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Results tagged with gt.geometric-topology
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user 23571
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).
2
votes
Accepted
Maps of balls with fixed value along boundary
This set of homotopy classes is in bijective correspondence with $\pi_3(M)$. More generally, let $[B^k,X;f]$ be the set of homotopy classes of maps $B^k\to X$ that restrict to a given $f:S^{k-1}\to X$ …
- 20k
30
votes
Accepted
Fibers of fibrations of a 3-manifold over $S^1$
There are simple examples with $M = F \times S^1$ for $F$ a closed surface of genus $2$ or more. Choose a nonseparating simple closed curve $C$ in $F$, then take $n$ fibers $F_1,\cdots,F_n$ of $F\time …
- 20k
10
votes
One question on cup product and torsion elements
Here's an example that's a 2-dimensional CW complex. Start with a 0-cell, then attach three 1-cells labeled $a$, $b$, $c$ to get a wedge of three circles, then attach a 2-cell via the word $aba^{-1}b^ …
- 20k
8
votes
distance formula in Farey graph?
The distance from $1/0$ to $p/q$ in the Farey diagram is some function of the continued fraction for $p/q$, which determines the strip of triangles in the diagram joining $1/0$ and $p/q$. If all the t …
- 20k
7
votes
Commutator subgroup of $\Gamma(2).$
The commutator subgroup of $SL(2,\mathbb{Z})$ has index 12 and $\Gamma(2)$ has index 6, but $\Gamma(2)$ does not contain the commutator subgroup since the quotient group $SL(2,\mathbb{Z})/\Gamma(2)$ i …
- 20k
32
votes
Accepted
What characteristic class information comes from the 2-torsion of $H^*(BSO(n);Z)$?
The basic fact is that the 2-torsion all has order exactly 2, so it injects into the mod 2 cohomology, forming a subalgebra of the polynomial algebra on the Stiefel-Whitney classes. This subalgebra c …
- 20k
15
votes
Accepted
Homotopy type of set of self homotopy-equivalences of a surface
A couple comments. For the result about diffeomorphism groups there is a very nice alternative proof due to A. Gramain in the Annales Scient. E.N.S. v.6 (1973), pp. 53-66, that uses no analysis, just …
- 20k
51
votes
Accepted
Triangulating surfaces
[Three years later …]
All the published proofs of triangulability of surfaces that I am aware of use the Schoenflies theorem, which is not exactly an easy thing to prove. There is however another line …
- 20k
22
votes
Accepted
Detecting homotopy nontriviality of an element in a torsion homotopy group
How about thinking about framed cobordism, which in this case gives an isomorphism between $\pi_4(S^3)$ and the group of cobordism classes of normally framed 1-manifolds in $S^4$. Since your map is c …
- 20k
14
votes
Periodic mapping classes of the genus two orientable surface
In the paper listed below there is a calculation of all the finite group actions on a genus 2 surface. There are 20 of them, with the groups ranging from order 2 to order 48. Nine of the actions are …
- 20k
10
votes
Accepted
Handle decompositions using only 1-handles
The second statement ought to be in the literature somewhere but I don't know a reference so I'll give an argument.
The result can be rephrased in terms of graphs. Let $S$ be a compact connected sur …
- 20k
2
votes
Accepted
Dehn-Nielsen-Baer Theorem for surfaces with boundary and punctures
The issue here is Dehn twists along curves parallel to the circles of $\partial S$. These usually generate infinite cyclic subgroups of ${\rm Mod}(S,Q)$, the only exceptions being when $S$ is a disk …
- 20k
15
votes
Satellite knot example
If by the symmetry group of a knot you mean the group of isometries of $S^3$ leaving the knot invariant, then this can only be cyclic or dihedral, apart from the special case of torus knots which can …
- 20k
16
votes
Topological $n$-manifolds have the homotopy type of $n$-dimensional CW-complexes
Every topological manifold has a handlebody structure except in dimension 4, where a 4-manifold has a handlebody structure if and only if it is smoothable. This is a theorem on page 136 of Freedman a …
- 20k
13
votes
Accepted
Two solid N_3 glued by its boundary
It is a general fact that a closed manifold of odd Euler characteristic cannot bound a compact manifold. This can be deduced pretty easily from the fact that a closed manifold of odd dimension has Eul …
- 20k