# Dualizing complex of the cone over a manifold

Let $$M$$ be a smooth (or just topological) closed manifold. Let $$C(M)$$ denote the cone over $$M$$, i.e. $$C(M)$$ equals to $$M\times [0,\infty)$$ with $$M\times \{0\}$$ contracted to a point. The image of $$M\times \{0\}$$ in $$C(M)$$ is called the origin.

What is the dualizing complex of $$C(M)$$? In particular what is its stalk at the origin?

The stalk of the dualizing complex at a point is the shift of reduced homology of the link at that point. In this case, the link is $$M$$ and so the homology of the stalk in degree $$i$$ is $$\tilde H_{i-1}(M)$$.
• @asv You can compute that the stalk $i^* \omega_X$ is dual to $i^! \mathbb Z$, and the homology of $i^!\mathbb Z$ is $H^*(U,U-p)$ for $U$ any neighborhood of the point. Now $U$ is contractible, and $U-p$ is the link. Sep 29 at 19:15