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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).
7
votes
$\mathbb{CP}(2)$ from gluing boundary of 4-ball
There was a gap in my first answer to the first question, discussed in the comments. I hope it's clear that this doesn't constitute a full answer to OP's question, which remains interesting.
First que …
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4
votes
Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
In fact, if $C = \text{rank}(I_*(S^3_0(K)))$, we have $$f(k) = \text{rank}(I_*(S^3_{1/k}(K))) = kC.$$ This holds with any coefficient field.
Floer's exact sequence gives us an exact triangle relating …
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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 …
- 9,580
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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4
votes
Accepted
Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?
No, not yet. The state of the art in computing instanton homology for Seifert spaces is in this paper. This is like knowing how to compute $\widehat{HF}(\Sigma)$, which is not enough: you also want to …
- 9,580
10
votes
Accepted
Topology change induced by small perturbation
The formal statement you are thinking of when you assert "The topology changes only when..." is Ehresmann's theorem: a proper smooth submersion is a fiber bundle, and hence all fibers are diffeomorphi …
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5
votes
Dual surfaces of a first cohomology class of a 3-manifold
$L(4,1)$ is a counterexample to your conjecture, taking $\alpha$ to be the nontrivial element of $H^1(L;\Bbb Z/2)$. Notice that this element squares to zero (the square is the same as the Bockstein, a …
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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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10
votes
0
answers
182
views
A minimal $\mathbb Z/2$-invariant Morse function on $U(n)$
Consider the group $U(2n)$ of unitary matrices. This has two standard and important homomorphic involutions. The most famous one $A \mapsto \overline A$ is complex conjugation, with fixed-set $O(2n)$, …
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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 …
- 9,580
8
votes
Accepted
Nonvanishing section of infinite-dimensional tautological bundle
Let $X$ be any paracompact space. Then Hilbert vector bundles over $X$ are classified by homotopy classes of maps $[X, BU(\mathcal H)]$. But when $\mathcal H$ is infinite-dimensional, the group $U(\ma …
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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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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 …
21
votes
0
answers
770
views
Is the mapping class group of $\Bbb{CP}^n$ known?
In his paper "Concordance spaces, higher simple homotopy theory, and applications", Hatcher calcuates the smooth, PL, and topological mapping class groups of the $n$-torus $T^n$. This requires an unde …
- 9,580
8
votes
Accepted
Topological Spin manifolds in dimension 4
The map $\Omega_4^{\text{Spin}} \to \Omega_4^{\text{SpinTop}}$ is taken isomorphically by the signature to the inclusion $16\Bbb Z \hookrightarrow 8\Bbb Z$, so that the groups are abstractly isomorphi …
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