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Results tagged with gt.geometric-topology
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user 41075
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).
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
$$
H^*(G_k(\mathbb{R}^m);\mathbb{Z}_2)=\mathbb{Z}_2[w_1,\cdots,w_k]/(\bar w …
- 838
0
votes
0
answers
84
views
packing numbers of the unit balls in Euclidean spaces and the dimensions
Let $k$, $m$ and $n$ be positive integers. Let $r$ be a positive real number.
The $n$-th ordered $r$-disk configuration space on the Euclidean space $\mathbb{R}^{mk}$ is
$$
F_r(\mathbb{R}^{mk} …
- 1,990
1
vote
1
answer
237
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 Cartesia …
- 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 …
- 15.5k
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 …
- 1,990
1
vote
1
answer
108
views
vector bundles induced by an action of a finite subgroup of $O(n)$
Let $M$ be a path-connected manifold. Let $G$ be a finite subgroup in $O(n)$ and suppose $G$ acts freely on $M$. Then we have an associated vector bundle
$$
\xi(M,G): \mathbb{R}^n\longrightarrow M\tim …
CommunityBot
- 1
6
votes
1
answer
183
views
non-orientability of vector bundles induced from a symmetric group action
Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $M$ be a manifold with a free $\Sigma_k$-action. Then we can form a $k$-dimensional vector bundle
$$
\xi:\mathbb{R}^k\longrightarrow M\times_{ …
- 1,990
1
vote
0
answers
101
views
coefficient of homology of configuration spaces over real projective spaces
In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an arb …
- 1,990
0
votes
1
answer
328
views
group actions on fibre bundles
Suppose that we have a group $G$ acting on the spaces $E$ and $B$. Suppose moreover that we have fibre bundles $\xi$ and $\eta$ in the following commutative diagram
If $\xi$ is a trivial bundle, i.e …
- 18.2k
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 …
- 118k
1
vote
1
answer
303
views
isotopy equivalence (topological meaning) between $CW$-complexes
Let $M$ and $N$ be $CW$-complexes.
Definition. (different from the isotopy notion in geometry of submanifolds). A (topological) isotopy is a fibre-wise continuous map
$$
F: M\times [0,1]\longright …
- 15.2k
4
votes
2
answers
370
views
homotopy equivalence between configuration spaces
Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary $\{0\}\times(0,1)^ …
- 6,813
9
votes
1
answer
383
views
embedding of quaternionic projective spaces
Let $\mathbb{H}P^m$ be the $m$-th quaternionic projective space. What is the smallest integer $N$ such that there exists an embedding
$$
\mathbb{H}P^2\longrightarrow \mathbb{R}^N?
$$
Are there any ref …
- 666
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 …
- 1,990
1
vote
1
answer
297
views
torsion part of the cohomology module of configuration spaces of manifolds
Let $M$ be a manifold and $C_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational homol …
- 35.9k