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age 25
visits member for 2 years, 11 months
seen Dec 14 at 17:39

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?).


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
Aug
13
revised Doubt about local flatness of low dimensional embeddings
added 94 characters in body
Aug
13
asked Doubt about local flatness of low dimensional embeddings
Aug
12
comment Question on “up to isotopy” when attaching two spaces
Do you know if there is any version of this theorem for complex manifolds?
Aug
8
asked Existence of orientation-reversing automorphisms
Aug
7
comment Is there any difference between a flat manifold and an affine space?
If you want to do affine combinations on differentiable manifold, then you have to require that the local charts are such that the change of charts are affine maps. This is what is called an affine manifold (see the wiki page for detailed discussion). Without this additional structure, the only "affine" aspects of a differentiable manifold is the lack of a preferred point, I guess.
Aug
7
comment Is there any difference between a flat manifold and an affine space?
If we agree that the main feature of an affine space is the availability of affine combinations, then I would say no. I agree that on a Riemannian manifold you can pick a chart around a point and perform affine combinations, but what if you pick a different chart around the same point? then the affine combination will not be preserved, because the general change of variable is not going to be an affine map. Tu put it in another way, affine combinations are not intrinsecally/well defined on a general Riemannian manifold.