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Results tagged with gt.geometric-topology
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user 14233
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).
18
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1
answer
1k
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Simply connected finite CW-complex with only finitely many nontrivial homotopy and homology ...
Let $X$ be a simply connected finite CW-complex such that all but finitely many of its homotopy groups and its homology groups (with $\mathbb Z$ coefficients) are 0.
Is $X$ then necessarily contracti …
- 11.9k
7
votes
2
answers
781
views
Automorphism of genus 2 surface with 5 fixed points
Is there a self-homeomorphism of a genus 2 (closed, orientable) surface, which has finite order and exactly 5 fixed points?
Of course, the same question can be asked replacing 2 by $g$ and $5$ by any …
- 11.9k
7
votes
3
answers
326
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Cyclic groups acting on balls, and interior fixed points
Let a finite cyclic group $G = \mathbb Z/n$ act continuously on an open $d$-ball $B^d$. Suppose further that this action extends to the closed ball $\overline{B^d}$. Is there necessarily a fixed point …
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6
votes
3
answers
657
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Smooth circle action, $\chi(M^{S^1}) = \chi(M)$
I need a reference for the following result (which I can prove myself, but my proof is rather ugly and I would prefer to just cite the statement instead or re-proving it):
Let $M$ be a closed smooth …
- 11.9k
9
votes
1
answer
226
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Is Homeo($M,D^n$) torsion-free?
Let $M$ is a connected smooth $n$-manifold, and suppose that $f$ is a self-homeomorphism of $M$ that has finite order, i.e. $f^k = \text{id}$ for some $k\geq 1$. Suppose moreover that $f$ fixes a non- …
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13
votes
4
answers
616
views
Is $\mathrm{Diff}_0(S_g)$ torsion-free?
Let $S_g$ be a closed oriented smooth surface of genus $g>1$, and let us consider $\text{Diff}_0(S_g)$, the identity component of the diffeomorphism group of orientation preserving diffeomorphisms of …
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25
votes
1
answer
2k
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Is every degree 1 self-map a homotopy equivalence?
In a rather obscure article, I found (without proof) the following statement:
If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence.
Is this rea …
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9
votes
1
answer
4k
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Universal covering of compact surfaces
Is there any elementary (i.e. without using analytical methods like the theory of Riemann surfaces or more elaborate results from differential geometry) way to show that the universal covering of the …
- 11.9k
19
votes
1
answer
786
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Which cohomology classes are detected by tori?
Given a space $X$, I am looking for a characterization of classes $\alpha \in H^n(X;\bf Q)$ such that there is a map $f\colon T^n \to X$ so that $f^{\ast} \alpha$ pairs non-trivially against the funda …
- 11.9k
6
votes
2
answers
318
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Almost free actions on simply-connected spaces
Let $G$ be a simply-connected compact topological group (you can think of $SU(n)$ if you like it more concrete), and let $X$ be a finite-dimensional simply-connected $G$-CW-complex. If we know that al …
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2
votes
3
answers
2k
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Minimal genus of Seifert surface of torus knot
Let $(p,q)$ be a pair of coprime (positive) integers. Consider the torus knot $T_{p,q}$. What is the minimal genus of an (embedded) oriented Seifert surface for this knot?
It is not had to convince o …
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8
votes
0
answers
262
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$\mathbb RP^n$ bundles over the circle, II
EDIT: I fixed the issue pointed out by Nicholas Tholozan, thanks for sheding light on this!
This question is written as a follow-up to this one.
Both answers there are great, but my impression is ther …
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8
votes
2
answers
349
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Configuration space like subspace of sphere product
For $k \geq 2, n \geq 1$ let $$M^{n,k} = \{(x_1,\dots,x_k) \in S^n \times \dots \times S^n \ | \ x_1 + \dots + x_k = 0\}$$ This is a compact CW-complex and almost, but not quite, a manifold. I general …
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7
votes
1
answer
429
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Z/p action on finite contractible complex
Let $p$ be a prime and $X$ a finite contractible CW-complex. Assume $\mathbb Z/p$ acts on $X$. Then it is easy to see that there has to be a fixed point. (E.g. use Lefschetz's fixed point theorem or g …
- 11.9k
13
votes
1
answer
552
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Realizing symmetric groups by diffeomorphisms
Let $M$ be a (closed, smooth) manifold of dimension $d$. For $n$ a positive integer, fix $n$ points $x_1, \dots, x_n \in M$. The group of diffeomorphisms of $M$ that permutes the points $x_i$ surjects …
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