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This might be something well-known.

For $1\le k\le n$, let $A(n,k)\subset\mathbb{R}^{n}$ be the set of points $x=(x_{1},...,x_{n}% )$ with at least $k$ distinct coordinates. Then what are the homologies of $A(n,k)$?

For example, $A(n,2)$ is equal to $\mathbb{R}^{n}\backslash\Delta$, where $\Delta$ is the diagonal, thus it is homotopy equivalent to $\mathbb{S}^{n-2}$.

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  • $\begingroup$ @YCor: Thanks for the comment. I realized that I was thinking of tuples, but asking about finite sets. I will edit the question. $\endgroup$
    – user118503
    Commented Oct 21, 2018 at 10:12
  • $\begingroup$ @YCor: Yes, indeed. Maybe it's better to ask a new question $\endgroup$
    – user118503
    Commented Oct 21, 2018 at 10:23
  • $\begingroup$ $A(n,1)=\mathbf{R}^n$. $A(n,n)$ is homeomorphic to $\mathbf{R}^n\times\{1,\dots,n!\}$. As you say, $A(n,2)$ is homeomorphic to $\mathbf{R}^2\times\mathbf{S}^{n-2}$. $\endgroup$
    – YCor
    Commented Oct 21, 2018 at 10:59
  • $\begingroup$ $A(n,3)$ 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. $\endgroup$
    – YCor
    Commented Oct 21, 2018 at 11:33
  • $\begingroup$ I guess $A(n,n-1)$ deformation retracts onto a graph but I don't have a proof now. $\endgroup$
    – YCor
    Commented Oct 21, 2018 at 12:00

1 Answer 1

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Let me try again after deleting an answer based on a misunderstanding of the question. (Thanks to @YCor for pointing this out.)

Edited thanks to observations of Dan Petersen.

Your space $A(n,k)$ is obtained from ${\mathbb R}^n$ by deleting a bunch of subspaces. Thus the theorem of Goresky and McPherson dealing with complements of unions of subspaces applies. This says that

$$\widetilde{H}^d(A(n,k)) \cong \bigoplus_X \widetilde{H}_{codim(X)-2-d}(\Delta(0,X)).$$

Here $X$ runs through the set of intersections of collections deleted subspaces, which is partially ordered by reverse inclusion, $\Delta(0,X)$ is the order complex of the interval $(0,X)$ ($0$ is the ambient space ${\mathbb R}^n$, which is not included in the sum), and $codim(X)$ is the usual codimension of the subspace $X$ in ${\mathbb R}^n$.

The intersection poset for this union of subspaces is isomorphic with the poset of set partitions of $\{1,\ldots,n\}$ having less than $k$ parts, ordered by reverse inclusion. As this is a rank selected subposet of the lattice of all set partitions, it is a Cohen-Macaulay poset. This means that the homology of $\Delta(0,X)$ is concentrated in degree $rank(X)-2$. On the other hand, if $X$ has $j$ parts, then $X$ has rank $k-j$ (in the intersection post) and codimension $n-j$.

It follows now from direct computation that the reduced homology of your space is concentrated in degree $n-k$. As Dan Petersen pointed out to me, if $k \leq n-2$, then $A(n,k)$ is simply connected, and it follows from the Hurewicz Theorem that $A(n,k)$ is homotopy equivalent to a wedge of spheres of dimension $n-k$.

Calculating the number of such spheres involves determining the M\"obius function on the intersection poset, which might or might not be tough.

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