TITLE: Relationships between homology maps of cobordant manifolds
QUESTION [16 upvotes]: Let $W$ be a cobordism between $n$-manifolds $M$ and $N$, and $f\colon W \mapsto X$ be a map to some manifold $X$. 
Does anybody know of any nice examples of general relationships between the images of the maps $g_*\colon H_*(M)\mapsto H_*(X)$ and $h_*\colon H_*(N)\mapsto H_*(X)$ induced by the restrictions $g=f\mathrel{|M}$ and $h=f\mathrel{|N}$, based on some given data about about these manifolds?  
For Example, if $M=N$ and $W=[0,1]\times M$, then $f$ is a homotopy between $g$ and $h$, so $g_*=h_*$. Also, if our manifolds are oriented and compact, $X$ is a connected $n$-manifold, and $f$ is smooth, then the degree of $f\mathrel{\mathrel{|}\partial W}=f\mathrel{\mathrel{|}M\coprod N}$ is zero, so the degrees of $g$ and $h$ are equal up to sign, as are the images of $g_*$ and $h_*$ in degree $n$ homology.

REPLY [10 votes]: If we consider cohomology with $\mathbb{Z}/2$-coefficients, if $M$ is a manifold $H^*(M;\mathbb{Z}/2)$ it satisfies Poincaré duality, and is an unstable algebra over the Steenrod algebra. Brown and Peterson in their paper "Algebraic bordism groups" (Annals of maths, 1964) had the wonderful idea to give an algebraic model of geometric bordisms in terms of algebraic bordisms of unstable algebras over the Steenrod algebra. 
They showed that this  algebraic bordism group $N^{alg}_*$ is isomorphic to Thom's bordism group of smooth manifolds. For any unstable algebra algebra $A$ they define an algebraic bordism group $N^{alg}_*(A)$. In fact, they proved that for any space $K$, Conner, Floyd's bordism group $N_*(K)$ of non-oriented smooth manifold over $K$ is isomorphic to $N^{alg}_*(H^*(K;\mathbb{Z}/2))$. The isomorphism is given by a natural transformation:
$$\phi_K:N_*(K)\rightarrow N^{alg}_*(H^*(K;\mathbb{Z}/2))$$
that sends a bordism class $f:M\rightarrow K$ to the algebraic bordism class of the induced map $f^*:H^*(K;\mathbb{Z}/2)\rightarrow H^*(M;\mathbb{Z}/2)$. 
Not only thanks to Wu formulas (that use Poincaré duality and Steenrod squares action on $H^*(M;\mathbb{Z}/2)$), you can recover Stiefel-Whitney numbers, and you can determine if two manifolds are cobordant but using the unstable algebra structure of $H^*(K;\mathbb{Z}/2)$ you can recover the geometric bordism group of $K$.