I'm interested in a concrete description of the "wrong way maps" in homology/cohomology.
$\textbf{Question 1:}$ Let $X, Y$ be compact smooth manifolds of dimensions $n, m$ respectively, and $\phi: X \rightarrow Y$ be a surjective $C^\infty$ map. Using the isomorphism with de Rham cohomology we can define a pullback map on homology classes. If $[C] \in H_i^{cellular}(Y, \mathbb{R})$ is a homology class in $Y$, then is the pullback $\phi^*([C]) \in H^{cellular}_{m-n+i}(X, \mathbb{R})$ given by $[\phi^{-1}(C)]$ if $C$ does not contain any of the critical values of $\phi$?
(In the case I'm most interested $X,Y$ are complex algebraic varieties and $\phi$ is a smooth map.)
$\textbf{Question 2:}$ Is the same thing true with $\mathbb{Z}$ coefficents, since you can't use comparison with de Rham cohomology. Is question 1 vastly more general?
(Sorry if this question is too easy. This isn't really my area – I am an arithmetic algebraic geometer.) References will be greatly appreciated.
Best Answer
Let me give you a geometric version of wrong way maps for smooth manifolds. Let us consider a smooth map $f:X\rightarrow Y$, for the sake of simplicity we assume that $X$ and $Y$ are closed and oriented (otherwise we need an orientability condition on the virtual normal bundle of $f$).
We know that a class $C\in H_i(Y,\mathbb{Z})$ can be represented by a pseudo-manifold $Z$ (see for example Mark Goresky's paper "Whitney stratified chains and cochains." Trans. Amer. Math. Soc. 267 (1981), 175-196). It means that:
we have a map $\psi:Z\rightarrow Y$,
every closed pseudo-manifold of dimension $i$ has a fundamental class $[Z]\in H_i(Z,\mathbb{Z})$ and we have $\psi_*([Z])=C$.
Now the pull-back in homology is defined by taking the geometric pull-back of $Z$ along $f$. We need to put $\psi$ and $f$ in transverse position, this can be achevied by some transversality lemmas for pseudo-manifolds. Thus when these two maps are in general position the geometric pull-back is again a pseudo-manifold and its fundamental class gives you a representative of $f^*(C)$.
If you don't feel comfortable with pseudo-manifolds, then you can write your cycle as a sum of $i$-simplexes. You put these simplexes in transverse position (same transversality lemmas), the pull-back of these $i$-cycles is a sum of manifolds with corners you pick a triangulation of each of them and get the pull-back cycle as a sum of simplexes.
The operation is well defined in homology.
Thus pull-back in homology is truly a geometric pull-back. Let me add that there are plenty of ways to define pull-backs in homology they all rely on Poincaré duality.
Maybe a nice reference is also M. Kreck's book "Differential algebraic topology" graduate studies in mathematics 110, AMS (chapter 12 and chapter 13). In this book he has a very good geometric model of homology in terms of stratifolds, its geometric definition of cohomology with its version of Poincaré duality will give you pull-back maps in homology which are exactly geometric transverse pull-backs of stratifolds.