3
votes
2answers
33 views

Complement of a solid genus-2-handlebody in $S^3$

I'm not sure if this is a stupid question or not but is the complement of a solid genus-2-handlebody in $S^3$ also a solid genus-2-handlebody? Thanks!
2
votes
1answer
45 views

When gluing maps are isotopic?

Let $M$ and $M'$ be compact orientable connected topological 3-manifolds. (One may need more conditions to answer the question.) Suppose we have two homeomorphisms $f$ and $g$ from the boundary ...
3
votes
0answers
32 views

Existence of orientation-reversing automorphisms

Let $M$ be a connected orientable manifold (I don't really care about which category of manifolds we should work in). Since I read that there are nice manidolds without orientation-reversing ...
4
votes
1answer
70 views

Mapping Class Group

$\newcommand{\MCG}{\mbox{MCG}}$Let $\alpha$, $\beta$ be non-isotopic, non-separating curves on a surface $S$ (meaning that "cutting along " them will not disconnect the surface). How do we show that ...
3
votes
1answer
93 views

For a Cantor set $\mathcal{C} \subset S^3$ such that $\pi_1(S^3 \setminus \mathcal{C})=0$, prove $S^3 \setminus \mathcal{C}$ can be split by a sphere.

I'm working from the paper Cantor Sets in $S^3$ with Simply Connected Complements by Richard Skora. On page 184 the second sentence states that any Cantor set $\mathcal{C} \subset S^3$ such that ...
1
vote
1answer
39 views

Number of connected components of the complement of a closed curve.

Let $\gamma:[0,1] \rightarrow \mathbb{R}^2$ be a continuous, closed curve (i.e. $\gamma(0) = \gamma(1)$). My question is about the number of connected components of the complement $\mathbb{R}^2 ...
3
votes
0answers
38 views

Why are homotopy spheres spin-cobordant in dimensions divisible by 4?

Manifolds are assumed to be smooth having dimensions $\geq5$. As usual, $\Omega^{Spin}_{*}$ denotes the spin bordism ring and $\Omega^{SO}_{*}$ the oriented bordism ring. Let $\Sigma^n$ be a closed ...
1
vote
1answer
76 views

Number of connected components of this complement

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?
1
vote
0answers
63 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
2
votes
0answers
37 views

Notation and shorthand of cobordism G=O, SO, U

This is really a basic question: From Wikipedia: "The basic examples are G = O for unoriented cobordism, G = SO for oriented cobordism, and G = U for complex cobordism using stably complex ...
5
votes
2answers
228 views

Does the Euler characteristic of a manifold depend upon the field of coefficients?

Define the Euler characteristic of a space $X$ to be $$\chi(X)= \sum_i \dim H_i(X, \mathbb Q)$$ This is obviously not necessarily well-defined for an arbitrary space $X$, so let $X$ be a manifold ...
1
vote
0answers
81 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
5
votes
1answer
80 views

Why are the total spaces of two Serre fibrations equivalent when the bases and the fibers are equivalent?

Suppose $B$ is a pointed space and suppose $f\colon E\to B$ and $f\colon E'\to B$ are two Serre fibrations. Let moreover a map $g\colon E\to E'$ be given such that $f=f'\circ g$ which is a weak ...
6
votes
1answer
225 views

Easier proof about suspension of a manifold

For what manifolds $M$ is the suspension $\Sigma M$ also a manifold? By the suspension of a topological space $X$ (not necessarily a manifold), I mean the space $$\Sigma X = (X \times [0,1])/{\sim}$$ ...
0
votes
0answers
58 views

Can Vanishing Cycles be Described as Fibers over Critical Points in a Lefschetz Fibrations?

I'm trying to see if it makes sense to see vanishing cycles in a Lefschetz fibration as the fibers over critical points. A Lefschetz fibration $f: M^4 \rightarrow X$ , where $M^4$ is a smooth ...
0
votes
0answers
25 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
1
vote
0answers
35 views

Open Book Decompositions of 3-manifold and Associated Heegard Splittings

In page 13 of the paper: http://arxiv.org/pdf/math/0510639v1.pdf It is stated that "An open book decomposition (S,h, K) , gives rise to a special Heegard decomposition of M ". Here, S is a surface , ...
0
votes
1answer
80 views

Manifolds Resulting from Gluing Tori

I'm trying to show that if solid tori $T_1, T_2; T_i=S^1 \times D^2$ ,are glued by homeomorphisms between their respective boundaries, then the homeomorphism type of the identification space depends ...
3
votes
1answer
81 views

Why are there cubics in a Calabi-Yau manifold?

