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I'm interested in the great amount of mathematics involved in the study of low-dimensional manifolds, above all the interaction between topology, geometry and algebra (and physics?).


17h
answered Dehn twist as isometries on hyperbolic surface
2d
accepted Commutator of hyperbolic isometries
Jan
28
asked Commutator of hyperbolic isometries
Jan
27
awarded  Civic Duty
Jan
7
accepted Complex structures on punctured disks.
Dec
30
awarded  Yearling
Dec
12
awarded  Notable Question
Nov
20
asked Divisor of meromorphic section of point bundle over a Riemann surface
Nov
19
comment Thin triangles vs Slim triangles in hyperbolic spaces
They are slightly different concepts, but turn out to encode the same notions of Gromov hyperbolicity, in the sense that: every geodesic triangle is $\delta$-thin iff every geodesic triangle is $\varepsilon$-slim, possibly with different constants. A great reference is Bridson, Haefliger, Metric spaces of non-positive curvature; have a look at def 1.1, 1.16 and Prop 1.17 in Chapter III.H
Nov
5
comment Heegaard splitting and mapping class group
In the smooth category this is a quite general result: gluing manifolds along isotopic diffeomorphisms gives diffeomorphic manifolds. Have a look at Hirsch, Differential Topology, chapter 8 sections 1 and 2, especially theorem 2.3
Oct
6
revised Branched coverings of unit disk
added hypothesis needed to use Riemann-Hurwitz formula
Oct
6
comment Branched coverings of unit disk
Sorry for the improper use of Riemann-Hurwitz. I probably want to consider covering of the closed unit disk which give unbranched covers of the boundary. I will edit the question accordingly.
Sep
26
comment Étalé space for sheaf of sections of a fiber bundle
Thanks for the nice references. I'm interested in the case $F$ is discrete. It looks like the equivalence of categories you talk about restricts to a correspondence between locally constant sheaves and fiber bundles with discrete fibers. I mean that the étalé space of a locally constant sheaf with stalks $A$ really looks like a fiber bundle with fiber $A$, if we put on $A$ the discrete topology. Does this work?
Sep
24
awarded  Autobiographer
Sep
23
asked Branched coverings of unit disk
Sep
3
awarded  Inquisitive
Sep
2
revised Étalé space for sheaf of sections of a fiber bundle
edited title
Sep
2
asked Étalé space for sheaf of sections of a fiber bundle
Aug
21
comment Doubt about local flatness of low dimensional embeddings
Ok this should follow from Thm 13 in chapter 10: if a subset of $\mathbb{R}^2$ is abstractly triangulable, then there is a homeomorphism $f$ of the plane taking it to a polyhedron of $\mathbb{R}^2$, which is of course locally flat. Thanks for the reference.
Aug
19
awarded  Popular Question