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Metrically homogeneous spaces as inverse limits

Let $(X,d)$ be a locally compact, separable, connected and $\sigma$-compact metric space such that the group of isometries $G$ acts transitively on $X$. The question is the following: Is $X$ ...
user44172's user avatar
  • 541
3 votes
1 answer
178 views

Sheaves on solenoids

Let $(X_n)$ be a tower of finite covering maps of compact smooth manifolds, with $f_{s,t} : X_t\to X_s$ the maps, and $\Lambda_n := f_{n,0}^{-1}\Lambda$, with $\Lambda$ the constant abelian sheaf on $...
user avatar
2 votes
1 answer
185 views

effectively distinguishing knots

It was proven, I think by Mijatović EDIT: by Waldhausen, that there is an effective algorithm for distinguishing knot complements (the effective constants were found by Coward and Lackenby). The bound,...
Dmitry Vaintrob's user avatar
8 votes
1 answer
595 views

Question about taking the Zariski closure in $\mathbb{A}_{\mathbb{R}}^n$

Let $\mathbb{A}_{\mathbb{R}}^n$ be $\mathbb{R}^n$ endowed with the Zariski topology, where closed sets are algebraic sets (in $\mathbb{R}^n$) defined by real polynomials. Suppose $V \subseteq \mathbb{...
Johnny T.'s user avatar
  • 3,625
12 votes
2 answers
778 views

Topological obstructions to existence of immersion

Let $M$ be a smooth, non-compact manifold. a) Can one always find a smooth, compact manifold $N$ with $\dim(N) = \dim(M)$ and a smooth embedding $i: M \to N$ ? b) If not, are there some concrete ...
H1ghfiv3's user avatar
  • 1,255
1 vote
0 answers
112 views

Question regarding the image of a polynomial map containing a small box

I have the following question, which intuitively seems it should be true but I wasn't sure how to prove it rigorously. Let $\delta, \varepsilon > 0$. Let $\Psi_i(w_1, w_2, \mathbf{v})$ be a ...
Johnny T.'s user avatar
  • 3,625
7 votes
1 answer
545 views

Could we always find a line to intersect transversally with a given compact manifold?

This problem may be an embarrassing one, but I could not prove it even for the $1$ dimensional case. Here is the problem: Question 1. $M$ is a compact $n$-dimensional smooth manifold in $R^{n+1}$. ...
Hu xiyu's user avatar
  • 697
7 votes
0 answers
504 views

Intersection form of logarithmic transformations

Now I want to calculate the intersection form of a logarithmic transformation which is defined as follows. Let $X$ be an oriented, closed, simply-connected 4-manifold and $T^2\subset X$ be an ...
Hopf Fibration's user avatar
2 votes
0 answers
35 views

If the set of non-0 stalks of F is relatively open, is the same true of its Verdier dual?

Let $X$ be a complex manifold, $F$ a bounded complex of $\Bbb C_X$-modules with constructible cohomology. If the set $\{x: F_x\neq0\}$ is relatively open (i.e. open in its closure), is the same true ...
Avi Steiner's user avatar
  • 3,079
5 votes
1 answer
258 views

Generating the topology of a manifold

Let $X$ be a topological manifold of dimension $d$, and let $F$ be a collection of continuous maps from $X$ into $\mathbf{R}^d$ such that: $F$ separates points of $X$, i.e. for any two distinct ...
erz's user avatar
  • 5,529
67 votes
22 answers
10k views

When has discrete understanding preceded continuous?

From my limited perspective, it appears that the understanding of a mathematical phenomenon has usually been achieved, historically, in a continuous setting before it was fully explored in a discrete ...
5 votes
1 answer
192 views

Equivariant cohomology defined by restrictions?

Suppose that $G=S^1$ acts on a smooth, connected, compact manifold with discrete fixed points, additionally assume that there is at least one fixed point. Let $\alpha \in H^{2}_{S^1}(M)$ be such ...
Nick L's user avatar
  • 6,995
3 votes
2 answers
297 views

Inverse limit space and $C^1$ topology

Let $f_n$ be a sequence of $C^1$-maps on closed manifold $M$. If $f_n \to f$ in $C^1$-topology. Does $M_{f_n}$ converges to $M_f$ in the Hausdorff distance? We define $M(f)=\{\bar{x}=(x_j) \in M^{\...
Wagner Ranter's user avatar
49 votes
3 answers
8k views

Thurston's 24 questions: All settled?

