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Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube manifolds, theory of retracts).

9 votes
1 answer
540 views

cohomology of classifying space of permutation groups

Let $\Sigma_k$ be the permutation group of order $k$. Let $r: \Sigma_k\to GL(k)$ be the regular representation by permuting the order of the standard basis of $\mathbb{R}^n$. Let $\rho: B\Sigma_k\t …
Shiquan Ren's user avatar
  • 1,990
9 votes
1 answer
383 views

embedding of quaternionic projective spaces

Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding $$ \mathbb{H}P^2\longrightarrow \mathbb{R}^N? $$ Are there any ref …
Shiquan Ren's user avatar
  • 1,990
8 votes
2 answers
844 views

quotient space of Eilenberg-MacLane space

Let $\pi$ be a group and $K(\pi,1)$ the Eilenberg-MacLane space. Let $G$ be a finite group acting on $K(\pi,1)$ such that the following is a covering map $$ K(\pi,1)\longrightarrow K(\pi,1)/G. $$ Ques …
Shiquan Ren's user avatar
  • 1,990
7 votes
2 answers
720 views

contractible configuration spaces

Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$. My question: is $F(S^\infty,k)$ …
Shiquan Ren's user avatar
  • 1,990
7 votes
0 answers
419 views

kernel of the mod $2$ Bockstein on the first cohomology group

Let $M$ be a path-connected finite $CW$-complex. Suppose the first integral homology group is $H_1(M;\mathbb{Z})= \mathbb{Z}_2^{\oplus r}\oplus A$ where $r\geq 1$ and $A$ is a finite abelian group of …
Shiquan Ren's user avatar
  • 1,990
6 votes
3 answers
2k views

classifying space of orthogonal groups

Let $O(n)$ be the $n$-th orthogonal group and $O$ be the direct limit of $O(n)$ with respect to $n$. Let $BO(n)$ and $BO$ be the classifying spaces. Question: Why $BO$ is an $H$-space? My supervisor …
Shiquan Ren's user avatar
  • 1,990
6 votes
1 answer
183 views

non-orientability of vector bundles induced from a symmetric group action

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle $$ \xi:\mathbb{R}^k\longrightarrow M\times_{ …
Shiquan Ren's user avatar
  • 1,990
5 votes
4 answers
3k views

integral or rational cohomology of real grassmannians

I have obtained that the cohomology rings $$ H^*(G_k(\mathbb{R}^\infty);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]. $$ Also $$ H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar w …
Shiquan Ren's user avatar
  • 1,990
4 votes
2 answers
370 views

homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary $\{0\}\times(0,1)^ …
Shiquan Ren's user avatar
  • 1,990
3 votes
1 answer
364 views

cohomology ring of configuration spaces

In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use …
Shiquan Ren's user avatar
  • 1,990
3 votes
1 answer
313 views

self-Whitney sum of the canonical vector bundle on Grassmannians

Let $G_{k}(\mathbb{R}^N)$ be the Grassmannian manifold consisting of $k$-subspaces in $\mathbb{R}^N$. There is a canonical $k$-dimensional vector bundle $$ \gamma_{k,N}: \mathbb{R}^k\longrightarrow E …
Shiquan Ren's user avatar
  • 1,990
3 votes
1 answer
546 views

the "Kahn-Priddy map" and "multiplicative $p$-local equivalence"

The following is a part of a paper that I need to understand I totally do not know the argument. Could you explain? Thanks. Let $\Sigma_n$ be the $n$-th symmetric group and $\Sigma_\infty$ be the …
Shiquan Ren's user avatar
  • 1,990
3 votes
2 answers
185 views

equivariant embeddings from the k-th configuration space to the k+1-th configuration space

Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in Co …
Shiquan Ren's user avatar
  • 1,990
2 votes
1 answer
658 views

homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe: $\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) Mo …
Shiquan Ren's user avatar
  • 1,990
1 vote
0 answers
125 views

cohomology ring of compact submanifolds of Euclidean spaces

Suppose we have a compact $m$-dimensional submanifold $M$ of $\mathbb{R}^N$ and we want to know the cohomology ring $H^*(M;\mathbb{Z})$. Let $\epsilon>0$ and a $m$-dimensional finite simplicial com …
Shiquan Ren's user avatar
  • 1,990

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