Topology of cell complexes and manifolds, classification of manifolds (e.g. smoothing, surgery), low dimensional topology (e.g. knot theory, invariants of 4-manifolds), embedding theory, combinatorial and PL topology, geometric group theory, infinite dimensional topology (e.g. Hilbert cube ...

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2
votes
1answer
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
0answers
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
3answers
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
2answers
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
1answer
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
1answer
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
1answer
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
3answers
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
0answers
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
1answer
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
2answers
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
0answers
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
5answers
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
2answers
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
3answers
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
1answer
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
1answer
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
0answers
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
2answers
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
3answers
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
1answer
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
1answer
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
0answers
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
3answers
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
1answer
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
0answers
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
1answer
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
25answers
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
0answers
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
1answer
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
1answer
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
3answers
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
0answers
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
1answer
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
1answer
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
4answers
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
0answers
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
0answers
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 ...