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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7
votes
1answer
428 views

Geometric interpretation for fourth coeficient of the polynomial for Hirzebruch–Riemann–Roch theorem

Hey guys :) I have a question on a theorem which is known as Tian-Yau-Zelditch theorem now. If there is some reference for following question, please let me know Let $(M,\omega)$ be a compact ...
1
vote
1answer
156 views

When $\frac{\text{Aut}(G/P,L)}{S^1}$ is discrete?

Let $(M,\omega)$ be a Kähler manifold with a pre-quantum Line bundle $L$ and $\text {Aut}(M,L)$ means the group biholomorphisms of $M$ which lift to holomorphic bundles maps $L\to L$. My question is ...
7
votes
0answers
100 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, ...
3
votes
1answer
127 views

Construction of a Bott manifold

I have been searching the literature for a construction of a simply connected spin manifold of dimension 8 with A-genus 1. I am not sure, but I think this is called a Bott manifold. Can anybody help ...
13
votes
2answers
341 views

Homotopy groups of spaces of embeddings

Let $\mathrm{Emb}(M, N)$ be the space of smooth embeddings of a closed manifold $M$ into a manifold $N$ equipped with smooth compact-open topology. Question 1. Are there conditions ensuring that ...
3
votes
1answer
125 views

Does every canonical decomposition of the intersection form come from a canonical homology basis?

Take a closed surface $X$ of genus $n$. By a canonical homology basis, I will mean a set of $2n$ homologically independent simple closed curves $\{\alpha_1,\ldots,\alpha_n,\beta_1,\ldots,\beta_n\}$, ...
1
vote
1answer
112 views

Structures on open surfaces

Let $\phi\in PSL(2,R)$ be hyperbolic and $\varphi\in PSL(2,R)$ be elliptic. Is it possible to find a local homeomorphism $f:H^2\rightarrow H^2$ such that $f(\phi(x))=\varphi(f(x))$ for all $x\in H^2$ ...
3
votes
2answers
156 views

Extending homotopy equivalence between manifolds with boundary

Let $f:N_1\to N_2$ be a homotopy equivalence between two simply connected manifolds with boundary of the same dimension. Can it be extended to a homotopy equivalence between closed manifolds $f:M_1\to ...
2
votes
3answers
227 views

Heegaard Floer Homology of double branched cover

The question is the following: Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...
4
votes
2answers
328 views

Jones polynomial of the concatenation of two braids

Let $\sigma_1$ and $\sigma_2$ be two braids with $n$-strings. Are there any formulas relating $J_{\widehat{\sigma_1\sigma_2}}(q)$, $J_{\hat{\sigma_1}}(q)$, and $J_{\hat{\sigma_2}}(q)$? Here, ...
10
votes
1answer
311 views

Did Milnor and Thurston write anything else about characteristic numbers for 3-manifolds?

In Characteristic numbers for 3-manifolds Milnor and Thurston define a characteristic number and this is cited in ch. 6 of Thurston's notes when discussing the Gromov approach to Mostow rigidity. The ...
2
votes
1answer
114 views

A smooth analog of the mapping cylinder?

Let $f:X\to Y$ be a homotopy equivalence between closed smooth manifolds. Is there a closed manifold $Z$ such that $X$ and $Y$ are (strong) deformation retracts of $Z$? (There is the mapping ...
3
votes
0answers
162 views

Action of the mapping class group on separating spheres in a connected sum of aspherical 3-manifolds

Let $M$ be a connected sum of $g$ closed aspherical 3-manifolds $M_1, \ldots, M_g$. [Update: I also assume that all the $M_i$-s are diffeomorphic, i.e. $M$ is a connected sum of copies of the same ...
1
vote
0answers
221 views

Functors with Mayer-Vietoris Sequences

Let $F$ be a contravariant functor from some category of spaces (e.g. smooth manifolds or (compact?) topological Hausdorff spaces), to Abelian groups. Assume that for any open sets $U, V \subseteq X$ ...
2
votes
1answer
59 views

Quasiconformal deformation

Given a finite set $A$ on the Riemann sphere and a homeomorphism $f$, may I say there exists a quasiconformal homeomorfism isotopic to $f$ relative to the set $A$?
1
vote
1answer
81 views

Sutured Manifolds and minimal genus

Is there a result relating sutured manifolds and surfaces of minimal genus? perhaps someone has a very clever point of view of these two notions that can share. In other matters, do we know how to ...
3
votes
1answer
124 views

Parity of knot signatures

I recently came upon a recursive formula for the (ordinary) signatures of torus knots. The formula, which I found in Murasugi's book "Knot Theory and Applications" (Springer, 2007), originally ...
5
votes
0answers
81 views

Distribution of random Projections on the Stiefel Manifold

Imagine we have an $m$-Frame in $\mathbb{C}^n$ uniformly distributed on the stiefel manifold. That is: $\mathbf{U}\in\mathbb{C}^{n\times m}$, $m<n$ and $\mathbf{U}^H\mathbf{U}=I$ and $U$ has a Haar ...
12
votes
2answers
872 views

Does there exist a closed manifold that can be given both a Euclidean and a Hyperbolic structure?

