**2**

votes

**1**answer

82 views

### characteristic classes of homotopy equivalent manifolds

Let $M,N$ be two manifolds of different dimensions. Suppose $M\simeq N$, i.e. $M$ is homotopy equivalent to $N$. Do the Stiefel-Whitney classes of the tangent bundles of $M$ and $N$ equal
$$
...

**3**

votes

**0**answers

73 views

### Are all transversely oriented foliations given by closed forms?

This paper, "AF-algebras and topology of 3-manifolds" seems to claim on page 5 that a [transversely] oriented measured foliation on a compact surface is given by a closed form. On the other hand, the ...

**10**

votes

**3**answers

2k views

### Teichmuller theory and moduli of Riemann surfaces

This is a sequel to my earlier question asking for references for Teichmuller theory and moduli spaces of Riemann surfaces.
In this connection, I have read Chapter 11 of the book Primer of mapping ...

**3**

votes

**2**answers

248 views

### Is there a nonabelian free group inside a group of positive rank gradient?

Let $G$ be a finitely generated residually finite group with positive
rank gradient, and let $F_2$ be the free group on $2$ elements. Must
there be an embedding $i \colon F_2 \to G$ ?
A group ...

**4**

votes

**1**answer

89 views

### cohomology ring of infinite iterated loop space

What is the cohomology ring
$$
H^*(\Omega^\infty \Sigma^\infty (S^m\vee S^n);\mathbb{Z}_2)?
$$
I already write out the graded-vector-space basis using Dyer-Lashof operations, but I do not know how to ...

**4**

votes

**1**answer

85 views

### unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then
$$
B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1},
$$
$$
B(S^n,2)\simeq \mathbb{R}P^n.
$$
Hence
$
(*)
$
$$
...

**4**

votes

**1**answer

127 views

### Current status of the linearity of mapping class group

In the paper A faithful linear-categorical action of the mapping class group of a surface with boundary it is claimed that it was (as of 2014) still unknown that mapping class group is linear, but the ...

**5**

votes

**3**answers

134 views

### Probability of random geodesics on the half-sphere intersecting

4 end points (a,b,c,d say) are chosen uniformly randomly and connected a to b and c to d by two geodesics on the 2-dim half-sphere. Here, uniform means that, probability that a point lies on a surface ...

**3**

votes

**0**answers

55 views

### Anosov representations and boundaries of (harmonic) maps

Let $\Sigma_g$ be a closed hyperbolic surface and $\rho\colon\pi_1\Sigma_g\to G$ an Anosov representation into a suitable Lie group. By definition of Anosovness, one has a $\rho$-equivariant ...

**3**

votes

**1**answer

226 views

### coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...

**7**

votes

**2**answers

268 views

### For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?

Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature?
An obvious case is when $Y$ ...

**2**

votes

**0**answers

40 views

### When do positively invariant subset contain a given set?

Non-triviality of the Conley index for an isolating neighborhood $N$ and a flow $\varphi$ can be used to prove non-emptyness of the related isolated invariant set. In particular, if $N$ doesn't ...

**6**

votes

**5**answers

2k views

### Lie groups admitting flat (bi)invariant metrics.

I would like to see an example of a non-abelian compact lie group admitting a bi/left/right-invariant flat metric.
Is there any non-abelian compact lie group admitting a flat metric that is bi or ...

**14**

votes

**2**answers

418 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 ...

**14**

votes

**3**answers

2k views

### What is Kirillov's method good for?

I am planing to study Kirillov's orbit method. I have seen Kirillov's method in several branch of mathematics, for instance, functional analysis, geometry, .... Why is this theory important for ...

**3**

votes

**1**answer

111 views

### Why almost every geodesic arc is generic？

In his paper "Geodesic laminations on surfaces", Bonahon gave the definition of generic arc and a property as following.
An arc $k$ is generic (with respect to simple geodesics) if it is transverse ...

**3**

votes

**1**answer

84 views

### Existence of tubular neighborohoods of locally flat topological embeddings

Suppose $X$ is a topological manifold and $Y \subset X$ is a locally flat submanifold. We know that $Y$ doesn't necessarily have a tubular neighborhood. My definition of a tubular neighborhood of $Y$ ...

**9**

votes

**0**answers

149 views

### Do homology classes have “special” representatives?

Recall that, according to Hodge, de-Rham cohomology classes of "nice enough" manifolds have "special" representatives - namely, harmonic forms.
Now, how does one choose a "special" one among ...

**4**

votes

**2**answers

162 views

### Fibered example of topologically slice knots

Is there any known example of fibered knot which is topologically slice but not (expected to be) smoothly slice?

