**5**

votes

**0**answers

48 views

### When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request.
In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...

**7**

votes

**1**answer

123 views

### Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$

Let $M$ be a connected noncompact parallelisable smooth $n$-manifold. (By Hirsch-Smale theory, $M$ can be immersed into $\mathbb{R}^n$.) I am interested in the following two properties:
P1: $M$ can ...

**0**

votes

**0**answers

47 views

### Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...

**5**

votes

**1**answer

222 views

### Finite group acting on sphere

Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?

**3**

votes

**1**answer

104 views

### In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In
Meyer, Werner
Die Signatur von ...

**7**

votes

**1**answer

375 views

### Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form.
My question is as ...

**1**

vote

**0**answers

47 views

### Rotation orientations in n-dimensions [migrated]

I'm doing a change of variables that involves doing simple rotations on the standard basis vectors in R^n, and I'm wondering what the standard orientations are in n dimensions are and why. For ...

**29**

votes

**11**answers

3k views

### Compelling evidence that two basepoints are better than one

This question is inspired by an answer of Tim Porter.
Ronnie Brown pioneered a framework for homotopy theory in which one may consider multiple basepoints. These ideas are accessibly presented in his ...

**5**

votes

**2**answers

86 views

### On trivalent spines of surfaces

Let $\Sigma$ be a surface with non-empty boundary labelled by finitely many points $x_i$.
For our purposes a spine $s$ for $\Sigma$ is a uni-trivalent graph embedded in $\Sigma$ whose 1-vertices are ...

**1**

vote

**0**answers

123 views

### Does there exist any subspace of R^n, homeomorhic to a manifold but not a C^0 submanifold of R^n?

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential ...

**8**

votes

**2**answers

326 views

### Covering the space by disjoint unit circles

Sierpinski has proved the following two interesting theorems.
Theorem 1. The Euclidean plane $\mathbb{R}^2$ is not a union of nondegenerate disjoint circles.
Theorem 2. The Euclidean space ...

**0**

votes

**1**answer

158 views

### A problem related to connectivity of analytic functions

Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$.
Question: Is the connectivity of ...

**0**

votes

**1**answer

264 views

### Is it possible to classify the boundaries of a manifold?

The open ball is a manifold, and the closed ball is a compact manifold-with-boundary which extends the open ball, in which the open ball is dense, in which all the new points are boundary points. Is ...

**4**

votes

**1**answer

184 views

### Example of collapsible complex which collapses to a non-collapsible complex

In their paper http://arxiv.org/abs/0907.2954 , Barmak and Minian claim that
"there exist collapsible complexes which collapse to nontrivial subcomplexes with no free faces"
but unfortunately do ...

**10**

votes

**1**answer

224 views

### Are the mapping class groups of manifolds finitely presentable?

The mapping class group of a manifold is the group $\pi_0 Diff(M)$ of components of the diffeomorphism group. There are several variations: oriented manifolds and orientation preserving ...

**12**

votes

**2**answers

496 views

### Low dimensional topological manifolds

There is a well known result that every one dimensional topological manifold without boundary is homeomorphic either to the circle or to the whole real line. However there is one detail hidden: ...

**3**

votes

**1**answer

104 views

### Comparing Contact Structures: What do we Mean when we Say that two Contact Structures are Homotopic/Eliashbergs Class. of OT structures

Please forgive me if this is too simple for MO; most of my posts on anything contact-structure-related in Math Stack, other sites, have barely received answers (maybe because I'm not an expert in the ...

**3**

votes

**0**answers

150 views

### Transversality in Bourgeois Oancea's non-equivariant contact homology

In their paper, "AN EXACT SEQUENCE FOR CONTACT- AND SYMPLECTIC HOMOLOGY", Bourgeois and Oancea defined, whenever there is sufficient transversality, a "non-equivariant contact homology". Essentially, ...

**7**

votes

**1**answer

115 views

### Hakenness of Heegard splitting

This is somewhat related to this previous quesiton. Suppose I give you a Heegard splitting of $M^3$ of genus $g$ with a gluing map $\phi.$ Is there some condition on $\phi$ which would guarantee that ...