I heard from a recent talk that the "number of $n$-degree curves" in a Calabi-Yau manifold is an invariant of the space. But what does that mean? (Specifically I would like to ask the following.) ...
3
votes
3answers
152 views

Framing Integer associated with a Framed Knot/Link

I have been looking for a clear definition of the n-framing of a knot unsuccessfully. A framing here refers to a choice of homeomorphism between a solid torus neighborhood (a.k.a, tubular ...
14
votes
1answer
208 views

Is the torus the union of two connected, simply-connected open sets?

Is the torus the union of two connected, simply-connected open sets? A routine computation with the Mayer-Vietoris sequence shows that if so, then their intersection must have exactly three ...
1
vote
2answers
106 views

Frame bundle of orthonormal frames orthogonal to a submanifold.

Suppose we have a smooth manifold $M$ of dimension $m$ with a Riemannian metric and a connected submanifold $N$ of dimension $n$ in $M$ with $n<m-1$. Let $n\le k<m-1$ and consider the bundle ...
0
votes
0answers
28 views

Intersection of closures of the Schubert cells

How can I determine does two closures of Schubert cells of Grassmannian $Gr(n,m) $ $e(\sigma_1,...\sigma_m)$ and $e(\sigma_1',...\sigma_m')$ intersect or not.
2
votes
0answers
21 views

Edges and genus in graphs

For a planar graph $G = (V, E)$ there is the well known bound $|E| \leq 3|V| - 6$. If instead of $S^2$ $G$ embeds in the orientable surface $S_g$ of genus $2 - 2g$ with minimal $g$, what can be said ...
3
votes
0answers
150 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
2
votes
1answer
112 views

The image of homomorphism of fundamental group of closed surface

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus $\geq 2$. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to ...
4
votes
2answers
473 views

Simply connected does not imply contractible. Is there a nice counter example in $R^2$?

The standard counter example to the claim that a simply connected space might be contractible is a sphere $S^n$, with $n > 1$, which is simply connected but not contractible. Suppose that I were ...
7
votes
1answer
110 views

Playing with the torus and semisimplicial sets (prove that $\phi$ and $\psi$ are not homotopic)

Recall that we can express the torus $|X.| \cong T$ as a square with edges $e$ and $f$, diagonal $g$, faces $T_1$ and $T_2$, and a single vertex $v$, and appropriate identifications. Let $Y.$ be the ...
3
votes
2answers
149 views

Generalization of the hairy ball theorem.

The hairy ball theorem of states that there is no nonvanishing continuous tangent vector field on even dimensional n-spheres. Can the hairy ball theorem be strengthened to say that there is no ...
2
votes
0answers
98 views

4-manifold: $0$-handle $\cup$ $2$-handles along a framed link in $S^3$ (intersection form = linking matrix = presentation matrix of $H_1(\partial M)$)

Let $L$ be a framed link in $S^3$, consisting of framed knots $L_1,\ldots,L_m$. Let $A=[a_{ij}]\in\mathbb{Z}^{m\times m}$ be its linking matrix, with $a_{ii}=$ framing of $L_i$ and $a_{ij}=$ linking ...
18
votes
0answers
362 views

In $n>5$, topology = algebra

During the study of the surgery theory I faced following sentence: Surgery theory works best for $n > 5$, when "topology = algebra". I don't know what is the meaning of topology=algebra. ...
5
votes
1answer
89 views

How hard is it to endow a $\textit{Spin}^{c}$ structure on four-dimensional manifolds?

I am in a certain math conference and we came across Seiberg-Witten equations. Since I am really novice in the field, I asked if all "reasonable" four manifolds carry a $\textit{spin}^{c}$ structure. ...
3
votes
1answer
74 views

Cohomology calculation for maps to the 2-sphere.

