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Results tagged with gt.geometric-topology
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user 40804
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).
10
votes
Accepted
Fundamental group of the space of immersions of the 2-sphere in 3-space modulo diffeomorphis...
The action of $\text{Diff}^+(S^2)$ on $\text{Imm}(S^2,\Bbb R^3)$ is free. For pick any immersion $i$ and diffeomorphism $f$; given $x \in \Bbb R^3$ $i^{-1}(x)$ is a closed discrete (because $i$ is an …
- 9,580
14
votes
0
answers
336
views
Are there Alexander-Whitney maps in geometric homology?
When $X$ is a smooth manifold, Lipyanskiy defines a chain complex whose homology is isomorphic to singular homology -
let's say $GC_*(X)$ - generated by maps $\sigma: M \to X$ from compact $k$-manif …
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6
votes
Accepted
smooth homotopy 4-balls with sphere boundary in dimension 4
Yes, you can perform that ambient isotopy: any oriented embedding $i: B^n \to M^n$ is isotopic to any other. (This is a lemma proven independently by Cerf and Palais1, but the idea is quite clear: shr …
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5
votes
Accepted
Space of simple closed curves in $S^2$
Yes; I find this easier than the whole Morse theory package needed for $LS^2$.
Thm. The projection map $\text{Emb}(S^1, S^2) \to (TS^2 \setminus 0)$, given by $\gamma \mapsto \gamma'(0)$, is a weak ho …
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4
votes
Accepted
Surface separating the boundary of a cylinder
I am going to focus on the oriented case here. A similar argument should hold in the unoriented case but the degree argument ought to use twisted coefficients, which I don't want to go through here.
W …
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7
votes
Singular homology of spin manifolds
There is only one constraint beyond those coming from Poincare duality, the one mentioned in Aleksandar's comment. So in total, we have:
The top homology is $H_n(M;\Bbb Z) = \Bbb Z$.
The second-to-to …
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12
votes
Accepted
Existence of normal microbundles
Not all locally flat submanifolds have a normal microbundle, but they do stably.
Rourke-Sanderson prove that there is a PL embedding $S^{19} \times I \to S^{29}$ with no topological normal microbundle …
4
votes
Homology spheres bounding homology balls but not embedding into $S^4$
The first examples (infinitely many of them) are given in Clayton McDonald's recent note. In particular, the double branched cover of the knot depicted in Figure 1 bounds a homology ball but does not …
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41
votes
Accepted
Exotic $R^4$ as the universal covering space
This is problem 4.79(A) of Kirby's 1995 problem list, contributed by Gompf. It is my impression that it is still open.
There is some small progress: Remark 7.2 in this article observes that their co …
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7
votes
Accepted
What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?
The explicit calculation of this Heegaard-Floer homology (at least, $HF^-$, and of the unique $\text{Spin}^c$ structure for which $c_1(\mathfrak s) = 0$) was carried out in the paper in which it was i …
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6
votes
Accepted
Possible orders of automorphisms for the Poincare homology sphere
Let $G$ act smoothly and orientably. The quotient $M/G$ is a spherical 3-orbifold. Therefore, by the elliptization theorem, it may be given a metric of constant curvature 1; pulling back, then $M$ is …
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8
votes
Accepted
Fixed point set of smooth circle action
Given two $n$-manifolds $M_i$ with a circle action and a choice of free orbit $\gamma_i$ we can define the fiber connected sum $(M_1, \gamma_1) \# (M_2, \gamma_2) = (M_1 \# M_2, \gamma)$ by taking a t …
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5
votes
1
answer
825
views
Why does the Gluck twist on a spun knot give the standard $S^4$?
Given a knotted arc $A \subset D^3$ (whose endpoints are, say, at $(\pm 1,0,0)$), the spun knot on this arc is $$\partial\left((D^3, A) \times D^2\right), = (\partial(D^3,A) \times D^2) \cup ((D^3,A) …
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16
votes
0
answers
427
views
Survey of known results on equivariant transversality
Most basic differential topology theorems carry over to the equivariant case with mild modifications; see for instance Wasserman's paper. One thing that fails (more or less obviously) is equivariant t …
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3
votes
Accepted
Is there an analogy of Austin-Braam approach to Bott-Morse type Hamiltonian Floer homology?
First, Austin and Braam did already apply their machine to a Floer-type theory: it was just instanton homology and not Hamiltonian Floer homology here. Their machine uses $\Bbb R$ coefficients, but yo …
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