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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8
votes
1answer
300 views

Is every degree 1 self-map a homotopy equivalence?

In a rather obscure article, I found (without proof) the following statement: If $M$ is a closed orientable manifold, every degree $1$ map $f: M \rightarrow M$ is a homotopy equivalence. Is this ...
2
votes
1answer
101 views

Braid wiring diagrams and matroids

recently I started reading some articles about the presentation of the fundamental group of lines arrangements in $\mathbb{C}^{2}$ via Wiring diagrams. I also found some relation with matroid theory. ...
1
vote
1answer
274 views

Isomorphism between a mapping class group and the fundamental group of a moduli space

For some fixed integer $d \geq 3$, let $M(0, d)$ be the mapping class group of self-homeomorphisms of the Riemann sphere which fix each of the $d$ points $0, 1, ... , d-2, \infty$. Let $X$ be the ...
4
votes
1answer
161 views

Are “Unions” of small exotic $\mathbb{R}^4$'s small?

Suppose $M$ is a smooth 4-manifold, and $U,V \subset M$ are exotic $\mathbb{R}^4$'s, i.e. homeomorphic to standard $\mathbb{R} ^4$. Further more suppose $U$ an $V$ intersect nicely sucht that $U \cup ...
11
votes
1answer
232 views

Thom conjecture in CP3

Thom conjecture, that was originally asked in $\mathrm{CP}^2$, and is now proven for symplectic 4 manifolds, states that complex curves (symplectic surfaces) are genus minimizing in their homology ...
6
votes
2answers
240 views

Is every $S^3$ block bundle over $S^4$ a fiber bundle?

I am interested in the difference between block bundle and fiber bundle. Let $K$ be a simplicial complex and $p: E\to |K|$ be a continuous map. A block diffeomorphism of $\Delta^p\times M$ is a ...
1
vote
1answer
63 views

Convex subcomplexes of CAT(0) cubical complexes

Is the following statement true? If so, can anyone provide a reference? Let $X$ be a CAT(0) cubical complex, and let $Y$ be a connected subcomplex of $X$. Then the following are equivalent: ...
11
votes
1answer
320 views

Is there a faithful transitive locally finite action of the modular group?

Is there a faithful transitive action of $G = \mathrm{PSL}_2(\mathbb{Z})$ on $\mathbb{Z}$ such that orbits under each $g \in G$ are finite?
5
votes
1answer
147 views

How many knots are there with hyperbolic volume less than a given constant

Are there any known upper bounds on: $$\#\left\{\text{hyperbolic knots }K\subseteq S^3\middle|\operatorname{Vol}(S^3\setminus K)<M\right\}$$ ? I expect this grows at least exponentially in $M$, ...
2
votes
2answers
194 views

F.g group with infinite ends not Q.I to a free group

Is there any easy example of a finitely generated group with a Cantor set of ends that is not quasi-isometric to a finitely generated free group? Thanks in advance.
10
votes
0answers
281 views

Smooth 4-manifolds with $E_8$ intersection form

Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on ...
5
votes
1answer
335 views

Are there spaces in which there are no fibered knots?

I am looking for orientable closed 3-manifolds in which there are no fibered knots. Although I know little about this, I think for links the answer to the question above is "no", and the result is ...
2
votes
0answers
135 views

3-manifold rigidity?

Defintion: a $n$-manifold $M$ is said rigid if any homotopy equivalence $M\rightarrow N$ is homotopic to a homemorphism, where $N$ is an $n$-manifold. The sphere $S^{3}$ and hyperbolic compact ...
7
votes
1answer
144 views

Does every embedded 2-sphere in $\mathbb{R}^n$ bound an embedded ball?

Fix $n \geq 3$ and let $S \subset \mathbb{R}^n$ be a smoothly embedded $2$-sphere. Must there exist a smoothly embedded $3$-ball $B \subset \mathbb{R}^n$ such that $\partial B = S$? This is true for ...
2
votes
0answers
147 views

Exotic actions of hyperbolic groups

Let $G$ be a hyperbolic group acting faithfully on $\mathbb{Z}$ such that: The action is highly transitive - it is $k$-transitive for each $k \in \mathbb{N}$. For every quasiconvex subgroup $H \leq ...
4
votes
0answers
98 views

Sampling from a Manifold

Suppose we were to obtain a uniform sample, $S=\{x_1,...,x_m\}$, of points on a closed Riemannian $n$-manifold $M$. Let $\Gamma(S)$ be the set of all geodesics between the points in $S$ and we are ...
9
votes
0answers
177 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 ...
-1
votes
0answers
87 views