Let $Y^3$ be a closed 3-manifold and $f\colon Y\to \operatorname{SO}(3)$, $g\colon Y\to S^2$ be smooth maps. Define $g'\colon Y\to S^2$ be the following composition: ...
1
vote
1answer
64 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, ...
0
votes
2answers
69 views

Surgery on $S^m$

On page 4 of the book "ALGEBRAIC AND GEOMETRIC SURGERY" by Andrew Ranicki, after the definition of surgery has written: Example View the $m$-sphere $S^m$ as $$S^m=\partial (D^{n+1} \times ...
6
votes
2answers
337 views

Uniqueness of Preferred Framing of a Solid Torus in $S^3$

One way to state my question tersely is: For a homeomorphism $f : S^1 \times \mathbb{D}^2 \rightarrow S^1 \times \mathbb{D}^2$, does $f|_{S^1 \times S^1}$ determine the isotopy class of $f$? This is ...
7
votes
1answer
123 views

Loop space and stable homotopy theory

The Bott periodicity theorem for unitary group $U(n)$ says that $$ \pi_{i-1}(U) \simeq \pi_{i+1}(U) $$ How can I prove, using this theorem, that $$ \Omega (U) \simeq BU \times \mathbb{Z} ?$$ What is ...
2
votes
0answers
63 views

How do we check if a covering of an orbifold is a manifold?

Let $X$ be an orbifold and suppose it is "good", i.e. its universal covering orbifold $\widetilde{X}$ has a trivial orbifold structure (it is "just" a manifold). It may be the case that some ...
4
votes
1answer
173 views

Representation of (co)homology classes of $3$-manifolds by embedded surfaces

Let $M$ be a closed oriented $3$-manifold. Theorems in algebraic topology allow us to identify $$H_2(M) \ \cong \ H^1(M) \ \cong \ \langle M,S^1\rangle$$ where (co)homology is meant with integer ...
1
vote
1answer
142 views

Simple Sphere Suspension Question

I have heard it said that the suspension of an $n$-sphere is an $n+1$-sphere. This is stated without proof in chapter 0 of Hatcher's book on algebraic topology. More generally, it seems that the ...
4
votes
2answers
251 views

How does Thurston's geometrisation conjecture imply Poincaré's conjecture?

I ran into the geometrisation conjecture a few days ago, and I started wondering how to prove Poincaré's conjecture. Let $M$ be a compact, simply connected, $3$-manifold. Clearly it is irreducible ...
12
votes
1answer
508 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
5
votes
0answers
120 views

$\operatorname{Spin}^c(n)$ is a Lie group?

Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.) Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
4
votes
0answers
135 views

Symmetric product of genus 2-surface

Let $\Sigma$ be the genus 2-surface. Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$. In the very ...
2
votes
0answers
68 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
1
vote
3answers
300 views

Infinitely sheeted covering spaces!

I was wondering what the fundamental group of an infinitely sheeted covering space of say some surface might be? I'm thinking it should be an infinite cyclic group, but this is more intuitively, i ...
2
votes
0answers
85 views

Relationship between the Hopf Fibration and Spinors on $S^2$

The unique spin structre for $TS^2$ is given by the Hopf fibration. We can trivialize the Hopf fibration over open sets $U_1 = S^2 \setminus \{N\}, U_2 = S^2 \setminus \{S\}$ where $N$ and $S$ are the ...
-1
votes
1answer
112 views

Compact surfaces and Fundamental Groups

Identify the compact surfaces $X$ for which there exist a proper subgroup $G$ of $\pi_1(X)$ such that $G\cong \pi_1(X)$. EDIT: Suggestions?
3
votes
0answers
138 views

Covering spaces!

If one wanted to find all connected covering spaces of a product of two spaces, say $S^1\times RP(3)$, how would you go about it? I'm thinking finding the fundamental group of $S^1\times RP(3)$, and ...
4
votes
1answer
351 views

Spin structures on $S^1$ and Spin cobordism

I'm trying to understand the 2 spin structures on the circle. Since the frame bundle for the circle is just the circle itself, Spin structures on $S^1$ correspond to double covers of $S^1$. There are ...