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Are there good references reporting the computation of homology of (non necessarily convex) open cones of $\mathbb{R}^n$ ? Here by cone we mean subset of $\mathbb{R}^n$ invariant by the natural action of $\mathbb{R}^*$.

The idea behind is to get a good intuition of it in order to compute the dual of the cohomology of the cone in the following sense :

let $k$ be a commutative field, for an open cone $\gamma$, let us denote by $\overline{\gamma}$ its adherence, $k_{\overline{\gamma}}$ the sheaf constant with stalks $\mathbb{R}$ on $\gamma$ and $R\mathrm{hom}$ the derived functor of the $hom$ bifunctor.

I want to compute the derived "dual" $R\mathrm{hom}(k_{\overline{\gamma}},k_{\mathbb{R}^n})$.

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    $\begingroup$ aren't cones contractible? $\endgroup$ – William Jan 3 at 19:50
  • $\begingroup$ @William no - take a quadratic cone : $q_0^2<q_1^2+q_2^2$ $\endgroup$ – epsilones Jan 3 at 21:42
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    $\begingroup$ @epsilones You should clarify what sense of cone you mean. Your tags come from a field where "cone" means "the cone on a topological space". $\endgroup$ – Kevin Arlin Jan 4 at 0:38
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    $\begingroup$ If a cone (by your definition that it is closed under non-zero scaling) contains $0$ then it deformation retracts onto $0$ via the homotopy $H(x, t) = (1 - t) x$. Otherwise it deformation retracts onto an open subset of $S^{n-1}$ via $G(x, t) = (1-t)x + t\frac{x}{||x||}$, and this subset will be either $S^{n-1}$ or homeomorphic to an open subset of $\mathbb{R}^{n-1}$. I feel like these can be rather complicated and there may not be a complete description of all of their cohmologies. $\endgroup$ – William Jan 4 at 4:25
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    $\begingroup$ @William sure my question was too large stated this way - what I have in mind is to look for a description of classes of cones such some which we could say to have signature $(p,n-p)$ in $\mathbb{R}^n$, $q_0^2+...+q_{p-1}^2<q_p^2+...+q_{n}^2$ $\endgroup$ – epsilones Jan 4 at 18:31

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