# Intersection of two homology classes

Studying the first pages of Gompf-Stipsicz's 4-Manifolds & Kirby Calculus forced me to worry about the geometric meaning of homology and cohomology classes; in particular page 7 contains the following sentence, which I feel quite obscure:

Let $X$ be a compact, oriented, topological 4-manifold. When $X$ is oriented, it admits a fundamental class $[X]\in H_4(X,\partial X; \mathbb Z)$ [which is, as explained before, a top-degree homology class generating $H^4(X,\partial X; \mathbb Z)$].

Definition. The symmetric bilinear form $$Q_X\colon H^2(X,\partial X;\mathbb Z)\times H^2(X,\partial X;\mathbb Z) \to \mathbb Z$$ defined by $Q_X(a,b)=\langle a\cup b,[X]\rangle$ is called the intersection form of $X$.

My first question is: how's that pairing $\langle-,-\rangle$ defined?

I think I have a slight confidence with the idea of homology-cohomology classes as geometric objects (immersed submanifold, cycles as sub-simplices...).

I'm also able to forsee that all the different pairing definable in various (co)homology theories can someway can someway be reconduced to a single idea.

So my second question is in fact a modified version of the first one: how can I link the former "intersection" pairing to the similar one $$\langle -,-\rangle_{dR}\colon H^k_{dR}(M)\times H^{n-k}_{dR}(M)\to \mathbb R\colon (a,b)\mapsto \int_M a\land b$$ in de Rham cohomology?

• You may want to look at cap product. Wedging ($k$-form) with a $(n-k)$ form and integrate over $M$ and capping a $k$-dimensional cohomology class with the fundamental class are the "same" thing, only that the latter one is the general construction for compact orientable (topological) manifolds.
– user325
Commented Jul 26, 2011 at 19:41

Recall that $H_k(X)$ is a subquotient of $C_k(X)$, the dimension-$k$ singular simplices in $X$, and that $H^k(X)$ is a subquotient of $C^k(X)=\mbox{Hom}(C_k(X),\mathbb{Z})$. So a $k$-dimensional cohomology class is represented by a cocycle (up to coboundaries), and its evaluation on a homology class is easily seen to be well-defined: if $[\alpha]\in H^k(X)$ and $[c]\in H_k(X)$, then $\langle \alpha+\delta \beta, c + \partial d \rangle = \langle \alpha, c \rangle$ since $\delta \alpha=0$ and $\partial c=0$. (Recall the adjunction $\langle \delta \gamma , e \rangle = \langle \gamma, \partial e \rangle$ -- this is in fact nothing more than the definition of $\delta$.)
• even more is true. if two cohomology classes $x,y$ are such that their poicaré duals are represented by embedded submanifolds $A,B$, then the cup product $x \cup y$ may be computed by intersecting A and B transversally and then forming the poincaré dual of this intersection. Commented Apr 29, 2012 at 9:00