# Tagged Questions

**2**

votes

**3**answers

179 views

### Diffeomorphism with prescribed behaviour

If $\gamma$ and $\eta$ are two smooth curves in a smooth manifold $M$, is it possible to find a diffeomorphism of $M$ such that $f \circ \gamma = \eta$? What if one removes the assumption of ...

**1**

vote

**1**answer

213 views

### (n-1)-dimensional normal currents and Smirnov's paper

I don't know much about currents, but I saw a paper of Smirnov which seems relevant to a problem I am working on. In the very last paragraph of page 848 of the following paper
...

**7**

votes

**3**answers

776 views

### Is the list of “known” 3D compact manifolds complete?

"it is an open question if the known compact manifolds in 3-D are complete."
This is a quote from Eric Weisstein's
CRC Concise Encyclopedia of Mathematics, Second Edition. 2010, p.480.
(Google ...

**0**

votes

**0**answers

43 views

### tensors invariant under irregular flow

Suppose $M$ is a compact smooth manifold, and let $R$ be a nowhere-vanishing vector field. Then in the irregular scenario, the closure of $R$-orbits is a $r$-torus $T^r$.
Now suppose there is a ...

**4**

votes

**2**answers

261 views

### Is it impossible for the dimension of a topological space to increase under a smooth map?

First let me make a definition. Let $M$ be a smooth manifold and
$S \subset M $ a topological subspace of $M$. We say that $S$ has
"dimenion" at most $k$ if $S$ is a subset of
$$ X_1 \cup X_2 \ldots ...

**9**

votes

**0**answers

277 views

### 3 manifolds with diffeomorphic unit tangent bundles

What can one say about two closed oriented 3-manifolds $M_1$ and $M_2$ such that $S^2 \times M_1$ is diffeomorphic to $S^2 \times M_2$?

**2**

votes

**1**answer

332 views

### Is $C^\nu(X,Y)$ a Banach manifold and a Lindelöf space?

Suppose that $X$ is a compact, finite dimensional manifold and $Y$ is an infinite dimensional, second countable ($C^\infty$-)Banach manifold. Let $\nu \in \mathbb{N}$.
Question: Is the space ...

**0**

votes

**0**answers

126 views

### Only finitely many fundamental groups in $M(n,k,v,D)$?

Let $M(n,k,v,D)$ denote the class of compact manifolds with $Ric \ge \left( {n - 1} \right)k,vol \ge v,diam \le D$.In 1990,M.Anderson proved that "There are only finitely many fundamental groups among ...

**7**

votes

**1**answer

312 views

### How unique is a conformal compactification?

I'm trying to understand the term "conformal compactification" which is often used in physics. I reckon that most places take this to mean a (sometimes specific) compact conformal completion. That is, ...

**5**

votes

**4**answers

682 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 ...

**6**

votes

**2**answers

332 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 ...

**2**

votes

**1**answer

292 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 ...

**8**

votes

**0**answers

274 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, ...

**57**

votes

**5**answers

3k 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 ...

**6**

votes

**2**answers

298 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 ...

**4**

votes

**1**answer

607 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 ...

**0**

votes

**0**answers

165 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 ...

**5**

votes

**1**answer

243 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 $g_\mathbb{R}$ ...

**13**

votes

**5**answers

854 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 ...

**1**

vote

**1**answer

216 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, ...

**2**

votes

**2**answers

557 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 ...

**18**

votes

**3**answers

2k 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 ...

**6**

votes

**2**answers

625 views

### Tangent bundle of the long line

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.
...

**6**

votes

**2**answers

820 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 ...

**6**

votes

**2**answers

648 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 ...

**9**

votes

**10**answers

2k 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 ...

**5**

votes

**3**answers

1k 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 ...

**3**

votes

**0**answers

227 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 ...

**6**

votes

**2**answers

420 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 ...

**7**

votes

**2**answers

526 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 ...

**2**

votes

**1**answer

904 views

### definition of the end of a manifold?

Hey everybody, I was hoping if somebody could help me out with the terminology. I've found that the "end of a manifold" is a function asigning to each compact set K a conected component e(K) of the ...

**3**

votes

**3**answers

801 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 ...

**4**

votes

**5**answers

925 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 ...

**16**

votes

**6**answers

1k 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 ...

**7**

votes

**1**answer

351 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 ...

**37**

votes

**2**answers

3k 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 ...