**6**

votes

**0**answers

141 views

### Linked circles in R3

Two circles in 3-D are linked iff each one passes through the interior of the other.
There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique ...

**2**

votes

**1**answer

141 views

### Surface curves equidistant from a simple closed geodesic

Let $S \subset \mathbb{R}^3$ be a surface embedded in $\mathbb{R}^3$,
let's say (to keep it simple) of genus zero.
Let $\gamma$ be a simple, closed, oriented geodesic on $S$.
Because $\gamma$ is ...

**4**

votes

**1**answer

118 views

### Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question:
If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...

**8**

votes

**1**answer

249 views

### Must a closed totally path-disconnected subset of the sphere have connected complement?

This question (which is more a curiosity than a research problem) originates from these two:
...

**6**

votes

**1**answer

473 views

### How can gauge theory techniques be useful to study when topological manifolds can be triangulated?

I was reading a review article arXiv:1310.7644 and it was explained there that in the last few years it was proven that there are topological manifolds of dimension greater than four that cannot be ...

**10**

votes

**1**answer

771 views

### Question about Thurston's paper “A norm for the homology of 3-manifolds”

I have a question about the proof of Theorem 5 in Thurston's paper "A norm for the homology of 3-manifolds". It's the theorem that asserts that the fibered faces are, well, fibered. More precisely, ...

**7**

votes

**2**answers

541 views

### Glue two solenoids along their boundaries

Here a solenoid is a dynamical system $(N,f)$ where $N$ is the solid torus $N=\mathbb{D}^2\times S^1$ with boundary $S^1\times S^1$, and $f:N\to N$ is a smooth embedding whose image is wrapped twice ...

**0**

votes

**0**answers

54 views

### Suitable references for Zappa-Szep products of groups

Can anyone provide references for the following: isoperimetric (Dehn) functions of semi-direct products of groups and, more generally, of Zappa-Szep products of groups?

**0**

votes

**0**answers

29 views

### definition of an avatar of a tree-automorphism group

Would anyone be able to show me where I can read about what the avatar of a tree-automorphism group is? This is so that I can understand Pierre-Emmanuel Caprace's article "Simple locally compact ...

**31**

votes

**1**answer

4k views

### Open map D⁴ → S²

Is it possible to construct an embedding $D^4\hookrightarrow S^2\times
\mathbb R^2$
such that the projection $D^4\to S^2$ is an open map?
Here $D^n$ denotes closed $n$-ball.
An open map D⁴ → S².
It ...

**2**

votes

**0**answers

98 views

### when is an irreducible SL_2(C) representation of a cusped hyperbolic 3-manifold scheme reduced or smooth

Let $M$ be an orientable cusped hyperbolic 3-manifold. Let $\rho \in hom(\pi_1(M),SL_2(\mathbb{C}))$ be an irreducible representation.
Is $\rho$ scheme reduced ?
What can prevent $\rho$ from being ...

**2**

votes

**0**answers

122 views

### Canonical trivialization of Stiefel-Whitney classes on the frame bundle

Let $X$ be a smooth manifold, perhaps oriented if necessary. The frame bundle $\pi:P \to X$ carries a canonical trivialization of the pullback of the tangent bundle of $X$ and thus a canonical ...

**2**

votes

**0**answers

103 views

### Cylinder isotopy relative to its bases

Is n-times twisted cylinder isotopic to 0-times twisted cylinder relative to its bases? (I hope I got the terms right)
That is, is it possible to twist (stretching, bending etc. is allowed) a cylinder ...

**4**

votes

**1**answer

204 views

### Subgroups of Gromov's hyperbolic groups

It's known that subgroups of Gromov's hyperbolic groups are not necessarily hyperbolic.
Is there any counter-example when the quotient is Abelian. More precisely, let $G$ be a Gromov's hyperbolic ...

**23**

votes

**8**answers

2k views

### Are there some other notions of “curvature” which measure how space curves?

I am learning differential geometry and have a few questions on curvature. -- Background:
Gauss invented "Gauss curvature" to measure how surface curves.
Riemann gives an ingenious generalization ...