**14**

votes

**3**answers

530 views

### Homology generated by lifts of simple curves

Let $\Sigma$ be a compact connected oriented surface and $p:\tilde{\Sigma}\to\Sigma$ a finite regular cover.
Consider the set $\Gamma$ of simple closed curves on $\tilde{\Sigma}$ obtained as a ...

**2**

votes

**1**answer

125 views

### Isometry Group of real Hilbert space?

Does the isometry group of a real separable infinite-dimensional Hilbert space have two connected components? Or, conversely, is the there even a Kuiper's theorem in the real case?
How does the ...

**13**

votes

**1**answer

996 views

### Looking for a simple proof that R^2 has only one smooth structure

So not so long ago, I asked for a simple proof that $\mathbf{R}$ has only one smooth structure. A proof that was communicated to me by Ryan Budney (link text) was the following:
So let me recall his ...

**16**

votes

**0**answers

394 views

### Polynomials with roots in convex position

Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...

**4**

votes

**3**answers

189 views

### Parameterizing rotations of a cube [closed]

For $g\in\mathrm{SO}(3),S\subseteq \mathbb{R}^3,$ define $g\cdot S:=\{g\cdot p : p\in S\}.$ In words, if $g$ is a rotation of $\mathbb{R}^3$, $g\cdot S$ is the set of elements of $S$ rotated by $g$. ...

**3**

votes

**1**answer

88 views

### Calculating Homology of the Boundary of a Handlebody

Given a manifold $M$ with boundary $W = \partial M$, I know that having a handle decomposition of $M$ allows one to compute its homology, at least in nice cases, by - for example - using the Morse ...

**1**

vote

**0**answers

81 views

### Gluing two diffeomorphisms and then smoothing

This question did not get an adequate answer on math.stackexchange.
Let $M_1,M_2$ be two $n$-dimensional closed manifolds and suppose that $M_i=\bar{U}_i^+\cup \bar{U}_i^-$ where $\bar{U}_i^\pm$ are ...

**2**

votes

**1**answer

182 views

### Perturb a given smooth function to a Morse function relative to fixed level sets, which are already fine

This question was not answered on math.stackexchange.
Let $M$ be a manifold (not necessarily compact) , for the sake of clearness embedded in $\mathbb{R^n}$ and $f\colon M\rightarrow \mathbb{R}$ a ...

**92**

votes

**25**answers

38k views

### Intuitive crutches for higher dimensional thinking

I once heard a joke (not a great one I'll admit...) about higher dimensional thinking that went as follows-
An engineer, a physicist, and a mathematician are discussing how to visualise four ...

**17**

votes

**0**answers

158 views

### Why, in terms of quantum groups, does the knot determinant appear as an evaluation of both the Jones and Alexander polynomials?

The Jones polynomial can be computed from the representation theory of $\mathcal{U}_q(\mathfrak{sl}(2))$. The Alexander polynomial has an analogous description in terms of the representation theory of ...

**4**

votes

**1**answer

69 views

### Immersed quasi-Fuchsian surfaces surviving Dehn fillings

In papers like, Cooper - Long - Some surface subgroups survive surgery or Li - Immersed essential surfaces in hyperbolic 3-manifolds the game is to find some quasi-Fuchsian immersed surface $Q ...

**7**

votes

**1**answer

128 views

### Questions on poincare homology spheres and branched covers

I have two questions:
Question 1. Suppose that $K$ is a knot in $S^3$. Let $\Sigma(K)$ be the double branched cover of $S^3$ branched along $K$. If $\Sigma(K)=\#_{i=1}^n\Sigma(2,3,5)$, then ...

**5**

votes

**0**answers

121 views

### Mapping the standard $(n!-1)$-simplex into $GL_n(\mathbb C)$

Let $\Delta^{n!-1}$ be the standard $(n!-1)$-simplex whose vertices are indexed by the elements of the symmetric group $S_n$, that is
\begin{equation}
\Delta^{n!-1} = \left\{ (t_\sigma)_{\sigma\in ...

**16**

votes

**0**answers

186 views

### Concordance and homology cobordism

If two knots $K_1$ and $K_2$ in $S^3$ are smoothly concordant, then for any rational number $r$, the $r$-surgeries $S^3_r(K_1)$ and $S^3_r(K_2)$ are homology cobordant. Is the converse true? What if ...

**9**

votes

**1**answer

627 views

### Orbifold fundamental group and configuration space

I'm not very familiar with (even simple examples of) orbifolds, so my first question is:
Let $C_2$ be $\mathbb{C}$ with one cone singularity at 0 of index 2. What is the fundamental group of $C_2$ ...

