I search for a chain of clean references, which lead the fact of topological manifolds of dimension $n$ having the homotopy type of a CW of dimension $n$.

Milnor's On spaces having the homotopy type of a CW-Complex proves that every topological manifold has the homotopy type of a countable CW-complex since it is an absolute neighborhood retract.

Theorem E of Wall's Finiteness Conditions for CW-complexes then gives me the desired answer for dimensions at least 3, as long as I know that the universal covering $\tilde{M}$ of a topological manifold $M$ of dimension $n$ has vanishing homology up to dimension $n$ and it holds $H^{n+1}(M,\mathcal{B})=0$ for all abelian coefficient bundles $\mathcal{B}$.

Theorem 3.26 and Proposition 3.29 of Hatcher's book gives me the first claim about the homology since universal coverings of topological $n$ manifolds are again topological $n$ manifolds since the fundamental group of a topological manifold is countable (a clean reference for this is Theorem 7.21 in Lee's book 'Introduction to Topological Manifolds')

Since I am happy to forget about the low dimensional cases, it leaves me with the search for a solid reference to the claim

Let $M$ be a topological manifold and $\mathcal{B}$ be an abelian coefficent bundle, then $H^{n+1}(M,\mathcal{B})=0$.

  • $\begingroup$ Are you trying to find a finite CW-complex? Then, what do you call a manifold? $\endgroup$ Apr 4 '15 at 9:04
  • $\begingroup$ An n-dimensional CW-complex is a CW-complex admitting a cell structure of (not necessarily finite) cells of maximal dimension n. A topological n-manifold is a locally euclidian (not necessarily compact) Hausdorff space with a countable base. $\endgroup$
    – Tom
    Apr 4 '15 at 12:29

Every topological manifold has a handlebody structure except in dimension 4, where a 4-manifold has a handlebody structure if and only if it is smoothable. This is a theorem on page 136 of Freedman and Quinn's book "Topology of 4-Manifolds", with a reference given to the Kirby-Siebenmann book for the higher-dimensional case. It is then an elementary fact that an $n$-manifold with a handlebody structure is homotopy equivalent to a CW complex with one $k$-cell for each $k$-handle, so in particular there are no cells of dimension greater than $n$. At least in the compact case a manifold with a handlebody structure is in fact homeomorphic to a CW complex with $k$-cells corresponding to $k$-handles; see page 107 of Kirby-Siebenmann. This probably holds in the noncompact case as well, though I don't know a reference.

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    $\begingroup$ That is overkill and probably circular. $\endgroup$ Apr 5 '15 at 15:34

Vanishing of cohomology above the top dimension follows from the Poincare duality, which is proved in a general form in Bredon's "Sheaf theory", Chapter 5, section 9. This works for all topological manifolds (compact or not) and for the local coefficent bundles as in Wall's paper (recast in sheaf theoretic language).

  • $\begingroup$ Wall's theorem requires singular cohomology as input, so you need another theorem, probably also in Bredon, comparing singular cohomology to sheaf cohomology, just using the manifold hypothesis. $\endgroup$ Apr 6 '15 at 15:04
  • $\begingroup$ Actually, Tom knows from Milnor that a topological manifold has the homotopy type of a CW complex, which gives the comparison theorem. $\endgroup$ Apr 6 '15 at 20:10
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    $\begingroup$ Yes, one has to unravel definitions and check the assumptions of various theorems. Actually, vanishing of cohomology beyond the top dimension holds in much more general setting, i.e. under mild assumptions there is a cohomological characterization of dimension (where cohomology is Bredon's sheaf cohomology). If memory serves for most metrizable spaces of interest the covering dimension and the cohomological dimension coinside. Of course $n$-manifolds have covering dimension $n$. $\endgroup$ Apr 6 '15 at 20:44
  • $\begingroup$ @IgorBelegradek: The book "General Topology" by Engelking has a chapter on topological dimension theory, giving that for all metrizable spaces "covering dimension" and small/large inductive dimensions coincide, with the latter equal to the "usual" dimension for cubes and then for paracompact topological manifolds. So as you say, that gives vanishing of Cech cohomology of any abelian sheaf beyond degree $n$ for a paracompact Hausdorff $n$-manifold (metrizable by Smirnov). Warner's textbook gives the agreement of Cech and derived functor sheaf cohomology for paracompact Hausdorff spaces. $\endgroup$
    – grghxy
    Jul 21 '15 at 2:30

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