The corpus of tools and results that arose from studying manifold theory using non-algebraic techniques, that is, as opposed to (algebraic-topology). The focus of the field tends to be on special objects/manifolds/complexes and the topological characterisation and classification thereof. A key ...

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2
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2answers
97 views

the knot surgery - from a $6^3_2$ knot to a $3_1$ trefoil knot

It is intuitive that one can simply doing a cut-gluing surgery to make a $6^3_2$ to a $3_1$ trefoil knot: e.g. from to All one needs to do it to cut the three intersections at the angle of ...
3
votes
0answers
64 views

John Hempel's proof of residual finiteness of surface groups

John Hempel proved that fundamental groups of $2$-manifolds are residually finite. I want to understand this proof so I have some questions: Why if we have $S(f)=\emptyset$ then $f$ represents ...
1
vote
1answer
36 views

Rotation orientations in n-dimensions

I'm doing a change of variables that involves doing simple rotations on the standard basis vectors in R^n, and I'm wondering what the standard orientations are in n dimensions are and why. For ...
1
vote
1answer
139 views

Extending homeomorphism of unit circle to unit disk

What is the best best way to prove that any homeomorphism of the unit circle onto itself can be extended to a homeomophism of the closure of the unit disk onto itself?
3
votes
1answer
219 views

A 3-manifold with fundamental group isomorphic to a surface group.

Let $M$ be a 3-manifold (the case I am interested is $M$ closed orientable connected hyperbolic); suppose $\pi_1 (M)$ is isomorphic to the fundamental group of a (closed orientable connected) surface ...
3
votes
0answers
76 views

handle moves: proof

In several 4-manifold textbooks, when handle moves (creation, cancellation, sliding) are discussed, they are explained using very helpful drawings. However, I would like to know if there is a ...
2
votes
0answers
45 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
127 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 ...
2
votes
2answers
118 views

Fundamental group of a Cayley graph

Could someone please indicate the proof of the following fact (I believe strongly, but I am not sure, that this is true). Let $F$ be a free group of finite rank, $N<F$ a normal subgroup and ...
3
votes
1answer
87 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.) ...
2
votes
1answer
46 views

Cancelling Handle Attachments

Let $(W, \partial W)$ be an $n$-dimensional manifold with boundary. Suppose that $(W', \partial W')$ is obtained from $(W, \partial W)$ by attaching a $k$-handle via an embedding $\phi: S^{k-1}\times ...
1
vote
0answers
69 views

Number of fixed order elements in the mapping class group of a closed surface

Let $S_g$ be a closed surface of genus $g\geq 2$. Given $r \in \mathbb{N}$, what is the number of elements of order $r$ in the mapping class group? Is it finite or infinite? If it is infinite is ...
5
votes
3answers
376 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 ...
0
votes
1answer
42 views

Another question about the classification of 1-Manifolds

I've been seeking for a proof of the classification of 1-Manifolds with very little success. In this case, a manifold is a Hausdorff, second countable, locally euclidean space. I know that every ...
1
vote
1answer
176 views

Is there any connected n-manifold such that $H_n(X,Z)=Z\times Z$?

I think the question is equal to whether a n-manifold has a n-submanifold which is compact in n-manifold. I feel there is not such manifold, but I don't know how to prove it. In fact, I just need some ...
14
votes
1answer
288 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
1answer
67 views

Definition of “Representing” a Handlebody (Lefschetz Fibration)?

Sorry, I could not find a clear explanation of the meaning of the word represented in the following:"any 4-dimensional 2-handlebody W can be represented by a topological (achiral) Lefschetz fibration ...
3
votes
2answers
53 views

Identification Space and Isotopy

Original Question: Let $X$ and $Y$ be topological spaces and let $f:X \to Y$ and $g:X \to Y$ be isotopic embeddings. Is it true that $X \cup_f Y$ is homeomorphic to $X \cup_g Y$? Edit: I meant to ...
4
votes
1answer
92 views

What's the difference between “crumpled cube” and “3-ball”?

Warning: My level of understanding of topology is very low. Small words would be appreciated. :) Browsing Wikipedia, I came to crumpled cube, defined as "a 2-sphere together with its interior". ...
3
votes
1answer
2k views

Understanding the Equation of a Möbius Strip

I am in HL Math and trying to finish my IA. My topic is the Möbius band. The only problem is, I do not understand the formula that defines it and everywhere I have looked has just given me a ...
2
votes
2answers
101 views

Homologous surfaces in three-manifolds

Let M be a 3-manifold. Let $S$ and $T$ be properly embedded surfaces in $M$ such that $[S] = [T] \in H_2(M, N(\partial S)) $. Is it true that we can isotope $\partial S$ so that it coincides with ...
1
vote
1answer
107 views

Torsion Subgroup of Mapping class group.