**22**

votes

**1**answer

1k views

### Example of 4-manifold with $\pi_1=\mathbb Q$

This might be well known for algebraic topologist. So I am looking for an explicit example of a 4 dimensional manifold with fundamental group isomorphic to the rationals $\mathbb Q$.

**6**

votes

**1**answer

238 views

### Satellite knot example

Can someone provide me with an example of a satellite knot with symmetry group which is neither cyclic nor dihedral?

**0**

votes

**0**answers

71 views

### Automorphism group of closed projective surface of negative Euler characteristic

Let $M$ be a smooth surface and $[\nabla]$ a projective structure on $M$, that is, an equivalence class of torsion-free connections on $TM$, where two such connections are called projectively ...

**13**

votes

**3**answers

660 views

### Simply-connected rational homology spheres

Every simply-connected rational homology sphere is, in fact, the usual sphere in dimensions $2, 3.$ Is this true in dimension 4? Where are the first counterexamples? (I know there are some in ...

**1**

vote

**0**answers

34 views

### A questions related to the Markus conjecture for special affine manifolds

An affine manifold $M$ is called special if there is a parallel volume form $\omega$ on $M$,
and a nowhere vanishing vector field $\mathcal{V}.$
Here we need to point out that any affine manifold of ...

**7**

votes

**1**answer

107 views

### Determine if an $n$-dimensional mesh of simplices is a non-manifold

In an $n$-dimensional space I have a set of simplices where each simplex consists of facets. Some of the simplices are 'connected' by sharing facets. Each facet is made up on edges, each consisting ...

**12**

votes

**0**answers

126 views

### $p$-Adic or arithmetic variants of Khovanskii's “low complexity $\Rightarrow$ tame topology” theory

This question is prompted by a remark I made in a comment to Is every polynomial a factor of a trinomial?, which was that Descartes's observation (cf. his rule of signs, etc.), that the number of real ...

**6**

votes

**1**answer

226 views

### Four-dimensional vector bundles over $S^4$, intuition?

I know that $\pi_3(SO(4)) = \mathbb{Z} \oplus \mathbb{Z}$. We can choose an explicit identification as follows: given $(i, j) \in \mathbb{Z}$, we have a map $\phi: S^3 \to SO(4)$ which sends a unit ...

**3**

votes

**0**answers

71 views

### What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? [migrated]

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...

**3**

votes

**1**answer

186 views

### Constructing a homology class of degree $d(d-1)/2$ in $H_3(S^3)$

There is a nice construction of a class of degree $d^2$ in $H_3(S^3)$. Take a class $h$ of degree $d$ in $H_1(S^1)$, and then take its join with itself: $h*h$ is degree $d^2$ in $H_3(S^1*S^1)$, and ...

**12**

votes

**0**answers

263 views

### Is Rasmussen's s-invariant of a knot an invariant of a 4-manifold?

Let $K$ be a knot in the 3-sphere $S^3$.
Here we denote by $s(K)$ Rasmussen's s-invariant for $K$,
and by $X_{K}(n)$ the 4-manifold obtained from the standard 4-ball $B^4$
by attaching a ...

**3**

votes

**1**answer

208 views

### Example of a triangulable topological manifold which does not admit a PL structure

I know there are some examples of manifolds which don't admit a PL structure (combinatorial triangulation), and that it has been recently proven that in dimension $n\geq5$ there are manifold which are ...

**2**

votes

**1**answer

47 views

### Shrinkable decompositions with uncountably many non-degenerate elements?

Let $\mathcal D$ be an upper semicontinuous decomposition of $\mathbb S^n$ and let $\mathcal D'\subset\mathcal D$ be the set of non-singletons. The decomposition space $^{\mathbb S^n}/_{\mathcal D}$ ...

**8**

votes

**4**answers

1k views

### When is a finite cw-complex a compact topological manifold?

I think the statement of the question is pretty straightforward. Given a finite $n$-dimensional CW complex, are there necessary and sufficient conditions for determining that it is also a compact ...

**2**

votes

**0**answers

116 views

### the homology of configuration spaces [closed]

In the paper ON THE HOMOLOGY OF CONFIGURATION SPACES, Section 5.4,
Why $C(M\times \mathbb{R};X)\simeq \Omega C(M; SX)$? Does this hold for $X$ not connected?

**1**

vote

**0**answers

81 views

### When Max(R) is Hausdorff space? [duplicate]

Let $R$ be reduce commutative ring with identity (a commutative ring such that $a^n$=0 ($a\in R$) implise $a=0$) and $Max(R)$ be the set of all maximal ideals of $R$. The hull-kernel (or Zariski ...