All Questions
Tagged with gn.general-topology dg.differential-geometry
127 questions
8
votes
4
answers
1k
views
Is a measurable homomorphism on a Lie group smooth?
Let $G$ be a Lie group, and let $\mathcal B(G)$ its Borel $\sigma$-algebra. Suppose that $f : G \to G$ is a Borel-measurable homomorphism. Is $f$ smooth?
Edit: My original question said "measurable ...
- 8,512
7
votes
2
answers
643
views
Proper maps and transversality
I'll begin with the question, which is intrinsically interesting:
Let M be a manifold with some submanifold Y. Suppose that $W \rightarrow M$ is a smooth, proper map. Does there exist another map $...
- 13.5k
2
votes
1
answer
407
views
Endomorphisms of degree d on a sphere with infinite fibers on a dense subset
Let $S^n$ be the sphere of dimension $n$. In order to construct a map $f:S^n\rightarrow S^n$ of degree $d\geq 2$ one has the following construction: Let $K$ be the complement of $d$
disjoint n-...
- 7,609
11
votes
1
answer
1k
views
Reference request for TQFT, functoriality
I am reading Turaev's blue book Quantum Invariants of Knots and 3-manifolds.
It is difficult for me to understand the proof of Theorem 1.9 in chapter 4, which says;
The function $(M, \partial_{-}M, \...
- 111
113
votes
4
answers
13k
views
Is there a sheaf theoretical characterization of a differentiable manifold?
I'm going through the crisis of being unhappy with the textbook definition of a differentiable manifold. I'm wondering whether there is a sheaf-theoretic approach which will make me happier. In a ...
- 22.1k
11
votes
2
answers
811
views
Higher dimensional Heegaard splittings?
Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing ...
- 732
6
votes
1
answer
1k
views
Some questions on Nicolai Reshetikhin's lectures on quantization of gauge theories.
This in reference to this fascinating lecture by Nicolai Reshetikhin-
http://staff.science.uva.nl/~nresheti/Holb-Quant-Gauge.pdf
Given what is said on page 13 in section 4.1 its not clear to me why ...
- 3,541
2
votes
0
answers
203
views
Faithful actions of finite groups on topological spaces
Suppose that $G$ is a finite group acting faithfully on a topological space $X$. In the smooth setting, one can deduce that for each $x$ in $M$, the induced map $$G_x \to Diff_x\left(M\right)$$ from ...
- 15.5k
5
votes
1
answer
294
views
Isotropic subspaces in cohomology
Hello,
Here is a problem I encountered in the study of Kähler manifolds but there is a natural generalisation of this for topological spaces.
If $X$ is a topological space, denote by $g_\mathbb{R}$ ...
- 339
17
votes
5
answers
2k
views
What abstract nonsense is necessary to say the word "submersion"?
This question is closely related to these two, but the former doesn't go far enough and the latter didn't attract much attention, and anyway I want to ask the question slightly differently.
Recall ...
- 54.6k
1
vote
1
answer
390
views
Isocontours of depth and magnitude of gradient
We are interested in characterizing a 2D surface $z(x,y)$, where $(x,y)$ is the regular 2D Cartesian grid. Let $\nabla z = (z_x, z_y)$ denote the gradient. The surface is a "general" one, that is, ...
- 13
2
votes
2
answers
657
views
Varieties, Frechet Completions, and Regular Functions
Take an algebraic variety $V$, and its set of smooth functions $C^{\infty}(V)$. One can endow $C^{\infty}(V)$ with a canonical locally convex topology (the seminorms are defined using the local ...
- 1,462
33
votes
4
answers
7k
views
Topology of function spaces?
Let $X,Y$ be finite-dimensional differentiable manifolds, and let's assume that they are connected. In fact, in applications I would like both $X$ and $Y$ to be riemannian manifolds.
Let $C^\infty(X,...
- 33.3k
14
votes
2
answers
1k
views
Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$?
Question: Is the tangent bundle of the long line $L$ homeomorphic to $L\times\mathbb R$? I'd guess that the answer doesn't depend on choice of differentiable structure, but maybe it does.
Motivation:...
- 787
16
votes
2
answers
2k
views
Compactification of a manifold
This is just a curiosity and the question is really foggy. I'm wondering if there can exist a notion of "minimal smooth compactification" (when I say minimal I think something like adding a finite ...
- 1,727
6
votes
2
answers
947
views
Compact cover of a Hausdorff compact space
In his book "Riemannian Geometry" do Carmo cites the Hopf-Rinow theorem in chapter 7. (theorem 2.8). One of the equivalences there deals with the cover of the manifold using nested sequence of compact ...
- 579
13
votes
11
answers
4k
views
Are nets and filters useful in geometry and topology?
Many results in topology can be restated using the concepts of nets and ultrafilters. This seems to be of interest for set theorists, maybe even logicians. But for geometers and topologists, those who ...
- 15.4k
16
votes
3
answers
8k
views
Defining Quotient Bundles
This is an extremely elementary question but I just can't seem to get things to work out. What I am looking for is a natural definition of the quotient bundle of a subbundle $E'\subset E$ of $\...
- 2,283
3
votes
0
answers
267
views
Maps of loop spaces with infinity-bounded differential.
I am currently working with loop spaces of manifold and finite dimensional manifolds approximating these and the following comes up very naturally:
In the following piece-wise smooth means smooth on ...
- 2,590
6
votes
2
answers
497
views
Can I detect the point of impact without looking at it?
I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...
- 26.8k
17
votes
1
answer
2k
views
Which Fréchet manifolds have a smooth partition of unity?
A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:
Which Fréchet manifolds have a smooth partition of unity?
How is the ...
- 4,519
7
votes
1
answer
3k
views
definition of the end of a manifold?
I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function assigning to each compact set K a conected component e(K) of the ...
- 71
4
votes
3
answers
1k
views
Morse theory and Euler characteristics
Suppose we have a space M with a real-valued, differentiable function F on M. Under what conditions on F will the Euler characteristic of M be expressed as a (signed) sum of Euler characteristics of ...
- 1,129
5
votes
5
answers
972
views
A walk on a compact 2D surface embedded in 3-space that never returns home
At the risk of asking an uninformed question...
Imagine an ant on a compact two-dimensional surface embedded in 3-space. The ant is placed at a point on the surface with random orientation. Once ...
- 811
31
votes
6
answers
6k
views
Least number of charts to describe a given manifold
Hello, I'm wondering if there is a standard reference discussing the least number of charts in an atlas of a given manifold required to describe it.
E.g. a circle requires at least two charts, and ...
- 2,901
8
votes
1
answer
688
views
Universal covers of domains in complex projective space
The Uniformization Theorem states that the universal cover of a Riemann surface is biholomorphic to the extended complex plane, the complex plane or the open unit disk. Each of these three is a domain ...
- 4,485
75
votes
3
answers
11k
views
Cohomology and fundamental classes
Let X be a real orientable compact differentiable manifold. Is the (co)homology of X generated by the fundamental classes of oriented subvarieties? And if not, what is known about the subgroup ...
- 14.7k