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Results tagged with gt.geometric-topology
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user 17913
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
0
answers
173
views
BKS pairing in the SU(2) Chern-Simons theory
I know that usually, the way to compare the Hilbert spaces arising from $SU(2)$ Chern-Simons theory with different Kähler polarizations is via the Hitchin connection. However, it should be possible, I …
- 1,025
7
votes
1
answer
748
views
SL(2,C) Chern-Simons theory in genus 1
In Link, Witten claims (p. 54) that to quantize the moduli space of flat $SL(2,\mathbb{C})$ connections on a torus, one can simply quantize the cotangent bundle of a real torus and take the part invar …
- 1,025
3
votes
0
answers
172
views
More questions about high-dimensional knot invariants
In a question yesterday I asked about the existence of algebraic invariants for embeddings of n-manifolds into n+2-spheres. The answers in the positive dimension all made certain assumptions about hom …
- 1,025
5
votes
0
answers
246
views
Ribbon knot presentations
Suppose $K$ is a $n$-knot, $n\geq 2$, which bounds two different ribbon disks $D_1, D_2$. These ribbon disks induce unique ribbon $(n+1)$-knots $K_1, K_2$ respectively. Is it known whether $K_1$ and $ …
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1
vote
1
answer
2k
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Homology and homotopy type for knot complements
I'm trying to understand a paper by Kawauchi and Matumoto, 'An estimate of infinite cyclic coverings and knot theory.' In one of their proofs they have a manifold $E$ which is the complement of a codi …
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1
vote
0
answers
437
views
Theta functions and Fourier transforms
Let $T_\tau$ be the 2-dimensional torus, with the complex structure induced by the lattice generated by $1$ and $\tau$. Then for a line bundle $L_k$ over $T$ with level $k$, there is an orthonormal ba …
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8
votes
2
answers
515
views
High-dimensional ribbon knots
Let us suppose that we have a ribbon embedding $S^n \rightarrow S^{n+2}$ for $n\geq 3$. Call this knot $K$. By a theorem of Levine (and Trotter for $n=3$ I believe) we know $K$ is unknotted if the com …
- 1,025
3
votes
High-dimensional ribbon knots
I found a paper which discusses this issue:
Ribbon knots and ribbon disks from Asano, Marumoto, and Yanagawa. They establish that for $n\geq 3$ a ribbon knot with infinite cyclic fundamental group is …
- 1,025
3
votes
2
answers
347
views
2-tangles and quantum groups and 2-groups
Turaev developed the notion of a quantum group by considering the category of tangles (thought of with objects as collections of 2$n$ points and with morphisms being braids between them with cups and …
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6
votes
1
answer
3k
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Connections on line bundles over the torus
If I understand correctly, every line bundle $L$ over the (2-dim) torus can be obtained from a quotient of $\mathbb{R}^2 \times \mathbb{C}$ by a $\mathbb{Z}^2$ lattice action. Different line bundles a …
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1
vote
0
answers
137
views
pairing theta functions for different complex structures
I apologize for my previous attempt to ask this, which was very badly written.
Let us start with $\mathbb{C}\times\mathbb{C}$. To form an Hermitian line bundle over a complex torus with complex struc …
- 1,025
10
votes
3
answers
835
views
Invariants of high-dimensional knots
In the study of knots in three dimensions, it can be shown that the fundamental group together with a specification of a meridian and longitude form a complete invariant for knots. What is known about …
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-4
votes
1
answer
464
views
Symplectic forms and 1-forms [closed]
Suppose we have a real symplectic manifold $(M,\omega)$. Under what conditions can we find a global 1-form $\alpha$ such that $\omega = \alpha \wedge\alpha$?
Obviously there are some simple obstructio …
- 1,025
11
votes
5
answers
1k
views
Symplectic structures from Lagrangians?
In Witten's paper 'Quantization of Chern-Simons Gauge Theory with Complex Gauge Group,' he makes the statement (p. 35) that the symplectic form on the moduli space of flat $G_\mathbb{C}$ connections o …
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