Thurston's 1982 article on three-dimensional manifolds1 ends with $24$ "open questions":       $\cdots$ Two naive questions from an outsider: (1) Have all $24$ now been resolved? (2)...
Joseph O'Rourke's user avatar
2 votes
0 answers
171 views

Existence of compatible almost complex structure of symplectic fibration with nice property

Let $\pi: E \to B$ be a fibration over a closed surface $B$ with fier $g(F) \ge 2$. Suppose that both of $B$ and $F$ are closed surfaces and $g(B) \ge2$ and $g(F) \ge 2$. Fix a Kahler structure $(...
trick1234's user avatar
  • 185
2 votes
0 answers
305 views

Are homotopy equivalent manifolds with homeomorpic boundaries themselves homeomorphic?

Let $f:M \to M′$ be a homotopy equivalence of topological manifolds with boundary such that $dim(M)=dim(M′)$ and $f:\partial M \to \partial M′$ is a homeomorphism. Does this imply the existence of a ...
Dean Barber's user avatar
16 votes
3 answers
3k views

Physical interpretations/meanings of the notion of a sheaf?

I fairly understand the fiber bundles, both the mathematical concept of fiber bundles and the physics use of fiber bundles. Because the fiber bundles are tightly connected to the gauge field theory in ...
wonderich's user avatar
  • 10.5k
2 votes
2 answers
134 views

Are two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D_{M})$ homeomorphic?

What is an example of a compact manifold $M$ without boundary which does not satisfy the following property: For every $a\in M$, two pairs $(M\times M, M\times \{a\})$ and $(M\times M, D_{M})$...
Ali Taghavi's user avatar
3 votes
0 answers
83 views

Is the increasing union of disk bundles a disk bundle?

Setup: Let $B$ be a $C^r$ $n$-manifold ($r \geq 1$) and $M$ a closed $k$-dimensional $C^r$ submanifold of $B$. Assume there exists a smooth retraction $p:B \to M$ which is also a submersion, so that $...
Matthew Kvalheim's user avatar
7 votes
1 answer
692 views

Homotopically trivial vs isotopically trivial diffeomorphisms

Let $M$ be a manifold. Let's say $M$ is smooth, connected, oriented. We can also assume that $M$ is closed if that makes things easier. Let $\mathit{Diff}(M)$ denote the group of diffeomorphisms of $...
seub's user avatar
  • 1,347
4 votes
2 answers
619 views

Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
Shinichiro Nakamura's user avatar
3 votes
1 answer
669 views

Do Peano curves provide a counterargument to Grothendieck's critique?

This question arose in the context of an earlier question on Grothendieck's critique of the traditional foundations of topology. Can the paper Group Invariant Peano Curves by Cannon and Thurston be ...
Mikhail Katz's user avatar
  • 16.6k
23 votes
1 answer
2k views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...
Matthew Kvalheim's user avatar
1 vote
1 answer
248 views

Tightening a loop

Consider two $d$-dimensional convex polytopes $c_1, c_2$ that share a $(d-1)$-dimensional face $f$. Let $M$ be a surface ($2$-manifold) that intersects each of $c_1$ and $c_2$ in a $2$-ball. Suppose ...
Blake's user avatar
  • 109
18 votes
3 answers
3k views

What are parabolic bundles good for?

The question speaks for itself, but here is more details: Vector bundles are easy to motivate for students; they come up because one is trying to do "linear algebra on spaces". How does one motivate ...
Dr. Evil's user avatar
  • 2,751
3 votes
1 answer
270 views

Holonomic splitting

I am reading the book "Introduction to the h-Principle" by Eliashberg and Mishachev. At the moment I try to understand the Section 1.7 Holonomic splitting on page 12 but without success. I do not ...
Ben's user avatar
  • 35
13 votes
1 answer
1k views

How is Ricci flow related to computer graphics?

I recently came across the book Ricci Flow for Shape Analysis and Surface Registration: Theories, Algorithms and Applications by Wei Zeng and Xianfeng David Gu. Because, I just saw the book on the ...
terett's user avatar
  • 1,099
2 votes
3 answers
351 views

Classification of open subset of $\mathbb{R}^{3}$ [closed]

There is a theorem which gives a classification of connected open sets of $\mathbb{R}^{2}$. Unfortunately, I don't remember the correct statement, but it looks like this Theorem ? Let $U\subset\...
google's user avatar
  • 277
2 votes
1 answer
1k views

Is it true that given any two point in $M$ if there exists an unique geodesic joining those two points, then $M \sim \mathbb{R^n}$ [closed]

This following doubt initially came to my mind while thinking the relationship between number of genus of a manifold and number of geodesic between given two points. DOUBT: Suppose $M\subset \mathbb{...
Anubhav Mukherjee's user avatar
5 votes
0 answers
207 views

Can the compactification of a (co)tangent bundle equipped with Saski metric be viewed as a "Wick rotation"?