**14**

votes

**1**answer

353 views

### Are there geometrically formal manifolds, which are not rationally elliptic?

Formality of a space is meant in the sense of Sullivan, i.e. a space $X$ is called formal, if it's commutative differential graded algebra of piecewise linear differential forms $(A_{PL}(X),d)$ is ...

**2**

votes

**0**answers

206 views

### Fundamental group of a Cayley graph [migrated]

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...

**29**

votes

**9**answers

5k views

### Nice proof of the Jordan curve theorem?

As a student, I was taught that the Jordan curve theorem is a great example of an intuitively clear statement which has no simple proof.
What is the simplest known proof today?
Is there an intuitive ...

**2**

votes

**1**answer

214 views

### undergraduate handle decomposition. Reference

As the title says, I'm searching for a nice textbook for introducing the theory of handle decomposition of manifolds to undergraduate students.

**5**

votes

**1**answer

99 views

### References to proofs of a theorem by Van Kampen-Flores

Theorem (Van Kampen-Flores 1930s) From any 7 points in four-dimensional space one can choose two disjoint triples such that the triangles with vertices at the triples intersect each other.
This ...

**0**

votes

**0**answers

67 views

### moduli space of equivariant holomorphic embeddings into the quintic

I'd like to understand better the following problem, whether it is mathematically well-posed, trivial, etc.
Fix a non-negative integer $g$ and consider
the space
...

**0**

votes

**0**answers

83 views

### Help understanding a proof in Stallings' Triangle of Groups paper

I'm trying to understand the proof of theorem 1 in Stalling's Non-positively Curved Triangle of Groups. I have specific questions, but is there anywhere someone has written out the proof in more ...

**3**

votes

**0**answers

109 views

### Lie Group Isomorphisms

I'm not sure of the difficulty of the question I'm about to ask. If it does not fit the criteria for this site then I apologize in advance, I'm rather new here.
So here it goes: Let $G$ be a Lie ...

**10**

votes

**6**answers

870 views

### Computing signature

I have a feeling that this might have already been asked, but can't find the question. Anyway, the question is: given a symmetric $n\times n$ matrix, is there a faster way to compute its signature ...

**-1**

votes

**1**answer

553 views

### monodromy defects and Chern-Simons

In the context of string theory I recently read "The formulation of Chern-Simons theory in terms of monodromy defects can be carried through all the dualities of the present paper, leading to ...

**9**

votes

**1**answer

433 views

### Including a Jordan arc into a Jordan loop (Can the Magi go home by another way?)

The title refers, of course, to Matthew (2:12) ''And being warned in a dream not to return to Herod, they departed to their own country by another way''. To be honest, it is not that specific ...

**3**

votes

**1**answer

68 views

### Transfinite sequence of contiguous simplicial maps

Recall that two simplicial maps of (abstract) simplicial complexes $f,g\colon K\to L$ are contiguous if $f(\sigma)\cup g(\sigma)$ is a simplex of $L$ for every simplex $\sigma\in K$. Contiguous ...

**2**

votes

**1**answer

605 views

### isotopy doesn't make sense (Milnor)

hello,
I am having a hard time following this isotopy put forth by Milnor in On the Total Curvature of Knots
For each $c$ and $p$ in
$\mathbb{R}^{n-1}$ such that $\|c-p\|
> < r$, there is ...

**6**

votes

**1**answer

328 views

### Examples of calculations of Turaev-Reshetikhin TQFT of cobordisms with boundaries have genera greater than 1

I am studying Turaev-Reshetikhin TQFT. I describe the definition of the invariant $\tau(M)$ of a cobordism $(M, \partial_{-}M, \partial_{+}M)$ in the previous question breifly. Framings in the ...

**4**

votes

**0**answers

94 views

### Does every null-homologous surface bound, part deux

This is a follow-up to this question, and Mark Grant's excellent answer. The answer shows that the answer is yes, but what if one were under a strange compunction to actually construct the submanifold ...

**3**

votes

**1**answer

99 views

### 0-homologous surface bounds

Given a map $f : S \to M^4$ from a compact closed not necessarily connected oriented surface to a compact oriented 4-manifold, such that $f_*([S])$ is zero in $H_2(M)$, is there a compact oriented ...