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Results tagged with gt.geometric-topology
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user 2349
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).
94
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1
answer
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The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to …
- 23.5k
61
votes
2
answers
3k
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The topological analog of flatness?
Recall that a map $f:X\to Y$ of schemes is called flat iff for any $x\in X$ the ring $O_{X,x}$ is a flat $O_{Y,f(x)}$-module.
Briefly the question is: what is the topological analog of this?
Many not …
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55
votes
3
answers
6k
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Kirby calculus and local moves
Every orientable 3-manifold can be obtained from the 3-sphere by doing surgery along a framed link. Kirby's theorem says that the surgery along two framed links gives homeomorphic manifolds if and onl …
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29
votes
5
answers
7k
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Proof of the Reidemeister theorem
While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can …
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25
votes
2
answers
3k
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Proofs of Kirby's theorem
Each orientable 3-manifold can be obtained by doing surgery along a framed link in the 3-sphere. Kirby's theorem says that two framed links give homeomorphic manifolds if and only if they are obtained …
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21
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3
answers
2k
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Cohomology of fibrations over the circle: how to compute the ring structure?
This question is inspired by Cohomology of fibrations over the circle Moreover, it can be considered a subquestion of the above, but somehow it seems to me that some of the more interesting points wer …
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19
votes
2
answers
887
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Lie algebra automorphisms and detecting knot orientation by Vassiliev invariants
Recall that there are knots in $\mathbf{R}^3$ that are not invertible, i.e. not isotopic to themselves with the orientation reversed. However, it is not easy to tell whether or not a given knot is inv …
- 23.5k
19
votes
7
answers
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CW-structures and Morse functions: a reference request
The following is probably well known, but I wasn't able to locate a reference in the literature.
Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a Rie …
- 23.5k
18
votes
1
answer
940
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Do chains and cochains know the same thing about the manifold?
This question was inspired by Poincaré quasi-isomorphism
Let $M$ be a closed oriented $n$-manifold. The cap product with the fundamental class of $M$ induces an isomorphism $H^i(M,\mathbf{Z})\to H_{n …
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18
votes
1
answer
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Fundamental groups of the spaces of rational functions
Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.
Let $X$ be a smooth complete complex curve (=a compact Riemann …
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16
votes
0
answers
324
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Rational equivalence of smooth closed manifolds
All spaces below will be assumed simply connected. A continuous map is a rational equivalence if it induces an isomorphism of the rational homology groups. Two spaces are rationally equivalent if they …
- 23.5k
15
votes
2
answers
967
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Infinity de Rham quasi-isomorphism
This question is similar to Do chains and cochains know the same thing about the manifold? in the sence that both deal with a natural "comparison" quasi-isomorphism that does not preserve the ring str …
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14
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4
answers
1k
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Mappings of mapping class groups
Let $X$ be a compact non-orientable surface, maybe with boundary, and let $\tilde X$ be the orienting cover of $X$. If I understand correctly, any smooth automorphism of $X$ lifts naturally to an auto …
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13
votes
3
answers
801
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Hyperelliptic loci in Teichmueller spaces
Let ${\cal M}_g$ be the moduli space of smooth complex genus $g$ curves, let ${\cal H}_g\subset {\cal M}_g$ be the hyperelliptic locus and set ${{\cal H}}'_g$ to be the preimage of ${\cal H}_g$ in the …
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13
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3
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Rational homotopy theory of a punctured manifold
Let $M$ be a smooth simply connected manifold and let $N$ be $M$ minus a point. Is it possible to construct an explicit Sullivan model for $N$ (i.e. a commutative differential graded algebra (cdga) wh …
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