All Questions
Tagged with symmetric-groups at.algebraic-topology
9 questions
15
votes
1
answer
629
views
Characteristic classes of symmetric group $S_4$
For the symmetric group $S_3$, it is classically known that \begin{equation} H^*(S_3;\mathbb{Z})\cong \mathbb{Z}[x,y]/(2x,6y,x^2-3y), \end{equation} where $|x|=2$ and $|y|=4$. Moreover, $x$ can be ...
- 439
13
votes
2
answers
556
views
Stabilization of representation of the symmetric group
In their seminal work on "representation stability" (https://arxiv.org/abs/1008.1368), Church and Farb deal with a stabilization procedure for representations (up to isomorphism) (over $\mathbf C$) of ...
- 2,521
7
votes
1
answer
272
views
Homotopy type of the semi-simplicial set of symmetric groups
Consider the collection of symmetric groups $\{\Sigma_n\}_{n\geq1}$ as a semi-simplicial set (i.e. a simplicial set without degeneracies) as follows. Consider $i\in\{1,\dots,n+1\}$ and $\pi\in\Sigma_{...
- 517
6
votes
0
answers
136
views
Second homotopy group of the symmetric power of a space
Let $X$ be a finite CW complex, $n \ge 2$, and $\Sigma_n$ be the permutation group on $n$ symbols. Let $X^{(n)}=X^n/\Sigma_n$ be the quotient of the natural action of $\Sigma_n$ on $X^n$. We call $X^{(...
- 311
5
votes
1
answer
419
views
canonical action of symmetric groups on orthogonal groups
There is a canonical faithful orthogonal representation of the symmetric group $S_{n+1}$, for $n\geq 1$:
$$
S_{n+1}\to O(n)
$$
given as follows.
(1). I regard $O(n)$ as the isometry group of the unit ...
- 543
4
votes
1
answer
759
views
cohomology ring of symmetric group of order $3$
Let $S_3$ be the symmetric group of order $3$. What is the cohomology ring
$$
H^*(S_3;\mathbb{Z})?$$
My attempt: I want to use mathematical induction on $n$ for $S_n$.
For $n=1$, $S_1$ is trivial. ...
- 1,990
3
votes
1
answer
341
views
Covering with Deck group $\mathfrak{S}_3$
I am looking for the easiest possible example of a connected covering $X\to X/\mathfrak{S}_3$ ($\mathfrak{S}_3$ the third symmetric group). More precisely, I want $X$ and $X/\mathfrak{S}_3$ to be ...
- 1,623
1
vote
1
answer
240
views
free group actions on a contractible topological space [closed]
Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $W$ be a contractible topological space with a free $\Sigma_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the ...
- 1,990
1
vote
0
answers
77
views
Singular chain complex of balanced products
Let $\pi\subseteq\Sigma_r$ and $V$ be a right $\pi$-space. We may assume that $V$ is free, if necessary. Consider the morphism of singular chain complexes (over a fixed commutative ring)
$$f:C_*(V) \...
- 1,623