We can equip the (co)tangent bundle of a Riemannian manifold (B,g) with a Saski metric $\hat{g}$ (see, for example, "On the geometry of tangent bundles" by Gudmunssun & Kappos) that looks like \...
harry's user avatar
  • 51
7 votes
1 answer
226 views

Are maps homotopic with respect to a uniform number of local homotopies

I've encountered the following problem that I'm sure someone more topologically inclined can answer: Say that a homotopy of maps $f:X\times[0,1)\to Y$ between two compact smooth manifolds $X$ and $Y$ ...
Jess Boling's user avatar
5 votes
2 answers
609 views

Exponential rule for Whitney-$\mathcal{C}^{\infty}$-topology

Let $M,N,X$ be smooth manifolds. Equip the space of smooth functions between two manifolds with the (strong) Whitney- $\mathcal{C}^\infty$-topology. The evaluation map $$ev\colon N\times\mathcal{C}^\...
Kathrin L.'s user avatar
2 votes
1 answer
1k views

Is the space of smooth maps $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology locally compact, if $M$ is compact

The title says it all: Let $M$ be a compact manifold and $N$ a (possible non compact) manifold. Equip the space of smooth functions $C^{\infty}(M,N)$ with the Whitney $C^{\infty}$ topology. (The ...
Kathrin L.'s user avatar
-3 votes
1 answer
330 views

Loop space of manifold [closed]

Question A: The free loop space of a manifold is also a manifold? Question B: The free loop space of an algebraic variety is also a algebraic variety ? Are these questions asked or answered anywhere ...
MyIsmail's user avatar
  • 189
3 votes
2 answers
459 views

Smooth paths on affine varieties

I have the following question which is in some way related to an application of Randell Isotopy Theorem to complex hyperplane arrangements. Let $h,k\geq1$ be integer numbers and let $F_{1},\ldots,F_{...
snaleimath's user avatar
5 votes
1 answer
744 views

Intersections of open balls in manifolds

This question is motivated by the post Uncountable intersections of open balls in a separable metric space. The general problem is the following: given a connected Riemannian manifold $M$, what are ...
coudy's user avatar
  • 18.7k
3 votes
2 answers
517 views

Connected complement manifold

I'm working on some problem in algebraic geometry. I need a reference to the following result: Let $h\in\mathbb{N}$ with $h\geq1$ and let $F\in\mathbb{C}\left[x_{1},\ldots,x_{h}\right]$ be a non ...
snaleimath's user avatar
1 vote
1 answer
370 views

Shrinking $\mathcal{C}_c^{\infty}(M)$ to obtain a first countable space

This is a follow-up to this question. Let $M$ be manifold (Hausdorff, countable base, finite dimensional, if it simplifies anything embedded in $\mathbb{R}^n$). I'm interested in the topological ...
Kathrin L.'s user avatar
8 votes
2 answers
2k views

Relating different topologies on $C^{\infty}_c(M)$

This is somehow connected to this question. I can think of at least four topologies to put on $C_c(M)$: Topologize $C^{\infty}_c(M)\subseteq C^{\infty}(M)$ as a subspace with the weak Whitney $C^\...
Kathrin L.'s user avatar
9 votes
2 answers
1k views

Why not develop a Hamiltonian-based Morse theory?

I have begun to learn the basics of Morse theory and Floer homology. I understand that Floer homology is the natural theory for symplectic manifolds, but from my preliminary knowledge of Morse theory ...
Tom's user avatar
  • 91
0 votes
1 answer
284 views

Creating topological spaces with portals [closed]

I'm trying to rigorously describe an object that I'm calling a "portal". The situation is easiest to describe in two dimension. I start with a line segment $pq$ in $\mathbb{R}^2$. I want to remove ...
user61430's user avatar
2 votes
0 answers
96 views

Branch point and alexandrov embeddedness

This is a question I have asked on mathstackexchange with a bounty but without any answer; it is probably more adapted to mathoverflow: Let us assume that $\Sigma_n$ is a sequence of topological ...
Paul's user avatar
  • 914
3 votes
3 answers
351 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 ...
Alex M.'s user avatar
  • 5,407
2 votes
1 answer
391 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 http://www.unige.ch/~...
A random mathematician's user avatar
8 votes
3 answers
1k 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 Books ...
Joseph O'Rourke's user avatar
4 votes
2 answers
414 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 ...
Ritwik's user avatar
  • 3,245
12 votes
0 answers
460 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 $S^2 \times M_2$?
Murat Saglam's user avatar
3 votes
1 answer
1k 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 $C^\nu(...
Dave's user avatar
  • 281
0 votes
0 answers
148 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 ...
jiangsaiyin's user avatar
15 votes
1 answer
2k 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, ...
Edward Hughes's user avatar