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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).
5
votes
Accepted
Can a 3-ball divide a standard 4-ball into two exotic 4-balls?
I think this is equivalent to the smooth Schoenflies conjecture; the executive summary is that this is true because smooth balls (in any dimension) are isotopic to standard ones.
Here are some detai …
- 19.4k
6
votes
Accepted
Extending homotopy equivalence between manifolds with boundary
The answer in general is no. Let $N_1$ be $S^2\times D^2$, and $N_2$ be the disk bundle of Euler class $1$ over $S^2$. Then both are homotopy equivalent to $S^2$. But no homotopy equivalence will ex …
- 19.4k
7
votes
Accepted
Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$
I was slightly confused by the wording of your question. I interpret it that you are asking first if an immersion into $R^n$ implies an embedding into $R^{n+1}$ (P1) and if so, if you get an embeddin …
- 19.4k
3
votes
Can cobordisms of 3 or 4 manifolds be visualized by moves on kirby diagrams?
I'm not sure what you mean by moves, but you can say something. As Marco says, a 5-dimensional cobordism can be decomposed into a series of handle additions, so you have to see what each handle additi …
- 19.4k
10
votes
Isotopy Invariants of 2-manifold
Well, the first invariant is the isotopy class of the boundary, so there's all of knot and link theory. If you fix the isotopy class of the boundary, then there are knots with inequivalent Seifert sur …
- 19.4k
10
votes
Accepted
A counter-example for the reversed direction of Casson-Gordon's theorem
Here's a particularly subtle counterexample, from the work of Kirk and Livingston (Topology Vol. 38, No. 3, pp. 663--671, 1999). They show that the pretzel knots $J = P(-3,5,7,2)$ and $K = P(5,-3,7,2) …
- 19.4k
8
votes
Accepted
Integral surgery on $S^2 \times S^1$
This is true, with some points to clarify. First, you are presumably talking about surgery along a knot that generates the first homology (and hence fundamental group) of $S^1\times S^2$. Then the res …
- 19.4k
6
votes
Rational homology cobordism invariants
There are other gauge theoretic obstructions to the existence of rational homology cobordisms. See eg Furuta (Invent. Math., 100 (1990), 339–355 Fintushel-Stern (Topology, 26 (1987), 385–393), Matic ( …
- 19.4k
7
votes
Accepted
Fickle's argument for Mazur manifolds
Call the manifold built in this way $W$. If you turn the handlebody for $W$ relative to $Y$ upside down, it becomes a handlebody with a single handle in indices $0$, $1$, and $2$. This is almost the s …
- 19.4k
10
votes
Accepted
Are the Morse inequalities sharp for 5-manifolds
Presumably you mean closed. Otherwise a non-trivial h-cobordism would not have the minimal number of critical points.
For closed simply connected manifolds, it seems to me that this is Corollary 2.2.2 …
- 19.4k
4
votes
Lefschetz duality for twist coefficient
Poincaré-Lefschetz duality for twisted coefficients is fundamental to surgery theory, and for the `universal' case of $Z[\pi_1(X)]$ coefficients is discussed in Chapter 2 of Wall's book, Surgery on Co …
- 19.4k
4
votes
Handle decompositions subordinate to an open cover
In an appendix to a paper of Tsuboi, On the uniform perfectness of the groups of diffeomorphisms of even-dimensional manifolds, you can find something rather close to what you are asking for. He build …
- 19.4k
7
votes
Accepted
Calculating Homology of the Boundary of a Handlebody
For $n>1$, manifold itself is determined by the linking numbers between the attaching spheres of your handles, and the framings of those handles. (For $n=1$, you have to take into account knotting and …
- 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
3
votes
manifolds with unusual rational cohomology rings
Doesn't Sullivan's paper On the intersection ring of compact three manifolds, Topology 1975; 14(3):275-277 characterize the cohomology ring? I think he shows that any skew tri-linear form is realized …
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