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Results tagged with gt.geometric-topology
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user 3460
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).
27
votes
Accepted
How can gauge theory techniques be useful to study when topological manifolds can be triangu...
The very short answer is that there is no direct connection between gauge theory (which is living on some perhaps hypothetical smooth 4-manifold) and triangulation of some high-dimensional topological …
- 19.4k
27
votes
Accepted
Is every degree 1 self-map a homotopy equivalence?
I believe that this is an open question in general, and the assertion is an old conjecture of Hopf. Some special cases were considered by Jean-Claude Hausmann, Geometric Hopfian and non-Hopfian situat …
- 19.4k
23
votes
Accepted
Manifold embedded in $R^{n+1}$ with a submanifold that doesn't embed in $R^n$
Here's another way to get examples, in codimension one and in low dimensions. There are lots of oriented closed 3-manifolds that don't embed in 4-space, for example any 3-manifold $M$ with $H_1(M) \co …
- 19.4k
23
votes
Accepted
What can we say about the Cartesian product of a manifold with its exotic copy?
Your question seems to be about simply connected exotic 4-manifolds, for which the answer is yes. That's because $M$ and $M^E$ are h-cobordant (by Wall), say via an h-cobordism W. Then $M \times W$ is …
- 19.4k
23
votes
Accepted
$S^3$ as cyclic branched cover of itself
The statement that for arbitrary K in $S^3$, if for some $n \ge 2$, the n-fold cyclic branched cover is $S^3$ (or in some versions, a homotopy 3-sphere) then K is the unknot, was known as the Smith co …
- 19.4k
22
votes
Very particular kind of 4-manifolds. Classification
I would say no. If M is simply connected, then it is contractible and hence determined topologically by its boundary. But there's no current smooth classification; the case when the boundary is $S^3$ …
- 19.4k
21
votes
Accepted
Acyclic Finite Groups
An acyclic finite group is trivial. In fact something even stronger is true. See Culler, Marc Homology equivalent finite groups are isomorphic. Proc. Amer. Math. Soc. 72 (1978), no. 1, 218–220.
- 19.4k
20
votes
Accepted
Is the normal bundle of a torus trivial?
You should be able to prove that the normal bundles in codimension $2$ are trivial as well. This is a little harder than codimension $1$; you need to know that such bundles are determined by their Eul …
- 19.4k
20
votes
Accepted
Homology spheres and fundamental group
See edit at bottom for further information answering the question in all dimensions.
In all odd dimensions $2k -1 > 3$, there are non-homeomorphic homology spheres with fundamental group G = the bina …
- 19.4k
18
votes
Accepted
Thom conjecture in CP3
This question was addressed in higher dimensions by Mike Freedman, Surgery on codimension 2 submanifolds. Mem. Amer. Math. Soc. 12 (1977), no. 191. (I think this was his PhD thesis.) Earlier work of T …
- 19.4k
17
votes
Accepted
Fibered example of topologically slice knots
Such a knot would yield a counterexample to one of two important conjectures in the area. A preliminary definition: a slice knot is homotopically ribbon if the inclusion of the knot into the slice di …
- 19.4k
17
votes
Accepted
Handlebody decomposition of an open 4-manifold
There are not that many explicit handlebody pictures of exotic open 4-manifolds, because they get awfully complex in short order. The ones that I know of are in work of Žarko Bižaca from the mid-90's …
- 19.4k
16
votes
Accepted
Can a homology $n-1$-sphere divide $\mathbb{S}^{n}$ into non-contractible components?
Yes; this is not hard to arrange. For instance, take any knot K in the 3-sphere with an n-fold branched cover, say M that is a homology sphere. A good example would be to take M to be the $k$-fold co …
- 19.4k
16
votes
Accepted
Existence of Morse functions on simply connected manifolds
It is still an open and very interesting question in dimension 4. Akbulut (The Dolgachev surface. Disproving the Harer-Kas-Kirby conjecture. Comment. Math. Helv. 87 (2012), no. 1, 187–241) showed tha …
- 19.4k
15
votes
Accepted
Characteristic class that cannot be represented by disjoint tori
In $H_2(CP^2)$, every class $nH$ where $H$ is a generator and n is odd is characteristic. However, if $n >3$, then such a class is not represented by a torus. It is not represented by a disjoint unio …
- 19.4k