Fundamental group of connected sum for non-orientable manifolds [migrated]

For orientable manifolds, $\pi_1(M\#N)=\pi_1(M)*\pi_1(N)$, fundamental group of connected sum is free product of fundamental groups. As far as I understand, for non-orientable manifolds connected sum ...
3
votes
1answer
277 views

Knots in 3-manifolds

Consider a closed $3$-manifold $M$ and a knot $K$ in $M$. Is it necessarily true that $\pi_2 (M \setminus K) = 0$? If not, are there any conditions on $M$ and/or $K$ to ensure the above 2nd homotopy ...
3
votes
1answer
270 views

Punctured 3-manifold

Suppose we have a closed 3-manifold $M$, not necessarily simply connected. What can I say about the homotopy groups of $M \setminus \text{pt}$? ($M$ punctured by one point) In particular, what ...
5
votes
1answer
238 views

Is there a Riemann existence theorem for orbifolds?

For smooth algebraic varieties $X$ over $\mathbb{C}$, the Riemann existence theorem establishes an equivalence of categories between the category of finite etale covers of $X$ and finite unramified ...
4
votes
1answer
322 views

Cap product à la Poincaré

Recently, it became apparent to me that I was not the only one who always first thought in terms of cap product before actually computing a cup product. There is no denying this is evil, but I found ...
8
votes
2answers
174 views

Intersection Cohomology and $L^2$ cohomology

In the study of singular spaces, topological methods like intersection cohomology have played an important role. They have led to the development of technology like perverse sheaves and these find ...
9
votes
4answers
639 views

Geometry of the space of circles in the Euclidean plane

We know that Mobius transformations, $z\to\frac{az+b}{cz+d}$, permutes circles and lines in the Euclidean plane, $(\mathbb{R}^2, dx^2 + dy^2 )$. It may even be possible to write an explicit formula ...
0
votes
1answer
104 views

Symplectic Submanifolds of Contact Manifolds and Contact Submanifolds of Symplectic Manifolds

We know every contact manifold admits a symplectic submanifold, e.g., by Giroux's bijection : if $\omega$ is a contact form for a 3-manifold $M^3$ , then $d \omega$ is symplectic on the fibers of the ...
5
votes
1answer
86 views

Continuity of taking collapse maps

Let $U$ and $V$ be open subsets of $\mathbb R^n$ and let $\mathrm{OEmb}(U,V)$ denote the space of open embeddings of $U$ into $V$ with the compact-opent topology. Let $\bar{U},\bar{V}$ denote their ...
0
votes
1answer
152 views

Principal bundle associated to a fiber bundle

Let $\pi : E\to B$ be a fiber bundle with fiber $F$ over a finite complex $B$ whose structure group is a compact Lie group $G$. How can we determine the principal $G$-bundle associated to $\pi$? For ...
5
votes
1answer
92 views

Does there always exist a sequence of handle moves between handle decompositions that does not increase index? (+ ref. request)

Reference request: Firstly, I'm looking for a proof of the following well-known result about handle decompositions: ($\ast$) Given two handle decompositions of a smooth $n$-manifold $M$, there ...
3
votes
1answer
375 views

What is wrong with the “naive” proof of the Hauptvermutung?

The Hauptvermutung is the statement that any two PL structures on a topological space have a common refinement. It is false in general, but (I think) true for some low dimensional manifolds. The ...
0
votes
0answers
60 views

Upper bound on number of cells created by varieties of co-dimension 1

Say I have polynomials $p_1,p_2,\dots,p_m$ in $\mathbb{R}^n$ (ie. over $n$ variables), each of degree $d$. Is there an upper bound on the number of "regions" created by the surfaces $p_i = 0$? Let's ...
6
votes
0answers
109 views

Do commuting homeomorphisms of the $2$-disk have a common fixed point?

Problem 2.20 (attributed to Lima) in Kirby's list of unsolved problems in low-dimensional topology asks: Are there commuting homeomorphisms of the $2$-ball $B^2$ without a common fixed point? ...
13
votes
1answer
522 views

Is any connected fibre of a fibration of a sphere also a sphere?

Is it known whether, for any fibration of a sphere with connected fibres, the fibre has to be a sphere? $S^1$, $S^3$ or $S^7$?
3
votes
1answer
222 views

Generalization of Giroux's Theorem for Higher Dimensions?

Just wanted to know if Giroux's theorem for 3-dimensional contact manifolds can be generalized: In contact geometry for manifolds of dimension 3 , we have Giroux's theorem , stating that for any ...
0
votes
0answers
36 views

Is there a straightforward way to define a differentiable structure on a localic manifold?

