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Results tagged with gt.geometric-topology
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user 7460
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).
58
votes
Accepted
Example of 4-manifold with $\pi_1=\mathbb Q$
Since any compact manifold has the homotopy type of a finite CW-complex (see this MathOverflow question: Are non-PL manifolds CW-complexes?) and $\mathbb{Q}$ is not finitely presented, the manifold $ …
- 66.3k
51
votes
Accepted
Do finite groups acting on a ball have a fixed point?
The answer is no.
A fixed point free action of the finite group $A_5$ on a $n$-cell was constructed by Floyd and Richardson in their paper An action of a finite group on an n-cell without stationary p …
- 66.3k
24
votes
Is there anything special about the Riemann surface $y^2 = x(x^{10}+11x^5-1)$?
Your curve is hyperelliptic.
If $X_g$ is a hyperelliptic curve of genus $g$, then $\textrm{Aut}(X_g)$ is a central extension of degree $2$ of one of the groups $$\mathbb{Z}_n, D_n, A_4, S_4, A_5,$$
se …
- 66.3k
18
votes
When the automorphism group of an object determines the object
The following result holds.
Theorem.
(1) $\,$ (Baer-Kaplanski) $\,$ If $G$ and $H$ are torsion groups with isomorphic endomorphism rings $\mathrm{End}(G)$ and $\mathrm{End}(H)$, then $G$ and $H$ are …
17
votes
Accepted
What is an example of an orbifold which is not a topological manifold?
It is quite easy to give an example in real dimension $4$.
In fact, it was shown by D. Mumford in the paper
The topology of normal singularities of an algebraic surface and a criterion for simplic …
- 66.3k
16
votes
Can every 3-manifold be triangulated?
Every $3$-manifold is triangulable.
This was proven by Edwin E. Moise in is paper "Affine structure in $3$-manifolds", Annals of Math. 56 (1952).
- 66.3k
10
votes
Line bundle on $S^2$
In general, there is a bijection $$\Phi \colon [S^{k-1}, \textrm{GL}_n^+(\mathbb{R})] \to \textrm{Vect}_+^n(S^k),$$
where $[X,Y ]$ denotes the set of homotopy classes of continous maps from $X$ to $Y$ …
- 66.3k
10
votes
Accepted
Are there any tests for knowing whether a topological space admits a CW structure?
Every compact topological manifold $M$ has the homotopy type of CW-complex. So, in the compact case, there is no such invariant of the type you are looking for (or, at least, such an invariant cannot …
- 66.3k
9
votes
Accepted
Example of a triangulable topological manifold which does not admit a PL structure
The answer is yes, see Rudyak's paper Piecewise linear structures on topological manifolds, Examples 21.4:
There are topological manifolds that can be triangulated as simplicial complexes but do …
- 66.3k
6
votes
Accepted
How small need a perturbation be to not change the diffeomorphism type of a variety?
Let me prove $(1)$.
First of all, I guess that $f, \, g$ are homogeneous polynomials of the same degree $d$, otherwise $Z(f+ \varepsilon g)$ is not well-defined as a subvariety of $\mathbb{RP}^k$.
Tha …
- 66.3k
4
votes
Automorphisms of Riemann surface and mapping class
Yes, this is an old result due to Hurwitz, and it is often used in Teichmuller theory.
It is cited, for instance, at p. 152 of this paper by P. Lochak. However, I do not know the original reference. …
- 66.3k
3
votes
Which topological spaces contain dense simply connected subspace?
Any real semialgebraic set $X \subset \mathbb{R}^N$ has a dense, open subset that is a submanifold: just take the complement of its singular set. In fact, the singular set is Zariski closed in $X$, he …
- 66.3k
3
votes
How to prove that $\phi: \;\mathrm Mod(S_g)\to \mathrm Sp(2g, \mathbb{Z})$ is an epimorphism?
This is well-known material, so insted of a "detailed answer" let me give you a standard reference. See
B. Farb and D. Margalit: A primer on mapping class groups, Theorem 6.4.
- 66.3k
3
votes
Topology of a smoothing of an isolated singularity
Edit. This does not answer the question, but it just provides examples of homotopically non-trivial smoothings $Y$ in every dimension $n \geq 1$. Perhaps someone might find it useful, hence I will not …
- 66.3k
2
votes
Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface wi...
The answer is actually yes, and here is an example with $n=6$ (inspired by Broughton's classification quoted by Danny and by Jim's answer).
Take the group $G = \mathbb{Z}/ 6 \mathbb{Z}$ presented as …
- 66.3k