Honology, Cohomology, Euler Number for Non Orientable Manifolds

in what follows, by cohomology I will refer to de Rham cohomology. Every manifold $M^m$ of dimension $m$ is assumed to be compact.

For orientable manifold I know the following results

1. From Poincaré duality: $H_k(M,\mathbb{R}) \simeq H^k(M)$;
2. Betti number $b_n$ is the dimension of the torsion-free part of the $n$th homology group and so, also the dimension of $n$th cohomology group;
3. Euler characteristi is the alterned sum of the betti numbers: $$χ(M)= ∑_{i=0}^m (-)^ib_i= ∑_{i=0}(-)^i\dim H^i(M)$$

I know also that Poincaré duality does not hold, at least in its original version, for non oriented manifold, as well as other results that for orientable manifold are taken for granted. My questions are:

1. What is the relation between $H^i(M)$ and $H_i(M,\mathbb{R})$ (the last being the torsion-free part of $H(M,\mathbb{Z})$ as a group. By knowing one of the two what can I infer for the other?
2. As a conseguence, Betti number, defined by the dimension of the torsion-free part of the homology, can also be thought as the dimension of cohomology?
3. Is it still true that the alternate sum of dimension of cohomology group yield the Euler characterstic?

I spot these points, computing the $\mathbb{Z}$-Homology, Cohomolgy, and Euler characteristic of Klein Bottle, Real Projective Plane and Moebus strip, and all these "conjectures" are verified, but I expect that this cannot be the general case.

I found that on books while for the oriented case the web of results is pretty clear, for non oriented cases is not like that. Could you also give me some reference (possibly not too technical since I'm a physicist)?

This is more a comment that a answer but it's a bit too long for a comment.

Poincaré duality holds for compact non-oriented manifold, if you take your coefficient to be $\mathbb Z/2 \mathbb Z$.

This is not very satisfying, and you can improve it with local coefficient. This is not very intuitive but it works. This is called "twisted Poincaré duality". Abstractly, they are defined in the following way : if $M$ is a $\pi = \pi_1(X)$-module, then the homology with local coefficient $M$ is defined as the homology of the complex $C_n(\tilde X) \otimes_{\mathbb Z \pi} M$. For non-oriented manifold you have a very adapted local system which is simply $\mathbb Z$ as a group, the structure of $\pi$-module given by $\gamma.1 = 1$ if going around $\gamma$ preserves orientation and $-1$ else. We call $\mathbb Z_w$ this local system.

Now (twisted) Poincaré duality can be stated the following way : we have an isomorphism $H^i(X, \mathbb Z) \cong H_{n-i}(X, \mathbb Z_w)$ and $H^i(X, \mathbb Z_w) \cong H_{n-i}(X, \mathbb Z)$ which related homology with local coefficient and cohomology with integer coefficient (and vice-versa). This is the version of Davis and Kirk, page 102.

For having a feeling about all this, you can compute the homology with local coefficient for $\mathbb RP^2$ and verifying this coincide with the cohomology of $\mathbb RP^2$.

Standard references about this are the book of Hatcher, the book of Davis and Kirk. They are probably lectures notes online for more precise statements or improvement of these theorem.

• Thank you for the comment. I found something similar in Wikipedia. So, as far as I understand there is no means to have a link between de Rham cohomology and Euler characteristic. At the very end is this what I'm looking for. – MaPo May 5 '17 at 12:46
• Maybe you can try "Twisted de Rham complex" seeing if this give you something. But on non oriented manifold, integrating differential forms is a bit tricky. – user171326 May 5 '17 at 13:14