An $n$-dimensional topological manifold $M$ is orientable if for each compact set $K \subset M$, there exists a homology class $ \langle o_K \rangle \in H_n(M, M \setminus K)$ with the property that for any $x \in K$, the natural map $H_n(M, M \setminus K) \to H_n(M , M \setminus x)$ induced by inclusion maps $ \langle o_K \rangle$ to a generator of $H_n(M , M \setminus x) \cong \mathbb{Z}$. This is equivalent to the fibre bundle $ \mathcal{O}(M)$ from last lecture being a trivial bundle.

We being this lecture we begin by proving the main result from last time: that the two functors $H_n(M, \square ;A)$ and $ \Gamma_c(M \setminus  \square ; A)$ are naturally isomorphic, and then use this to define orientability as above.

The lecture concludes with the introduction of (yet another) product, called the cap product. This works for arbitrary topological space and is a "mixed" product:

$$ H^{ \bullet}(X) \otimes H_{ \bullet}(X) \to H_{ \bullet}(X), \qquad \langle \alpha \rangle \otimes \langle c \rangle \mapsto \langle \alpha \rangle \frown \langle c \rangle.$$

Although useful in its own right, for us the most important property of the cap product is for Poincaré Duality, which states the following: if $M$ is a closed oriented topological manifold then $ \square \mapsto \square \frown \langle o_M \rangle$ is an isomorphism. We will prove this in Lecture 39.


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