**32**

votes

**0**answers

777 views

### Homotopy type of TOP(4)/PL(4)

It is known (e.g. the Kirby-Siebenmann book) that $\mathrm{TOP}(n)/\mathrm{PL}(n)\simeq K({\mathbb Z}/2,3)$ for $n>4$. I believe it is also known (Freedman-Quinn) that ...

**28**

votes

**0**answers

1k views

### Minimal volume of 4-manifolds

This question came up in a talk of Dieter Kotschick yesterday. The minimal volume of a manifold is the infimum of volumes of Riemannian metrics on the manifold with sectional curvatures bounded in ...

**26**

votes

**0**answers

930 views

### What is the three-dimensional hyperbolic volume of a four-manifold?

Every smooth closed orientable 4-manifold may be constructed via a handle decomposition. Before asking a couple of questions, I recall some well-known facts about handle-decompositions of 4-manifolds.
...

**21**

votes

**0**answers

766 views

### Manifolds admitting CW-structure with single n-cell

Let $M$ be a topological $n$-manifold, closed and connected (not necessarily oriented):
When does $M$ not admit (up to homotopy-type) a CW-structure with a single $n$-cell?
By classification of ...

**21**

votes

**0**answers

447 views

### Smooth thickenings of non-smoothable manifolds

It is known that any closed topological manifold is homotopy equivalent to an open smooth manifold.
Question 1. What can be said about the smallest
dimension of a smooth manifold
that is homotopy ...

**20**

votes

**0**answers

602 views

### Boundaries of noncompact contractible manifolds

It is known that a manifold $B$ bounds a compact contractible topological manifold if and only if $B$ is a homology sphere. The "only if" direction follows by excising a small ball in the interior of ...

**18**

votes

**0**answers

428 views

### The oriented homeomorphism problem for Haken 3-manifolds

Haken famously described an algorithm to solve the homeomorphism problem for the 3-manifolds that bear his name (fleshed out by many others, including Hemion and Matveev who fixed some gaps). But ...

**18**

votes

**0**answers

668 views

### Almost complex 4-manifolds with a “holomorphic” vector field

Main question. What is the class of smooth orientable 4-dimensional manifolds that admit an almost complex structure $J$ and a vector field v, that preserves $J$?
The following sub question is ...

**17**

votes

**0**answers

524 views

### Which manifolds decompose into pants?

In this nice paper Mikhalkin uses certain (more geometrical than algebraic) aspects of tropical geometry to prove that every complex projective hypersurface in $\mathbb C \mathbb P ^n$ decomposes as a ...

**16**

votes

**0**answers

398 views

### Monoid structure of oriented manifolds with connect sum

Take the class of all compact, connected, boundaryless, smooth oriented $n$-dimensional manifolds, each taken up to orientation-preserving diffeomorphism. This is a commutative monoid with operation ...

**15**

votes

**0**answers

394 views

### Do there exist exotic 4-tori?

More precisely: are there known manifolds which are homeomorphic, but not diffeomorphic to the standard 4-torus? Are there any nice invariants distinguishing such manifolds?
Related: if such a ...

**14**

votes

**0**answers

606 views

### Horrible sets and blowups in Hubbard's Teichmuller theory

Edit: I can rephrase this question this way: When blowing up every point in the $x$-axis in $\mathbb{C}^2$ by means of an inverse limit of finite blowups, how can anything be 'left over'? The horrible ...

**14**

votes

**0**answers

222 views

### Shortest Casson tower containing a slice disk for the attaching curve

A Casson tower is obtained as follows: Start with a properly immersed disk in $\mathbb{B}^4$ - a regular neighborhood of such a disk is called a kinky handle. The boundary of the core disk ...

**13**

votes

**0**answers

308 views

### Asymptotics for the number of triangulations of a manifold M

In Gromov's talk at the Clay Math Research from 23:23 to 25:55 Gromov says (slightly paraphrased)
I want to emphasize a problem which
comes from mathematical physics which
is unsolved which is ...

**12**

votes

**0**answers

150 views

### What do tangles teach us about braids?

A braid is a smooth level-preserving embedding $f\colon\, \{1,2,\ldots,n\}\times[0,1]\hookrightarrow \mathbb{R}^2 \times [0,1]$ such that $f(k,0)=(k,0)$ and $f(k,1) \in \{1,2,\ldots,n\} \times ...

**12**

votes

**0**answers

327 views

### What is the determinant of Poincare duality?

For a complex $C^{\bullet}$ of finite dimensional vector spaces, one has a determinant
$$|C^\bullet|:= \bigotimes \left(\Lambda^{top} C^i\right)^{(-1)^i}$$
functorial with respect to ...

