Timeline for homologies of some subsets of ${R}^{n}$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
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| Oct 21, 2018 at 14:10 | answer | added | John Shareshian | timeline score: 3 | |
| Oct 21, 2018 at 13:14 | comment | added | YCor | A related paper (from a deleted answer) is The homology of "k-equal'' manifolds and related partition lattices. Adv. Math. 110 (1995), no. 2, 277–313. sciencedirect.com/science/article/pii/S0001870885710122 It computes the homology of $M_{n,k}\subset\mathbf{R}^n$, defined by the condition that no $k$-uple of coordinates is constant, so it's a bit distinct, but of the same flavor. | |
| Oct 21, 2018 at 12:00 | comment | added | YCor | I guess $A(n,n-1)$ deformation retracts onto a graph but I don't have a proof now. | |
| Oct 21, 2018 at 11:33 | comment | added | YCor | $A(n,3)$ is homeomorphic to the product of $\mathbf{R}$ with ($\mathbf{R}^{n-1}$ minus $2^{n-1}-1$ lines through zero), which itself is homeomorphic to the product of $\mathbf{R}^2$ with ($\mathbf{S}^{n-2}$ minus $2^{n}-2$ points); $\mathbf{S}^{n-2}$ minus $2^{n}-2$ points is homeomorphic to $\mathbf{R}^{n-2}$ minus $2^{n}-3$ points for $n\ge 2$. The homology is not hard to compute: for $n\ge 3$ it's the homology of a bunch of $2^n-3$ $(n-3)$-spheres. | |
| Oct 21, 2018 at 10:59 | comment | added | YCor | $A(n,1)=\mathbf{R}^n$. $A(n,n)$ is homeomorphic to $\mathbf{R}^n\times\{1,\dots,n!\}$. As you say, $A(n,2)$ is homeomorphic to $\mathbf{R}^2\times\mathbf{S}^{n-2}$. | |
| Oct 21, 2018 at 10:53 | history | edited | YCor | CC BY-SA 4.0 |
deleted 7 characters in body; edited tags
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| Oct 21, 2018 at 10:33 | history | edited | user118503 | CC BY-SA 4.0 |
deleted 35 characters in body; edited title
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| Oct 21, 2018 at 10:23 | comment | added | user118503 | @YCor: Yes, indeed. Maybe it's better to ask a new question | |
| Oct 21, 2018 at 10:12 | comment | added | user118503 | @YCor: Thanks for the comment. I realized that I was thinking of tuples, but asking about finite sets. I will edit the question. | |
| Oct 21, 2018 at 9:55 | history | asked | user118503 | CC BY-SA 4.0 |