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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).
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
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
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
28
votes
Accepted
Is the space of diffeomorphisms homotopy equivalent to a CW-complex?
Here is an example where ${\rm Diff}(M)$ with the compact-open topology is not homotopy equivalent to a CW complex. Take $M$ to be a surface of infinite genus, say the simplest one with just one nonco …
- 20k
28
votes
Accepted
Unique smooth structure on 3-manifolds
An alternative to Moise's paper for the existence and uniqueness of piecewise linear (PL) structures on topological 3-manifolds is the paper "The triangulation of 3-manifolds" by A.J.S. Hamilton in Qu …
- 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
22
votes
CW-structures and Morse functions: a reference request
The result you are looking for is Theorem 4.18 in "An Introduction to Morse Theory" by Yukio Matsumoto, published by AMS in 2002 (translated from Japanese). The connections between Morse functions, ha …
- 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
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
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
15
votes
Accepted
Mapping class groups of small Seifert-fibred 3-manifolds
The determination of mapping class groups of small Seifert manifolds was completed by M. Boileau and J.-P. Otal in a paper in Invent. Math. 106 (1991), 85-107. They give references for cases previous …
- 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
14
votes
Accepted
Mapping class group of certain 3-manifolds
Since you write ${\rm Diff}_+(M)$ you are probably assuming $M$ is orientable and diffeomorphisms of $M$ are orientation-preserving. Every diffeomorphism of $M$ can be isotoped to take fibers to fiber …
- 20k
13
votes
Accepted
About the proof of Wajnryb's finite presentation of Mod(S)
It is still an open problem to find a short and simple way to derive a finite presentation for the mapping class group. The book by Farb and Margalit (in the recent preliminary version 4.00) gives a c …
- 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