I originally asked this on math.stackexchange, where I asked if there could exist a closed manifold that could be given different geometric structures of constant curvature (not at the same time, of ...
19
votes
1answer
450 views

good covers of manifolds

It is well-known and easy to prove (see for instance this post) that every smooth manifold admits a "good cover", i.e. a locally finite cover by open balls such that all nonempty intersections of the ...
2
votes
1answer
132 views

homology of punctured manifolds

Let $M$ be an $n$-dimensional closed manifold. Choose $x \in M$. Using long exact sequence of pairs $(M,M - x)$, we have $$H_k(M - x, \mathbb{Z}) \cong H_k(M, \mathbb{Z})$$ for $k<n-1$. For ...
3
votes
2answers
141 views

Circle Bundles of surfaces

Let S be a surface with a metric of constant curvature and finite area. Is there a classification of the circle bundles over S?
4
votes
2answers
300 views

a question on rank of fundamental group

Assume $G$ be the fundamental group of a closed orientable hyperbolic 3-manifold. Let $G_{1} = \langle a_{1},...,a_{k} \rangle$ be a free subgroup of $G$, and let $G_{2}=\langle a_{k+1} \rangle$ be a ...
2
votes
0answers
35 views

Tait conjectures for alternating w-links

The Tait Conjectures are useful in knot tabulation. For alternating knots and links, two of them state: Any reduced diagram of an alternating link has the fewest possible crossings. Any two reduced ...
10
votes
1answer
305 views

Finitely presented group in which every element is conjugate to its square

Does there exist a nontrivial finitely presented group in which every element is conjugate to its square? Is this an open problem? Motivation: Jahren proved in [Geom Dedicata (2010)] that if $M$ is a ...
2
votes
2answers
104 views

linking number and covering

Recently I read Dale Rolfsen's paper –A surgical view of Alexander’s polynomial. This is a good paper. But there is a lemma which I don’t know how to prove.The lemma is following: Lemma:In the ...
6
votes
0answers
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 ...
2
votes
2answers
203 views

manifolds with unusual rational cohomology rings

I'm looking for examples of 3-manifolds with unusual rational cohomology rings. I'm curious about what the cup product structure can actually look like, and I'd like some examples to play with. Does ...
9
votes
2answers
477 views

The Alexander polynomial of a slice knot, Reidemeister tosion, Whitehead group

My question is about the Alexander polynomial of a slice knot. For a slice knot $K$, Fox-Millnor and Terasaka proved that $$ \Delta_{K}(t) \doteq f(t) f(t^{-1})$$ for some polynomial $f(t) \in ...
7
votes
1answer
113 views

Relation between the Alexander module of a link and intermediate free abelian covers

I'm working through McMullen's paper "The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology" and have a question concerning the following setup: Given a link complement $(X, ...
4
votes
1answer
415 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 ...
2
votes
1answer
174 views

The behaviour of holomorphic mapping of curves

Given a polynomial, (or rational function, transcendental entire (meromorphic) function) $f$, and a smooth closed Jordan curve $\gamma$, can we give a complete description of the image of $f(\gamma)$ ...
1
vote
0answers
48 views

Groups of equi-quasi-isometric diffeomorphisms of a Riemannian surface of bounded geometry

Let $M$ be an open Riemannian surface of bounded geometry. Let $\Gamma$ be a group of diffeomorphisms of $M$. Suppose that $\Gamma$ is equi-quasi-isometric; i.e., its elements are (differentiable) ...
1
vote
0answers
97 views

Combinatorial Pin structure

David Cimasoni and Nicolai Reshetikhin have a paper on the combinatorial description of spin structure http://arxiv.org/abs/math-ph/0608070, where it shows the equivalence of spin structure to the ...
4
votes
1answer
104 views

Rationality of translation lengths in hyperbolic groups

Recall that the translation length $\tau(g)$ of an element $g \in G$ is the limit $d(1, g^n)/n$, where $d$ is the word metric on $G$ with resepct to some generating set. It is a theorem of Gromov ...
7
votes
1answer
239 views

Casson invariant

Part of the definition of the Casson invariant is that if you have an integer homology sphere $\Sigma$ and a knot $k,$ then $$\lambda(\Sigma + \frac{1}{m} k) - \lambda(\Sigma + \frac{1}{m+1} k)$$ does ...
7
votes
2answers
202 views

Are there infinitely many commensurable classes of finite-covolume hyperbolic Coxeter groups?

