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Results tagged with gt.geometric-topology
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user 23193
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).
12
votes
CW structures on unitary sphere of a banach/Hilbert space
The answer is no in infinite dimension (and yes in the finite dimensional case, of course). Because the sphere in a Banach space is Baire. But an infinite dimensional CW-complex is not, being a counta …
- 2,283
1
vote
Accepted
Homotopy of Unitary sphere in a Banach space and finite dimensional spheres
Well, it is well-known that both of $X$ and $Y$ are contractible (hence they are homotopy equivalent, this should answer your question). Notice that $Y$ is exactly the set of unit vectors of $E$.
To …
- 2,283
8
votes
Open immersions of open manifolds
This is the special case announced in the comment. We assume $M$ to be an oriented 4-manifold with the homotopy type of a 2-complex. We prove that $M$ immerses in $\mathbb{C}P^2$.
In the case of comp …
- 2,283
7
votes
Accepted
Can cobordisms of 3 or 4 manifolds be visualized by moves on kirby diagrams?
I think that a complete set of moves for cobordisms are the following: Kirby moves (that preserve the 4-manifold), handle trading (dotted circles become 0-framed 2-handles), addition/deletion of pairs …
- 2,283
3
votes
Accepted
Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface wi...
The answer is yes, as it has been remarked in the previous answers. Here is a very explicit construction.
Let $T$ be the sphere $S^2 \subset \Bbb R^3$ with three disks removed. Take these three disks …
- 2,283
0
votes
2
votes
Mapping class between coverings of Riemann surfaces
This is a simple exercise and the answer is no (cf. comments of Lee Mosher and Misha). Consider the two degree-4 coverings of the torus to itself with monodromies $\omega_1, \omega_2 : \pi_1(T^2) \to …
- 2,283
4
votes
How to specify a compact topological 4-manifold with a finite amount of data
In my understanding, I guess that the following strategy could be be attempted. Let $M$ be a closed connected orientable 4-manifold (while nonorientable 4-manifolds are doubly covered by orientable on …
- 2,283
20
votes
2
answers
1k
views
Manifolds with homeomorphic interiors
Suppose that two compact topological manifolds with boundary have homeomorphic interiors. Can we conclude that the two manifolds are homeomorphic? What happens in the smooth category?
- 2,283
8
votes
1
answer
380
views
Second homology of mapping class group of genus 3
In a survey paper of Korkmaz it is stated that $H_2(\mathrm{Mod}_3)$ is either $\Bbb Z$ or $\Bbb Z \oplus \Bbb Z_2$, but I was not able to find out a precise computation of this group (resolving the a …
- 2,283
2
votes
Algorithm for computing the Arf invariant of a knot
This is too long for a comment, it's just an idea for this computation. Start with a Seifert surface $F \subset \Bbb R^3$ for the knot $K$. Up to isotopy, we can assume that $F$ projects regularly to …
- 2,283
7
votes
The boundary of a domain whose interior is diffeomorphic to the ball
First, your assumption imply that $\bar D$ is a compact smooth manifold with boundary a topological sphere (because is a simply connected homology sphere). So, $\bar D$ is a topological $n$-ball, by t …
- 2,283
6
votes
0
answers
197
views
Surgering locally flat tori in 4-manifolds
Is there a locally flat torus in some not smoothable topological 4-manifold such that surgering on it produces a smoothable 4-manifold? Surgering means removing a tubular neighborhood and reattaching …
- 2,283
11
votes
Morse theory in TOP and PL categories?
For TOP Morse function a reference is the classical book of Kirby and Siebenmann "Foundational essays on topological manifolds, smoothings, and triangulations" (1977).
The key point is to consider the …
- 2,283
12
votes
1
answer
812
views
Handlebody decomposition of an open 4-manifold
Let $M$ be the fake $CP^2$ (namely the closed topological 4-manifold which is homotopy equivalent but not homeomorphic to the complex projective plan). It is well-known that $M$ admits no smooth struc …
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