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?


1 Answer 1


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 $\tilde H_{i-1}(M)$.

  • $\begingroup$ It would be helpful to have a reference. Thanks. $\endgroup$
    – asv
    Sep 29 at 18:42
  • 2
    $\begingroup$ @asv Sorry, I don't have a reference off the top of my head. I would try looking at any of the standard books on sheaves (Kashiwara-Schapira, Iversen, Bredon, etc.). $\endgroup$ Sep 29 at 19:12
  • 3
    $\begingroup$ @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. $\endgroup$ Sep 29 at 19:15

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