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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44
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540 views

Topological cobordisms between smooth manifolds

Wall has calculated enough about the cobordism ring of oriented smooth manifolds that we know that two oriented smooth manifolds are oriented cobordant if and only if they have the same ...
36
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0answers
930 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 ...
33
votes
0answers
2k 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 ...
30
votes
0answers
1k 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. ...
27
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0answers
1k 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 ...
23
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233 views

Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic ...
22
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0answers
494 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 ...
21
votes
0answers
690 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 ...
21
votes
0answers
772 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 ...
19
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0answers
212 views

“High-concept” explanation for proof of a theorem of Ochanine?

See Akhil Mathew's notes on Ochanine's theorem for elliptic genera here and here. Let $\phi: \Omega_{SO} \to \Lambda$ be a genus. We might ask when $\phi$ satisfies the following multiplicative ...
19
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0answers
345 views

Topological description of inverting a knot

The first figure shows an offset overhand knot. To tie it, you simply bring the two cords together and make an overhand knot in them. It's more secure than it looks, and several climbers have been ...
19
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0answers
662 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 ...
19
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459 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
0answers
503 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 ...
18
votes
0answers
719 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
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422 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 ...
16
votes
0answers
454 views

What is the determinant of Poincaré 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 ...
16
votes
0answers
673 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 ...
16
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0answers
502 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
0answers
493 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 ...
14
votes
0answers
323 views

pizza lemma (topology)

given six real-analytic arcs in the unit disk $D$, each of which connects the origin to a boundary point, and no two arcs meet anywhere except at the origin, and the arcs meet at equal (60 degree) ...
14
votes
0answers
338 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 ...
13
votes
0answers
204 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 ...
13
votes
0answers
345 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 ...
13
votes
0answers
299 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 ...
12
votes
0answers
126 views

Which spherical space forms embed in $S^4$?

Is there any hope of getting a classification of which 3-dimensional spherical space forms are smoothly embeddable in $S^4$? I read that lens spaces cannot embed in $S^4$, but some other spherical ...
12
votes
0answers
192 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 ...
12
votes
0answers
368 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 ...
11
votes
0answers
226 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 ...
11
votes
0answers
359 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 $S^2 \times M_2$?
11
votes
0answers
732 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 Lovász conjecture for Cayley graphs, I started noticing ...
11
votes
0answers
418 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 ...
10
votes
0answers
190 views

Mapping class groups in high dimension

Let $M$ be a $1$-connected, closed, smooth manifold with $dim(M)>4$ and let us set $MCG(M)=\pi_0(Diff(M))$. Dennis Sullivan proved that $MCG(M)$ is commensurable to an arithmetic group. I was ...
10
votes
0answers
181 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 ...
10
votes
0answers
131 views

Kernel of “Hat to Plus” in Heegaard Floer Homology

Given a 3-manifold $M$, there is a map of Heegaard Floer groups $$f:\widehat{HF}(M) \to HF^+(M)$$ induced by the inclusion $$\widehat{CF}(M) \to CF^+(M)$$ of the respective chain complexes. Given a ...
10
votes
0answers
211 views

Is the quotient map of the action of homeomorphisms on embeddings well-behaved?

It is well known that if $M$ and $N$ are smooth manifolds, the diffeomorphisms $Diff(M)$ act continuously on the smooth embeddings $Emb^{C^\infty}(M,N)$ by precomposition, if both are given the ...
10
votes
0answers
157 views

Excluding exotic PL structures on S^4

Suppose you have a finite group $G<SO(5)$ such that $S^4/G$ is homeomorphic to $S^4$ and such that $S^4/G$ is a PL manifold with respect to a PL structure induced by a standard structure on $S^4$. ...
10
votes
0answers
179 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 ...
10
votes
0answers
227 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 ...
10
votes
0answers
354 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: ...
10
votes
0answers
508 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 ...
9
votes
0answers
154 views

Detecting a module for the free group algebra on a finite quotient

Let $F_2$ be the free group on two generators $x,y$ and let $R$ be the group algebra $\mathbf{Q}[F_2]$. Let $a,b,c$ be integers. Then define a right $R$-module $M = R / (ax + by + c) R$. I am ...
9
votes
0answers
137 views

Why should we regard $PL(M)$ as a simplicial group?

Let $M$ be a manifold. If $M$ is smooth, it is clear what $\text{Diff}(M)$ should be, namely it should be the set of diffeomorphisms of $M$ equipped with the topology in which a sequence of ...
9
votes
0answers
184 views

Doubly slice knots and an embedding of 4-manifold

It is well-known that the existence of topologically slice knot which is not smoothly slice implies the existence of exotic $\mathbb{R}^4$. For example, see the answer of K. Davis. To prove the above ...
9
votes
0answers
220 views

Alexander polynomial in branched covers

Suppose I am given a 3-manifold as a double branched cover over a link. Let a null-homologus knot in this space be given as a lift of an arc with endpoints on the link (it is automatically ...
9
votes
0answers
218 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 Poincaré conjecture in ...
9
votes
0answers
255 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
0answers
304 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
0answers
586 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 ...
9
votes
0answers
324 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 ...