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 manifolds,...

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5
votes
2answers
445 views

Can a Morse function define a unique structure on a closed manifold?

I was thinking about the doubt that if $M$ and $N$ are closed manifold and if there exists two Morse function $f$ and $g$ respectively on $M$ and $N$ with the following property that they both have ...
7
votes
1answer
156 views
+50

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure?

Does $\mathbb{CP}^2$ admit a Riemann surface lamination structure? Every paper or article I looked at, talk only about singular laminations on $\mathbb{CP}^2$. I was wondering why. If you know ...
5
votes
2answers
196 views

Generalizations of the handle trading techniques

As Theorem 8.1 in "Lectures on the h-cobordism theorem (written by J.Milnor)" show, we can choose a handle decomposition of cobordism (satisfying some connectivity and dimensional assumptions) with no ...
14
votes
1answer
307 views

Are homology spheres stably trivial?

A homology sphere is a closed smooth $n$-dimensional manifold with the same homology groups as $S^n$. Igor Belegradek's answer to a previous question of mine shows that the smoothness hypothesis is ...
1
vote
1answer
186 views

free group actions on a contractible topological space [closed]

Let $\Sigma_k$ be the symmetric group on $k$-letters. Let $W$ be a contractible topological space with a free $\Sigma_k$-action (from the left). Let $X$ be a $CW$-complex and let $X^k$ be the ...
6
votes
2answers
225 views

Homeomorphic but Non-Conjugate Mapping Tori

Suppose we fix a genus $g$ closed surface $S$. Let $f, g \in Map(S)$ be conjugate, for $Map(S)$ the mapping class group of $S$. Then I know that $M_f$ (the mapping torus of $M$ with monodromy $f$) is ...
2
votes
1answer
247 views

induced group actions and covering maps on Eilenberg-Maclane space

Let $M$ be a finite $CW$-complex. Let $\Sigma_k$ be the symmetric group acting on $k$-letters. Suppose there is a free action of $\Sigma_k$ on $M$. Then we have a covering map $$ f:M\to M/\Sigma_k. ...
6
votes
0answers
91 views

mod $p$ homology module of unordered configuration spaces of the projective plane

Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
5
votes
0answers
107 views

Essential surfaces in knot complements

Given any knot $K \subset \mathbb{S}^3$, one can find a closed oriented embedded surface $S$ such that $K \subset S \subset \mathbb{S}^3$. Moreover, pick such an $S$ that has minimal genus. One can ...
2
votes
0answers
120 views

Gaussian Integrals and Pseudo-Anosov Maps

The hep-th section of arXiv if often filled with beautiful semi-rigorous computations on Mathematics. However sometimes it is very difficult to understand what is being stated. Here I take from: ...
2
votes
1answer
107 views

homeomorphism type of punctured real projective spaces

Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{*\}$ be the punctured space. I observe: $\mathbb{R}P^2\setminus\{*\}$ is homeomorphic to a (open) ...
8
votes
2answers
168 views

Interior periodic points of area preserving homeomorphisms of a pair of pants

A celebrated result of Franks shows that any area preserving homeomorphism of the closed annulus $A$ with at least one periodic point (possibly along the boundary) has infinitely many interior ...
0
votes
0answers
92 views

asymptotic behavior of minimum dilatations on punctured surfaces

Let $l_{g,n}$ be the logarithm of minimum dilatation for pseudo-Anosov homeomorphisms on surface of genus $g$ with $n$ punctures. Let $n$ be fixed and $g$ varies. Is the asymptotic behavior of $l_{g,n}...
2
votes
2answers
286 views

Is it true that all sphere bundles are some double of disk bundle?

Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you ...
11
votes
1answer
259 views

Does every dimension $n\geq4$ admit a manifold with an exotic smooth structure?

It is known that $\mathbb{R}^4$ has exotic smooth structures, and there are many such examples in higher dimensions, such as the famous 7-sphere. My (probably very naive) question is, for every $n\...
5
votes
1answer
349 views

Are two equivariant maps between aspherical topological spaces homotopic?

Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi_1(X)$ acts on the universal cover of $X$ via deck transformations, and on ...
15
votes
1answer
306 views

Classification of fake (quaternionic, octonionic) projective spaces

If $X$ is a closed $n$-manifold, a fake $X$ is another closed manifold homotopy equivalent to $X$. There is some interest in classifying manifolds (up to, say, homeomorphism) homotopy equivalent to a ...
7
votes
2answers
568 views

Geometric or topological results from group theory

Do you know interesting examples of purely geometric or topological results which can be proved using group theory? To make precise what I have in mind, let us consider the two following examples: ...
1
vote
0answers
48 views

Change of length of curve when Fenchel-Nielsen length coordinate increase

Let $F$ be a hyperbolic surface of finite type. Suppose $\alpha$ is a simple closed geodesic and $\beta$ is any closed geodesic intersecting $\alpha$. Consider a Fenchel-Nielsen coordinate of the ...
0
votes
1answer
70 views

Sequence of translation surfaces and length of saddle connections

A sequence of translation surfaces $(X_n,\omega_n)$ is said to "go to infinity" if it leaves every compact set in the space of translation surfaces as $n$ goes to infinity. I know that this is ...
2
votes
1answer
167 views

Prove that a metric space is intrinsic

Let $(X,d)$ be a general locally compact metric space (in particular not a Riemannian manifold). Suppose we don't know if $(X,d)$ is complete. To prove $(X,d)$ is intrinsic. I have to compute the ...
4
votes
0answers
66 views

Points of failure in definition of X- and A-moduli spaces for arbitrary G

In their work [0] on defining notions of higher Teichmüller space for local systems on surfaces, Fock and Goncharov require split reductive Lie groups, and sometimes also require simple-connectedness. ...
1
vote
1answer
206 views

Addition of two homology classes is zero in construction of Poincare Sphere

I ask here the question since it hasn't been answered in Math Stack Exchange. I am working through Greenberg and Harper, Lecture notes on Algebraic Topology, and I am having trouble with one ...
1
vote
0answers
87 views

Does the link of a hypersurface singularity determine its analytic type?

Consider a hypersurface $V(f) \subseteq \mathbb{C}^{n+1}$ with an isolated singularity at the origin. If $L := V(f) \cap S^{2n+1}_\epsilon$ is the link of $V(f)$ (with $S^{2n+1}_\epsilon$ a ...
3
votes
1answer
90 views

Morse function on slicing disk complement determines ribbon?

It is well-known that given a ribbon knot and the corresponding slicing disk in the 4-ball, the distance function (maybe squared) to the origin defines a Morse function in the complement of the ...
4
votes
1answer
101 views

Non-Haken hyperbolic 3-manifolds without nonorientable surfaces

It is well known that there exist infinitely many (non homeomorphic) non-Haken closed hyperbolic 3-manifolds. These can be obtained for example doing Dehn surgery on the figure eight knot complement. ...
8
votes
0answers
158 views

Holomorphic contractibility of GL(H)?

Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later ...
4
votes
2answers
159 views

Looking for examples of large hyperbolic two-generator knots or 3-manifolds

We say a knot $K$ in $S^3$ is small if its exterior contains no closed properly embedded incompressible surfaces and we say $K$ is large otherwise. Does anyone know of an example of a large ...
10
votes
2answers
180 views

Triangulation with simplices of same volume

Let $M$ be a Riemannian smooth compact manifold. It is known that $M$ has a triangulation, for any dimension. But do we know if there exists a triangulation such that all simplices have same volume ? ...
6
votes
1answer
248 views

“structure group” for fibration

Regarding "fibration" as a homotopy analogue of "fiber bundle",I want to see parallel notions of "structure group" and "fiber change" in "fibration". Does it make sense to talk about "structure group"...
3
votes
0answers
101 views

Can we use the “size” of smooth structure set to predict the information geometry or other topological information?

The "size" can mean the number of elements or the diameter of the set of smooth structures. Y. Shikata defined a distance function on it and proved that it is a distance. He then used it to prove that ...
3
votes
1answer
107 views

What is the Heegaard Floer Homology of a connect sum of $S^2 \times S^1$s?

There are many conjecturally-equivalent three-manifold Floer homologies, of which my understanding is the most-computable is Heegaard Floer homology. What is the (Heegaard) Floer homology of a ...
20
votes
2answers
548 views

Is there a smooth manifold which admits only rigid metrics?

Does there exist a (finite dimensional) smooth manifold $M$, such that every Riemannian metric on $M$ has no isometries except the identity? Of course, such a manifold must not admit a diffeomorphism ...
3
votes
1answer
188 views

Is there a degree one map from a product $B\times S^1 \to \#_n S^2 \times S^1$ for any n

For any $n \geq 1$, let $\Sigma_n$ denote the closed orientable surface of genus n. In http://arxiv.org/abs/1202.6302, the authors showed that for any $n$, there is a degree two, $\pi_1$-surjective, ...
5
votes
1answer
107 views

Are square tiled surfaces dense in the moduli space of translation surfaces?

I'm reading the survey "An introduction to Veech surfaces" by Pascal Hubert and Thomas Schmidt. At page 19 they state "In any fixed stratum, the set of square-tiled surfaces of that stratum is dense....
5
votes
0answers
180 views

Does $\#_n S^2×S^1$ really admit a map of non-zero degree from $B×S^1$?

