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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
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 $ …
Blake's user avatar
  • 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 …
Blake's user avatar
  • 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 …
Blake's user avatar
  • 1,025
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 …
Blake's user avatar
  • 1,025
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 …
Blake's user avatar
  • 1,025
1 vote
1 answer
2k views

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 …
Blake's user avatar
  • 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 …
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  • 1,025
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 …
Blake's user avatar
  • 1,025
-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 …
Blake's user avatar
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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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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 …
Blake's user avatar
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6 votes
1 answer
3k views

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 …
Blake's user avatar
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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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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 …
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