I'd ideally like a categorical definition of differentiability that can then be trivially translated into locales. Barring this, I'm still interested in whether the notion make sense for locales.
2
votes
0answers
113 views

Jones Polynomial as an Equivariant Index

In Khovanov homology the Jones polynomial of a given link is computed as the graded Euler characteristic of a certain complex associated with the link. From other side the Atiyah-Singer theorem ...
1
vote
2answers
155 views

Handle body of 3-manifold with boundary

We know from Morse theory that smooth manifold(with or without boundary) is a handlebody. However, I found a paper "Three-dimensional manifolds with boundary of nonnegative Ricci curvature" by Ananov, ...
2
votes
1answer
107 views

Does $A_{j,k}$ commute with all its conjugates in homotopy braid groups?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group. I am ...
2
votes
0answers
101 views

Teichmuller geodesics vs. geodesics in the hyperbolic plane

Geodesics in $\mathbb H^2$ have the following properties: For every two points in the plane there exists a unique geodesic joining them. Every geodesic determines exactly two points on the ...
2
votes
0answers
142 views

Example of symplectic 4-manifolds with no Lefschetz fibration structure?

I just read about Donaldson's result on existence of Lefschetz pencil structure on symplectic manifolds (Donaldson 1999). However, one has to blow up the base locus to get a Lefschetz fibration ...
1
vote
0answers
63 views

Kontsevich integral for 2-bridge knots

Are there any articles that explain a formula for Kontsevich integral of 2-bridge knots?
19
votes
0answers
256 views

Can one properly embed a differential manifold into numerical space of double dimension? [duplicate]

If $X$ is a $ C^\infty$ differential manifold of dimension $n$, then there exists an embedding $f:X\to \mathbb R^{2n+1}$. This is a not too difficult theorem due to Whitney, proved in many textbooks. ...
2
votes
1answer
193 views

Counting Ribbon graphs

Let $G$ be a ribbon graph (sometimes called fat graph) with $v$ vertices and $e$ edges. Furthermore each vertex is of degree $d$. Q) What is the number of $G$ with the above properties? I mean does ...
8
votes
2answers
209 views

Configuration space like subspace of sphere product

For $k \geq 2, n \geq 1$ let $$M^{n,k} = \{(x_1,\dots,x_k) \in S^n \times \dots \times S^n \ | \ x_1 + \dots + x_k = 0\}$$ This is a compact CW-complex and almost, but not quite, a manifold. I ...
4
votes
0answers
185 views

Topology of the space of foliations on a 3-manifold

Denote by $\mathcal{P} (M)$ the space of smooth plane fields(oriented and transversely oriented) on a given closed and orientable 3-manifold $M$ with the $C^{\infty}$ topology, and by $\mathcal{F}(M)$ ...
2
votes
3answers
422 views

Is true that $\left[\frac{\hat{A}(\mathbb HP^m)} { \hat{M}(\mathbb HP^m) }\right]_{4m} = 0$?

There are two questions: How to prove that in general $[\hat{A}(\mathbb HP^m)]_{4m} = 0$ It is possible to verify it for low values of $m$. How to prove that in general ...
0
votes
0answers
107 views

An integrality theorem for immersions of quaternion projective spaces in the euclidean space

There are three questions: Please let me know your proof of the following theorem: If $\mathbb HP^2$ can be immersed in $\mathbb R^{12}$ with an Euler class $W_{4}(\nu)$ for the normal bundle of ...
0
votes
1answer
161 views

An integrality theorem for immersions of complex projective spaces in the euclidean space

There are three questions: Please let me know your proof of the following theorem: If $CP^3$ can be immersed in $R^8$ with an Euler class $W_{2}(\nu)$ for the normal bundle of $CP^3$ respect to ...
2
votes
1answer
104 views

Open Books $( \Sigma, \Phi) $ living in Lefschetz Fibrations over the disk $D^2$

I have a question about open books and Lefschetz fibrations over the 2-disk $D^2$. Please let me set it up first, before going on. Setup: Say we have a Lefschetz fibration $f: W^4 \rightarrow D^2 $ ...
5
votes
0answers
195 views

Gompf's invariant of $2$-plane fields

I am interested in low dimensional contact topology. These days I read "Handlebody construction of Stein surfaces" written by R. E. Gompf, and study an invariant $\theta (\xi)$. This invariant is ...
3
votes
1answer
159 views

Immersed versus embedded surfaces representing a same homology class

I am working on the Gromov norm of submanifolds in the total space E of surface bundles over surfaces. So I am interested in knowing the minimal genus of a surface representing a given homology class ...