the comments; indeed, I will need to impose conditions on $X$. I am not willing to weaken the homeomorphism, however. – John Wiltshire-Gordon Jul 25 '14 at 21:00

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