# Meaning of a certain Kronecker pairing.

Let $M$ be a compact oriented manifold of dimension $n$ with fundamental class $[M]$. Consider the image of $[M]$ under the map $H_n(M)\rightarrow H_n(M\times M)$ induced by the diagonal map, and let $\alpha\in H^n(M\times M)$ be the Poincaré dual. Does the Kronecker Pairing $<\alpha\cup\alpha,[M\times M]>$ carry a topological meaning?

Edit: It has been claimed in the comments that this is $\chi(M)$, so I am looking for a reference including a proof of this NOT USING smooth manifolds.

• It is the Euler characteristic of the manifold, for which there is a plethora of topological interpretations. – Thomas Rot Nov 10 '16 at 0:03
• @ThomasRot do you happen to know a reference for this fact? I didn't find this in any book on algebraic topology – Juan Fran Nov 10 '16 at 15:26
• I would refer to bott and tu which is smooth manifolds. Another guess is that is in Bredon's book. – Thomas Rot Nov 10 '16 at 19:56
• The usual proof interprets this class as the intersection of the zero section of the tangent bundle with any nonzero class transverse to it. – Thomas Rot Nov 10 '16 at 20:14
• Which is not available in the non smooth setting – Thomas Rot Nov 10 '16 at 20:15