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Results tagged with gt.geometric-topology
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user 1465
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).
15
votes
1
answer
514
views
Where is the Steenrod Realization problem at?
I'm wondering if there is a more modern reference out there for the Steenrod Realization Problem than the book of Connor and Floyd?
Realizing homology classes in a manifold via embedded submanifolds, …
- 44.4k
9
votes
1
answer
374
views
Mapping class groups are finitely generated
Let $N$ be a compact smooth manifold. By "mapping class group" I will mean
$$\pi_0 \operatorname{Diff}(N)$$
i.e. the isotopy-classes of diffeomorphisms of $N$.
My presumption is that this mapping cla …
- 44.4k
5
votes
3
answers
234
views
First usage of the terms pseudo-isotopy and concordance in manifold theory
I am hoping I can use the collective knowledge of the forum to piece together some history. I'm wondering where the terms pseudo-isotopy and concordance originated, in their modern forms as used in ma …
- 44.4k
3
votes
0
answers
134
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Isotopy-classes of oriented Schoenflies spheres in $S^4$ is a group under oriented connect-sum
Given two oriented, smoothly-embedded copies of $S^3$ in $S^4$ (called Schoenflies spheres), one can take an oriented connect-sum of the pairs $(S^4, M_1) \# (S^4, M_2)$. This puts a monoidal structu …
- 44.4k
5
votes
0
answers
133
views
Smoothing tame topological knots, from an analytic perspective
A tame topological knot (for the purpose of this question) is a topological embedding of $S^1 \times D^2$ into $\Bbb R^3$.
Tame topological knots are known to be isotopic to smooth knots. This questi …
- 44.4k
12
votes
1
answer
639
views
Revisiting Gordon-Luecke theorem
$\DeclareMathOperator\Diff{Diff}\DeclareMathOperator\GL{GL}$Here is an proof-sketch of a strengthened Gordon-Luecke theorem. This is presumably known, but is it written down somewhere? I am also cur …
- 44.4k
25
votes
2
answers
841
views
Which homotopy classes $S^3 \to S^2$ lift to embeddings $S^3 \to S^2 \times D^3$?
The question is, for a smooth embedding
$$f : S^3 \to S^2 \times D^3$$
one can compose the map $f$ with projection $\pi : S^2 \times D^3 \to S^2$, giving the map $\pi \circ f : S^3 \to S^2$.
Which hom …
- 44.4k
8
votes
0
answers
310
views
Smale's relative h-cobordism theorem
In Smale's On the structure of manifolds paper there is his relative version of the h-cobordism theorem, specifically Theorem 3.1 (and 1.4). Roughly speaking this concerns the situation where one has …
- 44.4k
8
votes
0
answers
173
views
Stratification of space of labelled circles in the plane
Consider the space of $n$ round circles in the plane to be the open subset of $\mathbb R^{3n}$:
$$C_n = \{ (v_1, v_2, \cdots, v_n, r_1, r_2, \cdots, r_n ) : v_i \in \mathbb R^2, r_i \in (0, \infty) \ …
- 44.4k
14
votes
0
answers
500
views
Structure of Gordian graph of knots
The Gordian graph of knots has the knot isotopy classes as it's vertices, and an edge whenever you can pass from one knot to a other via a "finger move", equivalently if for some diagram of the knot y …
- 44.4k
10
votes
0
answers
127
views
Compatibility of spherical and hyperbolic geometry for fibred knots
Hyperbolic knots and links have a lovely peculiarity that you can always find a position for them in $S^3$ making two groups the same, one defined using the spherical geometry of $S^3$, and the other …
- 44.4k
7
votes
1
answer
399
views
High dimensional generalized Poincare hypothesis without the h-cobordism theorem?
The generalized PL Poincare hypothesis states that in dimension $n$ there is a unique PL manifold that has the homotopy-type of $S^n$. It's known to be true in all dimensions except perhaps $n=4$.
…
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5
votes
1
answer
1k
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The cyclic branched covers of "simple" knots in $S^3$
Is there a convenient place in the literature where the geometric decompositions of cyclic branched covers of $S^3$ branched over "small" knots is recorded?
By small knots, I'm referring to things l …
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13
votes
1
answer
1k
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Mapping class groups of small Seifert-fibred 3-manifolds
Are computations of the mapping class groups of small Seifert-fibred 3-manifolds recorded in some convenient location?
For most Seifert manifolds working out the mapping class group is easy-enough ( …
- 44.4k
11
votes
2
answers
531
views
Simplicial replacements in smoothing theory
As far as I can tell, ever since Milnor's Microbundles and differentiable structures (1961) paper, whenever people talk about $Diff(\mathbb R^n)$ or $PL(\mathbb R^n)$ or $Homeo(\mathbb R^n)$, they alw …
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