**12**

votes

**0**answers

131 views

### Approximating homeomorphisms of 2-disk by diffeomorphisms

Any homeomorphism of a compact surface can be approximated by diffeomorphisms.
Is there a parametrized version of this result, where the parameter space is an $n$-disk?
In other words, if $S$ is a ...

**12**

votes

**0**answers

278 views

### rigidity of representations of dehn fillings on the figure eight knot

Consider a Dehn filling of the figure eight knot with coprime parameters (p,q) denoted by M.
I am interested in the representation space of M in $SU_2$.
I wonder wether all such representations are ...

**11**

votes

**0**answers

382 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**11**

votes

**0**answers

694 views

### Hamiltonian cycles and fundamental groups

I'm interested in the interplay between the Hamiltonian cycles of graphs and the compact surfaces they embed in. I was doing some reading on the LovÃ¡sz conjecture for Cayley graphs, I started noticing ...

**10**

votes

**0**answers

173 views

### Fox re-imbedding theorem in dimension four

Fox re-imbedding theorem states the following:
A compact 3-manifold $M$ with boundary that embeds in the three-sphere $S^3$, can be re-imbedded in $S^3$ so that its complement is a union of ...

**10**

votes

**0**answers

483 views

### Once differentiable, piecewise degree three polynomials on triangulated planar domains

Here is an easily described, but very difficult, problem that I
(and a number of other people) really would like to see solved during
our life times. The basic problem is to compute the dimension of ...

**10**

votes

**0**answers

376 views

### Codimension 2 foliations on simply connected 4-manifolds

Are there examples of codimension 2 foliations on simply connected compact 4-manifolds such that
Every leaf is diffeomorphic to $\mathbb R^2$
Every leaf is dense?
Same question for 5-manifolds ...

**9**

votes

**0**answers

239 views

### Provide a citation for the “spine lemma”?

I'm looking for a citable reference for the following (perhaps folkloric?) result on topological field theories.
(There are obviously generalizations to other dimensions; I'm happy with just the ...

**9**

votes

**0**answers

181 views

### Embedding of products of projective spaces into a projective space

Does anybody know estimates on the minimal dimension $k$ for which the product $P^n \times P^m$ can be smoothly embedded into $P^k$? I am interested in projective spaces over $\mathbb RR$ and $\mathbb ...

**9**

votes

**0**answers

286 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 $S^2 \times M_2$?

**9**

votes

**0**answers

281 views

### Is the connected sum of knots an isometry?

Take $X$ as the set of knots in the 3-sphere (i.e. smooth embeddings of $S^1$ in $S^3$ up to smooth isotopy), endowed with the Gordian distance $d$.
For a fixed knot $K$ we can define the map ...

**9**

votes

**0**answers

313 views

### Covering a subset by submanifolds

Let $K$ be a compact subset of a Euclidean space of very large dimension $N$. Assume
that any point $x\in K$ has a neighborhood $U\subset K$ which is contained in a smooth
$l$-dimensional submanifold ...

**9**

votes

**0**answers

300 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 ...

**8**

votes

**0**answers

161 views

### Consequences of Zeeman's conjecture

Recall the Zeeman's conjecture: if $K$ is a contractible polyhedron of dimension 2, then $K\times I$ has a collapsible subdivision.
Zeeman showed that this implies the PoincarÃ© conjecture in ...

**8**

votes

**0**answers

104 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 ...

**8**

votes

**0**answers

139 views

### Does the shortest path between two braids pass through string links?

One of the fundamental facts underlying the application of braid theory to knot theory is that braids inject into string links.
This means that braids $B_1$ and $B_2$, considered inside a cube $I^3$, ...

**8**

votes

**0**answers

126 views

### Naive Reidemeister-Schreier for $\mathbb Z$ quotients

I have a question about a "standard" variant of the Reidemeister-Schreier algorithm used by topologists when manipulating manifolds they either know or suspect are fibre-bundles over $S^1$.
Say you ...

**8**

votes

**0**answers

128 views

### What sort of geometry does the Whitehead manifold have as a hypersurface in $\mathbb{R}^4$?

If I understand correctly, the standard $\mathbb{R}^4$ is diffeomorphic to $\mathbb{R}\times W$ where $W$ is the Whitehead manifold (i.e., is an open three-manifold that is contractible and not ...

**8**

votes

**0**answers

240 views

### Can a composition with itself of a universal self-map be non-universal?

I have formulated (and published) the notion of a universal map (and of a universal morphism), and the problems below, in the early 1960-ies.
DEFINITION A continuous map $u: ...