In Proposition 4 on page 6 of this paper, http://arxiv.org/abs/1202.6302, the authors claim to produce a degree 2 $\pi_1$-surjective map $f$ between $M=S^1 \times \Sigma_2$ and $N=\#_2 S^2 \times S^1$ ...
3
votes
1answer
173 views

Is the number of intervals you get by slicing the closed region under a nice graph continuous from below?

Let $I$ be a closed interval in $\mathbb{R}$, and let $f:I \to \mathbb{R}$ be a bounded function, smooth except at finitely many points (piece-wise smooth). There are also only finitely many critical ...
6
votes
1answer
137 views

Can a surface group act on a finite-valence simplicial tree?

Question. Let $S$ be a closed surface of genus $> 1$. Can $\pi_1(S)$ act faithfully and minimally on a simplicial tree of finite valence? Here "minimal" means that there is no invariant sub-tree. ...
11
votes
1answer
289 views

Are there non-smoothable homotopy/homology spheres?

A homotopy sphere is a topological $n$-manifold $M$ which is homotopy equivalent to $S^n$. A homology sphere is a topological $n$-manifold $M$ such that $H_i(M) \cong H_i(S^n)$ for all $i$. Note, by ...
23
votes
2answers
698 views

Why is it true that if two 4-manifolds are homeomorphic then their squares are diffeomorphic?

Near the top of the second page of this paper, it is claimed that if two 4-manifolds $X$ and $Y$ are homeomorphic, then their squares $X \times X$ and $Y \times Y$ are diffeomorphic. Why is this true? ...
1
vote
0answers
59 views

Laplace eigenvalue and floer theory

I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow. Any ...
0
votes
0answers
20 views

Extending a symplectomorphism of a surface with boundary, properly embedded in a 4-disk, to the ambient space

Let $S$ be a surface with boundary that is properly embedded, via a map $f$ say, inside a $4$-disk. When is the embedding isotopic to the identity? -- All embeddings and isotopes are in the symplectic ...
9
votes
1answer
274 views

Elements of infinite order in the topological mapping class group

Let $M$ be a closed topological manifold, and let $\operatorname{MCG}(M):=\operatorname{Homeo}(M)/\operatorname{Homeo}_0(M)$ denote the topological mapping class group of $M$ ($\operatorname{Homeo}_0(...
8
votes
1answer
172 views

Are knots determined by their complements within a homotopy class?

Suppose $M$ is a closed 3-manifold and $K_1,K_2$ are two homotopic knots in $M$. That is, they are two embeddings $f_1,f_2\colon S^1 \to M$ such that there exists a homotopy $h\colon S^1 \times [0,1] ...
10
votes
1answer
187 views

Open (resp., closed) balls homeomorphic to open (resp., closed) discs on the plane

Let $\Sigma$ be a compact (smooth) surface, with a geodesic metric $d$ (compatible with the topology of $\Sigma$). Let $x \in \Sigma$, and suppose you have the following: for every $r<1$, the open ...
7
votes
0answers
163 views

Biholomorphic neighborhoods of the boundary of Stein domains

Let $(X_1,J_1)$ and $(X_2,J_2)$ be Stein domains with the same contact boundary $(Y,\xi)$. Under what conditions does there exist a biholomorphism between a neighborhood of their respective boundaries ...
5
votes
0answers
89 views

$H(M)$ necessarily highly non-integrable, i.e. forms contact structure?

Let$$M^{2n - 1} = \{z \in \textbf{C}^n : \textbf{h}(\textbf{z}, \textbf{z}) \equiv \textbf{z} \cdot \overline{\textbf{z}} = \textbf{1}\}$$be the unit sphere in $\textbf{C}^n$. Consider the real-...
5
votes
0answers
94 views

Images of the $3$-dimensional solvable geometry

This is more of a mathematical art question. Most people are familiar with the mathematical movie "Not Knot", which explores a hyperbolic $3$-manifold. My question is: are there any images along the ...
4
votes
1answer
78 views

Locally nilpotent algebraic section of tangent bundle is complete?

Suppose $X$ is a smooth affine algebraic variety over $\mathbb{C}$ and let $V$ be an algebraic vector field (i.e. an algebraic section of the tangent bundle). If $V$ is locally nilpotent, meaning that ...
21
votes
1answer
316 views

Is the normal bundle of a torus trivial?

Question: Let $T^k \subseteq \mathbb{R}^n$, $ n > k$, be a smoothly embedded $k$-torus. Is its normal bundle trivial? What about the normal bundle of $S^k \subseteq \mathbb{R}^n$, $n > k$, the $...