What is the cardinality of finite order elements in Mapping class group of a surface $S_{g,n}$ of genus g and n boundary components. 1) If it is infinite then how can I generate a collection of ...
2
votes
2answers
110 views

Quasi-isometric embedding and Quasi-isometry

Let $X$ and $Y$ be geodesic metric spaces. Suppose there are quasi-isometric embeddings $f:X \rightarrow Y$, $g:Y \rightarrow X$. Then, can we say there is a quasi-isometry from $X$ to $Y$? I tried to ...
1
vote
1answer
41 views

Why is the dividing set nonempty when a convex surface has Legendrian boundary?

I am an undergrad and curious about the following question. Let (Y,ξ) be a contact manifold, and L⊂(Y,ξ) be a Legendrian knot which is the boundary of a convex surface Σ embedded properly in Y. Why ...
1
vote
2answers
128 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 ...
17
votes
1answer
341 views

The kernel of free group map to surface group

$G$ is a surface group of genus $g\geq 2$ (the fundamental group of closed orientable surface of genus g). $F$ is a free group of rank $2g$ with basis $\{x_1,\dots,x_{2g}\}$. $\phi$ is a surjective ...
2
votes
0answers
23 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 ...
0
votes
1answer
48 views

The deficiency of surface group

Let $G$ be the fundamental group of a closed surface of genus $g$. We know $G$ has a presentation $$\langle a_1,b_1,a_2,b_2,\dots,a_g,b_g \mid [a_1,b_1][a_2,b_2]\dots[a_g,b_g]=1 \rangle.$$ The ...
3
votes
0answers
163 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: ...
1
vote
1answer
56 views

How to enclose a ball more than once with a surface homeomorphic to $S^2$? In 3D.

In 3 dimensional space, how to enclose a monopole (either point-like or ball-like, both types exist.) more than once with a surface homeomorphic to $S^2$?
2
votes
1answer
128 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 ...
5
votes
2answers
849 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 ...
5
votes
1answer
158 views

How does a left group action on the fiber of a principal bundle induce a right action on the total space?

Suppose I define a "principal $G$-bundle" as follows: A principal $G$-bundle is a fiber bundle $F \to P \overset{\pi}{\to} X$ with a left group action of $G$ on $F$ that is free and transitive, ...
0
votes
0answers
87 views

3-Manifolds from identifying faces

I have a question about two polyhedra and getting manifolds out of them. The first of these is a tetrahedron. When calculating the Euler characteristic I got $\chi(X)=1-3+2-1=-1$. I believe 3 edges ...
0
votes
1answer
58 views

Boundary of a compact 3-dimensional manifold with boundary is a compact manifold of 2 dimensions.

I have been able to prove to myself that the boundary of a 3-dimensional manifold is indeed a compact set. I am stuck however proving that it is a 2 dimensional manifold. Specifically why the ...
4
votes
0answers
59 views

What is a 2-surgery on a disk?

I am confused by a certain point in Scharlemann's paper "Sutured Manifolds and Generalized Thurston Norms", which seems important enough to not just skip it. I mean the "2-surgery on disks" in the ...
1
vote
1answer
222 views

Handlebody decomposition and intuition

I am trying to get an intuition of how to approach handle body decompositions. I understand that a Torus can be decomposed into a 1 0-handle, 2 1-handles, and a 2-handle. The 0-handle is a hole you ...
8
votes
1answer
116 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
174 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
80 views

Sufficient conditions for quasi-isometric embeddings of Cayley graph

I would like to know more about the assumptions under which the Cayley graph of a given group embeds quasi-isometrically into the space where the group is acting. For instance, if a group $G$ acts by ...
0
votes
1answer
67 views

Deforming disks into other disks

Right now I'm casually reading through Carson's "Topology of Surfaces, Knots, and Manifolds." I don't have a strong background in topology, and I was told that this was a very accessible and ...
4
votes
1answer
139 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 ...
25
votes
0answers
696 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. ...
6
votes
1answer
259 views

How can I tell if two functions are conjugates in the homeomorphism group of $\mathbb{S}^n$?

Suppose we have two functions $f,g:\mathbb{S}^n\to\mathbb{S}^n$ which are bijective, continuous, and have a continuous inverse (aka bicontinuous). They are conjugates in the homeomorphism group when ...
5
votes
1answer
98 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
80 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
74 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, ...
2
votes
0answers
64 views

Kirby diagrams for nonorientable $4$-manifolds

In http://www.math.msu.edu/~akbulut/papers/akbulut.lec.pdf, which is a (still developed) set of lecture notes on 4-manifolds by Selman Akbulut, in section 1.5 there is a way to draw a non-orientable ...
5
votes
1answer
120 views

Showing every knot has a regular projection using diff top

My question is: Can we use differential topology to prove that every smooth knot has a regular projection? Here is some background: Let $\gamma : S^1 \rightarrow \mathbb{R}^3$ be a smooth unit-speed ...
0
votes
2answers
74 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 ...