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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.

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3 Answers 3

The wrong-way maps in homology are in fact less mysterious than they look. Suppose that $ f: X \to Y $ is a continuous map of oriented closed topological manifolds. Then there is a composition of maps $$ {H_{i}}(Y;\mathbb{Z}) \stackrel{\cong}{\longrightarrow} {H^{\dim(Y) - i}}(Y;\mathbb{Z}) \longrightarrow {H^{\dim(Y) - i}}(X;\mathbb{Z}) \stackrel{\cong}{\longrightarrow} {H_{\dim(X) - (\dim(Y) - i)}}(X;\mathbb{Z}), $$ where the arrows on the left and right are given by Poincaré Duality, and the one in the middle is the usual pullback in cohomology. The orientations on $ X $ and $ Y $ are required to specify the Poincaré isomorphisms. (Here, $ \dim $ stands for the dimension of a manifold.)

There is a version of this in the case when $ X $ and $ Y $ are oriented but no longer compact. In this case, one has to use Borel-Moore homology instead of ordinary homology.

Note also that in a similar way, one can construct a wrong-way cohomology map $$ {H^{i}}(X;\mathbb{Z}) \longrightarrow {H^{i - \dim(X) + \dim(Y)}}(Y;\mathbb{Z}), $$ which is nothing but the Gysin map when $ f $ is a closed embedding.

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algori, Thanks for your great answer! –  6672 Dec 8 '12 at 20:01
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6642 -- welcome! By the way, the gemoetrical construction you describe in your posting is precisely the same: if we have a homological class $c$ represented by an oriented submanifold $M$ of $Y$, then the pullback of the Poincar\'e dual class of $c$ is the Poincar\'e dual of the preimage of $M$ (under some transversality assumptions). The reason for this is the construction of the Poincar\'e dual cohomological class of $M$: it can be described as the class of the say singular cochain that takes a chain $\sum a_i \sigma_i$ where $a_i\in\mathbb{Z}$ and $\sigma_i$ are singular simplices ... –  algori Dec 8 '12 at 21:00
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... transversal to $M$ to $\sum a_i \varepsilon_i$ where $\varepsilon_i$ is the sign of the intersection $\sigma_i\cap M$. This cochain is, of course, defined on a subcomplex of $C_*(Y)$ but this subcomplex is quasiisomorphic to the whole of $C_*(Y)$ so one can extend it, noncanonically, to $C_*(Y)$. –  algori Dec 8 '12 at 21:03

to Leonard-

An important example of homology pull-back is given by the intersection product. Let $M$ be a closed oriented manifold then we define this product as: $$H_i(M)\otimes H_j(M)\stackrel{\times}{\rightarrow} H_{i+j}(M\times M)\stackrel{\Delta^{*}}{\rightarrow} H_{i+j-d}(M)$$ where $\Delta^*$ is the homology pull-back of the diagonal $\Delta:M\rightarrow M\times M$. The intersection product was first introduced by means of geometric transverse intersection of cycles. Bredon "Geometry and Topology" explains it and makes the relationship with the cup product which is its Poincaré dual. He does not explain how to do when the cycle is not representable by a manifold but this is a good starter.

Fore general cycles, as I explained in my other answer either you play with simplexes viewed as manifolds with corners or you play with manifolds with singularities. M. Kreck in his book "Differential algebraic topology" did it for a particular type of manifolds with singularities and he proves all the tranversality results you want in order to define homology pull-backs. M. Goresky in his PhD thesis studied it in the case of stratified pseudomanifolds and also build a geometric homology and a geometric cohomology thanks to what he called Whitney chains and cochains. Here you need transversality for stratified pseudomanifolds (it works inductively strata by strata). This is in the smooth world, if you are more a PL-guy then you should have a look at:

S. Buoncristiano, C. P. Rourke and B. J. Sanderson "A geometric approach to homology theory", Cambridge Univ. Press, Cambridge, Mass., 1976

Another good reference is: Jakob, Martin "Bivariant theories for smooth manifolds." Papers in honour of the seventieth birthday of Professor Heinrich Kleisli (Fribourg, 2000). Appl. Categ. Structures 10 (2002), no. 3, 279–290.

It is another geometric approach to homology theories where homology pull-backs are geometric pull-backs. In fact this all goes back to

Quillen, D. "Elementary proofs of some results of cobordism theory using Steenrod operations." Advances in Math. 7 1971 29–56 (1971).

I like and recommand Jakob's paper because it is related to the framework of

Fulton, W.; MacPherson, R. "Categorical framework for the study of singular spaces." Mem. Amer. Math. Soc. 31 (1981), no. 243

which explains how to mix homology covariant and contravariant morphisms in order to have a good framework to get Riemann-Roch types theorem.

To conclude this geometric interpretation of homology pull-backs certainly goes back to the first days of homology in algebraic topology before cohomology was invented.

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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.

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Would you be able to provide more references for the fact that pullback in homology corresponds to pullback in geometry? Thanks! –  Leonard Dec 8 '12 at 8:44
    
David C, Thanks for your great answers! –  6672 Dec 8 '12 at 20:01

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