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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).
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
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
2
votes
Books for learning branched coverings
You can have a look at Makoto Namba's book "Branched coverings and algebraic functions".
- 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
0
votes
How would a topologist explain "every Riemann surface of genus $g$ is hyperelliptic if and o...
The fact that every curve of genus $2$ is hyperelliptic comes from the fact that the canonical $g_2^1$ induces a hyperelliptic involution.
If $g \geq 3$, the general curve is not hyperelliptic. In fac …
- 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
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
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
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
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
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
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
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
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