**8**

votes

**0**answers

456 views

### Categorification of finite type invariants

Recall that finite type invariants are those numerical knots invariants vanishing on knots with enough singularities. The value of an invariant on a singular knots is computed using the Vassiliev ...

**8**

votes

**0**answers

307 views

### What is (explicitly) known about the SL(n,C) character varieties of 3-manifolds ?

The $SL(2,{\bf C})$ character variety of a 3-manifold with 1-cusp $M$ (like a knot complement in the 3-sphere) essentially coincide with the variety defined by the A-polynomial. Those polynomials are ...

**8**

votes

**0**answers

449 views

### Pseudocycle definition of open Gromov-Witten invariants

I decided my original question was unnecessarily long, and have edited to simply ask the desired question directly (the below is much more direct than my original post, though it may seem just as ...

**8**

votes

**0**answers

297 views

### is a group $G$, that admits finite $k(G, 1)$ and has no Baumslag-Solitar subgroups, necessarily hyperbolic?

This is the first question asked in Bestvina's article "Questions in Geometric Group Theory". Does anyone know if there has been any progress made on this problem? Is the question answered if $G$ is ...

**7**

votes

**0**answers

99 views

### The homology of the braid group with coefficients in the Burau representation

Let $B_n$ denote the braid group with $n$ braids. The Burau representation $B_n\to GL_n(\mathbb{Z}[t^{\pm1}])$ makes $(\mathbb{Q}[t^{\pm1}])^n$ a $B_n$-module. I am curious in knowing what $H_i(B_n, ...

**7**

votes

**0**answers

174 views

### Reference Request: Topological h-cobordism theorem in higher dimensions

I think this question on math.stackexchange is more appropriate on mathoverflow. Correct me, if you don't think so.
The h-cobordism theorem is true in the topological and in the smooth category in ...

**7**

votes

**0**answers

148 views

### Why is it hard to obtain improved $L^6$ bound of eigenfunction of Laplacian on 2-dimensional compact Riemannian manifold?

Consider the $L^p$ estimate of the Laplacian on a compact boundaryless Riemannian manifold, suppose that $-\Delta_ge_{\lambda}=\lambda^2e_\lambda(x), x\in M$.
C.D. Sogge proved that we have the ...

**7**

votes

**0**answers

187 views

### Right-angled polytopes

%This question is motivated by the little discussion here at the bottom.
The following thing are known about hyperbolic right-angled polytopes:
Compact hyperbolic right-angled polytopes do not ...

**7**

votes

**0**answers

134 views

### PL surface projections - is there a theory of folds and cusps?

For smooth surfaces, the generic singularities of a map of one surface to another are folds and cusps (Whitney). It is a standard result in singularity theory that the generic isotopy of such a map is ...

**7**

votes

**0**answers

217 views

### What is the historical connection between Zeeman's twist spinning and Fox's Examples?

Both Ralph Fox and (at that time, yet to be knighted) Sir Christopher Zeeman attended the 1961 Georgia topology conference. Fox's paper from that conference was his seminal work, "A Quick Trip through ...

**7**

votes

**0**answers

207 views

### What is known about the Yang-Mills stratification over 3-manifolds?

Rade proved in his thesis (Crelle's Journal, 1992, available here: digizeitschriften.de/dms/toc/?PPN=PPN243919689) that if $E\rightarrow M$ is a $U(n)$-bundle over a 3-manifold, then the gradient flow ...

**7**

votes

**0**answers

464 views

### Embeddings without nonvanishing normal vector fields

For which values of $n$ does there exist an embedding of a smooth compact manifold $M\hookrightarrow R^n$ into $n$-dimensional Euclidean space such that the normal bundle to $M$ has no nonvanishing ...

**7**

votes

**0**answers

269 views

### Directed arcs on a surface

This question is a little odd. I have specific class of structures on a surface, which satisfy several nice properties, and I want to know if they are more natural geometric structures in disguise ...

**6**

votes

**0**answers

92 views

### Two-relator products of cyclic groups

In "A proof of the Scottâ€“Wiegold conjecture on free products of cyclic groups" Howie proved that every one-relator product of three cyclic groups is nontrivial. Is there a now proven theorem that says ...

**6**

votes

**0**answers

127 views

### Characterize spin cobordism invariants in dimer models

The paper by Cimasoni and Reshetikhin http://arxiv.org/abs/math-ph/0608070 shows that one can map problems about spin structures on a Riemann surface into problems about dimmer configurations on a ...