In the lecture on algebraic topology, which I am currently attending, we tried to proof as much as possible using just the Eilenberg-Steenrod axioms for homology theory before we constructed singular homology. In particular we introduced the topic of homological orientation of manifolds by saying that a topological manifold is orientable, if it admits a fundamental class. We were not told however, how $H_n(M)$ can be calculated, so it is not clear to me that fundamental classes should exist at all.
Researching homological orientation I found the definition of orientability via local orientations (noting that $H_n(M,M\setminus \{x\}) \cong H_n(\mathbb S^n) \cong R$ for $H_\bullet$ being a ordinary additive homology theory with coefficient ring $R$).
Is there a proof of the fact that for a connected closed manifold $M$ we have $H_n(M) = R$ if and only if $M$ is $R$-orientable, which avoids details of the construction of singular homology, i.e. uses tools from formal homology theory only (e.g. Mayer Vietoris sequence, the triple exact sequence or excission)?
The proofs of Hatcher or Greenberg-Harper use a rather technical lemma, which seems to rely on the explicit representation of cycles. For me this is a problem in the sense that we worked quite alot with fundamental classes before even constructing singular homology. As I prefer to learn things in order, I would have to reorder the whole lecture when preparing for the exam, so I am keen to have at least the implication $R$-orientable $\Rightarrow$ top homology is nonzero cyclic $R$-module.
PS: I would be interested in a formal (i.e. following from the Eilenberg-Steenrod axioms and not involving the construction of singular homology) way to proof any of the following facts:
- For an $n$-dimensional topological manifold $M$ it holds that $H_k(M) = 0$ for $k>n$
- For a connected closed manifold $M$ it holds that $H_n(M) = 0$ if and only if $M$ is not $R$-orientable
- For a connected noncompact manifold $M$ it holds that $H_n(M)=0$
Thank you all in advance.
Update
I want to give an update on what I have tried so far. Please be aware, that I am currently attending the lecture, so I have no advanced understanding in the tools involved and what turned out to fail for me may actually work.
- I tried my best to apply any of the long exact sequences onto canonical choices of subsets of the manifold, but to no avail.
- Another approach was to cover the compact manifold $M$ with finitely many open subsets homeomorphic to $\mathbb R^n$ and try to use Mayer Vietoris to calculate the top homology group. However I could not find a version of Mayer Vietoris using a cover with more that $2$ sets and I could not come up with a way to inductively circumvent this.
- My last approach was to find a CW-structure on the manifold. Besides being circular in the sense that we used degrees (and in particular orientations) to show that cellular homology is isomorphic to a given ordinary homology theory, this would have been nice to at least get the result that the homology in degrees greater than $n$ vanish. It turns out though, that one does not actually know, whether each topological manifold admits a CW-structure. The best I could find was some kind of approximation by CW-structures, but this seems to be linked to homotopy groups and hence (as far as I know) to singular homology. Even assuming that higher homology groups vanish did not enable me to show my original question, so it would not have helped anyway.
