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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9
votes
3answers
344 views

Quotient of the hyperbolic plane with respect to commutator group of $\pi_1(\Sigma_g)$

Let $\Sigma_g$ be a Riemann surface of genus $g\geq 2$ and $G=\pi_1(\Sigma_g)$. Let $\pi\colon \mathbb{H}\to \Sigma_g$ be the universal covering map. What kind of surface is $\mathbb{H}/[G,G]$? ...
5
votes
1answer
197 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
285 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 ...
39
votes
1answer
4k views

Open map D⁴ → S²

Is it possible to construct an embedding $D^4\hookrightarrow S^2\times \mathbb R^2$ such that the projection $D^4\to S^2$ is an open map? Here $D^n$ denotes closed $n$-ball. An open map D⁴ → S². It ...
7
votes
1answer
124 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
605 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 ...
5
votes
1answer
81 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 ...
8
votes
2answers
387 views

Number of the Reidemeister moves needed to transform one diagram into another one

A recent question Random Reidemeister moves to unknot contains a link to the paper http://www.ams.org/journals/jams/2001-14-02/S0894-0347-01-00358-7/S0894-0347-01-00358-7.pdf, in which J. Hass and J. ...
10
votes
0answers
327 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: ...
0
votes
1answer
92 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 ...
3
votes
0answers
270 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
1answer
137 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 ...
0
votes
0answers
59 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 ...
5
votes
1answer
79 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 ...
6
votes
0answers
102 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
504 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
159 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
34 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.
37
votes
2answers
1k views

Can knot diagrams be monotonically simplified using under moves?

It is well known that knot diagrams cannot be monotonically simplified using Reidemeister moves. For instance, the Goeritz unknot cannot be directly simplified. On the other hand, there is a stronger ...
2
votes
0answers
102 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 ...
2
votes
1answer
104 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 ...
4
votes
2answers
589 views

A question about Marsden-Weinstein reduction theory

Let $G$ ba a compat Lie group and $\frak g$ be its Lie algebra, then by Marsden-Weinstein reduction theory we know that if we take $M=T^*G$ and $J:M\to \frak g^*$ be its moment map then the reduced ...
0
votes
1answer
246 views

A problem related to connectivity of analytic functions

Let $f(z)\in A(\mathbb D)$, where $A(\mathbb D)$ is the space of analytic functions on the open unit disk $\mathbb D$ and continuous on $\overline{\mathbb D}$. Question: Is the connectivity of ...
1
vote
2answers
143 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
3answers
416 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 ...
1
vote
1answer
111 views

Singular leaf of Strebel differential

Let $R$ be a Riemann surface.Let $\gamma$ be a loop which is non-trivial in $H_{1}(R,\mathbb{Z})$. By the Jenkins–Strebel Theorem we know the following: there exists a holomorphic quadratic ...
6
votes
0answers
259 views

Presentation of Homotopy Pure Braid Group?

Let $\tilde{B_n}$ be the homotopy braid group; namely, in the deformation of braids, a braid string is allowed to self-intersect itself. Similarly let $\tilde{P_n}$ be the homotopy pure braid group. ...
2
votes
0answers
96 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
126 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
61 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
250 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
188 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
199 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
176 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 $\mathcal{F}(M)$ ...
0
votes
0answers
103 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 ...
24
votes
3answers
2k views

A “meta-mathematical principle” of MacPherson

In an appendix to his notes on intersection homology and perverse sheaves, MacPherson writes Why do we want to consider only spaces $V$ that admit a decomposition into manifolds? The intuitive ...
0
votes
1answer
160 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
97 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 $ ...
4
votes
1answer
176 views

Are homotopy braid groups residually nilpotent?

A group is called residually nilpotent if given any non-identity element, there is a normal subgroup not containing that element, such that the quotient group is nilpotent. It is known that pure braid ...
5
votes
0answers
187 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
143 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 ...
-2
votes
1answer
85 views

Definition of Milnor exact sequence and complex-oriented generalized cohomology of $\mathbb{C}P^{\infty}$

Consider a complex-oriented multiplicative generalized cohomology theory $h^{*}(X)$. It is complex-oriented, if by the definition the following two conditions hold: 1) There exists an element $t\in ...
6
votes
2answers
311 views

Surgery diagram for the Seifert-Weber space

In every reference I see, the Seifert-Weber space is presented as an identification space (specifically identify opposite faces of a dodecahedron by a 3/10ths twist). What I can't seem to find is a ...
4
votes
2answers
440 views

HNN extensions which are free products

which HNN-extensions are free products? this question is related with another still unsolved about Nielsen-Thruston-reducibility and connected-sum-irreducibility of 3d-torus- bundles...
4
votes
0answers
92 views

Existence of a torus fibration with given vanishing cycles

Suppose I have a torus fibration over the disc with $n$ nodal singular fibers $F_1,\dots,F_n$ over the points $p_1,\dots,p_n$. I was specifically thinking about a Lagrangian fibration, but I'd be ...
11
votes
1answer
525 views

Thurston geometries in dimension 4

In the sense of W. Thurston here, there is 3 geometries in dimension 2 and there is 8 geometries in dimension 3. Question: How many different geometries (in the sense of Thurston) do we have in ...
0
votes
1answer
101 views

Unseparability of two linked rings in higher dimensions [closed]

I am not familiar with topology. We know that in $R^3$, we cannot separate two "rings": two copies of $S^1$, if they are "linked". I wonder that is there any similar results for two copies of ...
14
votes
1answer
259 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 ...
1
vote
1answer
124 views

Topology of surfaces and mean curvature

The Gauss-Bonnet theorem characterizes topology of surfaces by their Gaussian curvature. Do there exist results characterizing topology of surfaces embedded in $\mathbb{R}^3$ by their mean ...
7
votes
1answer
352 views

Characterisation of Q-rank 1

I'm looking for a reference and/or the original source for the following fact: An irreducible non-uniform lattice in a semisimple Lie group without compact factor has Q-rank 1 if and only if it does ...