# references for cones' homology

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" $$R\mathrm{hom}(k_{\overline{\gamma}},k_{\mathbb{R}^n})$$.

• aren't cones contractible? – William Jan 3 at 19:50
• @William no - take a quadratic cone : $q_0^2<q_1^2+q_2^2$ – epsilones Jan 3 at 21:42
• @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". – Kevin Arlin Jan 4 at 0:38
• 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. – William Jan 4 at 4:25
• @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 $\mathbb{R}^n$, $q_0^2+...+q_{p-1}^2<q_p^2+...+q_{n}^2$ – epsilones Jan 4 at 18:31