2 edited body

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 manifold 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 cochainshere . 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 morphism morphisms in order to have a good framework to have get Riemann-Roch types theorem.

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

1

to Leonard-

An important example of homology pull-back is given by the intersection product. Let $M$ 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 manifold with singularities and he proves all the tranversality results you want 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 morphism in order to have a good framework to have Riemann-Roch types theorem.

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