Allcock(2006) proved that there are infinitely many finite-covolume (resp. cocompact) Coxeter groups acting on hyperbolic space $H^n$ for every $n\le 19$ (resp. $n\le 6$). His main technique of ...
9
votes
2answers
370 views

Vector bundle for prescribed Stiefel-Whitney classes

I hope this is not trivial. Let $B$ be a nice topological space (paracompact, CW-complex or whatever you think is nice) For $i=1,\ldots,n$ let $x_i \in H^i(B,\mathbb{Z}_2)$ be certain cohomology ...
0
votes
0answers
109 views

Manifolds supporting finite order diffeomorphisms (a local construction?)

The following question is mainly inspired by this previous one Which manifolds admit a diffeomorphism of order $n$? and some answers given there. For $d\geq 2$, let $\mathbb{B}^d$ denote the closed ...
2
votes
1answer
197 views

Meaning of “ Open Book cannot be Stabilized Further”?

I'm going over some old notes on Giroux's theorem on the equivalence ( bijection, actually) between open books ( up to positive stabilization) for 3-manifolds and contact structures ( up to isotopy.) ...
1
vote
0answers
113 views

The relations between some 3-components links and trefoil knots [closed]

It is intuitive to see that the 3-components links (under Alexander–Briggs notations) $6^3_1, 6^3_2, 6^3_3$ are closely related to each other; in a sense by doing a cut-gluing or sew-gluing surgery, ...
8
votes
0answers
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 ...
3
votes
0answers
96 views

Cancellations in products of two elements of a hyperbolic group

Let $G$ be a non-abelian free group with the standard generating set and the corresponding word metric. If we take two elements $g,h\in G$ and compute their product $gh$, some letters might cancel, ...
9
votes
1answer
239 views

Immersions of $n$-manifolds in $\mathbb{R}^n$ versus embeddings in $\mathbb{R}^{n+1}$

Let $M$ be a connected noncompact parallelisable smooth $n$-manifold. (By Hirsch-Smale theory, $M$ can be immersed into $\mathbb{R}^n$.) I am interested in the following two properties: P1: $M$ can ...
4
votes
1answer
291 views

What are the low-degree group cohomology of the mapping class group of a surface

Let $MCG_g$ be the mapping class group of genus $g$ closed surface. (Say $MCG_1=SL(2,Z)$). I would like to know what is the group cohomology of $MCG_g$ with coefficients in Z, such as ...
5
votes
1answer
269 views

Finite group acting on sphere

Let $G$ be a finite abelian group (of odd order if it's significant) acting on sphere $S^2\subset\mathbb{R}^3$. So my question: is it true that $G$ has a fixed point?
3
votes
1answer
122 views

In H_2 of Sp(2g,Z), why does Meyer's signature cocycle give 4 times a generator?

Fix some $g \geq 2$, let $\Gamma_g$ be the mapping class group of a genus $g$ surface, and let $\pi : \Gamma_g \rightarrow Sp(2g,\mathbb{Z})$ be the projection. In Meyer, Werner Die Signatur von ...
6
votes
2answers
116 views

On trivalent spines of surfaces

Let $\Sigma$ be a surface with non-empty boundary labelled by finitely many points $x_i$. For our purposes a spine $s$ for $\Sigma$ is a uni-trivalent graph embedded in $\Sigma$ whose 1-vertices are ...
1
vote
0answers
158 views

Does there exist any subspace of R^n, homeomorhic to a manifold but not a C^0 submanifold of R^n?

At first I thought that if a subspace of $\mathbb{R}^n$ is homeomorphic to a manifold, then it is a $C^0$ submanifold of $\mathbb{R}^n$. But I found an asterisked exercise in the book Differential ...
8
votes
2answers
376 views

Covering the space by disjoint unit circles

Sierpinski has proved the following two interesting theorems. Theorem 1. The Euclidean plane $\mathbb{R}^2$ is not a union of nondegenerate disjoint circles. Theorem 2. The Euclidean space ...