-2
votes
1answer
76 views

Cw complex $\Sigma_g$

Consider the oriented connected compact surface $\Sigma_g$ of genus $g$ with its standard CW structure. How do I write down the attaching map for the single $2$-cell and how can it be proven that it ...
3
votes
1answer
41 views

Does a pseudo-Anosov homeomorphism of a punctured surface possess infinitely many periodic points?

In A Primer on Mapping Class Groups by Farb and Margalit theorem 14.19 implies that every pseudo-Anosov homeomorphism $f:S \rightarrow S$ on a compact surface $S$ possesses infinitely many periodic ...
1
vote
1answer
26 views

Examples of finding norm-minimizing surfaces and Thurston polytope

Let $Y$ be a (compact, oriented, connected) $3$-manifold. Thurston introduced a norm on $H_2(Y, \partial Y)$, which is defined as follow: any class $x \in H_2(Y, \partial Y; \mathbb{Z})$ is ...
1
vote
1answer
52 views

Is it true there exists $f:S^{2n}\longrightarrow S^{2n}$ making the diagram commutative?

Let $g:\mathbb R\mathbb P^{2n}\longrightarrow \mathbb R\mathbb P^{2n}$ be a continuous map where $\mathbb R\mathbb P^{2n}=\mathbb S^{2n}/\{\pm x\}$. Is it true there exists $f:\mathbb ...
0
votes
1answer
27 views

The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices

"The line between the centers of two k-simplices crosses through the centers of lower dimensional simplices." Is this always true? The barycentric centers are equidistant from the vertices. k=2 So ...
2
votes
1answer
55 views

What are the formulas for topological transformations? How to obtain them?

I'm reading Flegg's From Geometry to Topology, the author says that in Euclidean geometry, translation and rotation are: $$T:(x,y)\to(x+a,y+b)$$ $$R:(x,y)\to(x \cos \phi - y \sin \phi, x \sin \phi +y ...
2
votes
2answers
90 views

homotopy between two functions

Let us discuss this problem: Let $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{n}\}$ and $C=\{c_{1},c_{2},\ldots,c_{n-1}\}$ be discrete finite sets embedded in a unit sphere ...
2
votes
1answer
63 views

whether gluing the all faces of finite tetrahedrons in pairs would yield a manifold?

In fact ,I have asked this question in this web.But after read the book recommended to me ,I found this question still can't be solved。 According to the Theorem 10.1.1 ,Theorem 10.1.2 and the ...
0
votes
1answer
73 views

genus of a Mobius strip.

So I understand that one can create a closed loop (inside/on) a mobius strip. And cutting if we cut along the loop what remains will still be a connected surface. Meaning that the genus is $\geq 1$. ...
1
vote
1answer
46 views

A diagram which is not the torus

At page 17 of Munkres' Elements of Algebraic topology it says, referring to fig. 3.6, that "the diagram does not determine [the torus]. It does more than paste opposite edges together": Is it so? I ...
2
votes
3answers
107 views

finding a topological group with specific conditions

I have a question, it sounds difficult. The question is the following: Let $X$ be a topological group such that the binary operation defined on it is $*$. For any two points $a$ and $b$ in $X$ ...
3
votes
1answer
86 views

How many triangles are there in each “layer” of Poincaré disk?

Assume we grow from a single triangle layer by layer to get the whole disk. Every time a new ring of triangles makes all the vertices of the triangles already in the picture surround by seven ...
1
vote
1answer
48 views

Identity in Thom spaces.

Let $T$ be the one-point compattification, $E$ a real vector bundle, $\epsilon$ the trivial line bundle and $\Sigma$ the suspension operation. How can I prove that $$ T(\epsilon \oplus E) \simeq ...
4
votes
3answers
204 views

Geometry and Physics

I have to do a presentation on Geometry and Physics. I am asking it here (rather than physics.se) because I have to focus on Geometry More than Physics. The intended audience is Undergraduate Seniors ...
4
votes
2answers
143 views

Textbooks with exposition done mostly in proof outlines or exercises?

As the title indicates, I'm trying to find books where the exposition of the main course of thought is done entirely or mostly in outlines of proofs, or as exercises with or without hints. I'm trying ...
3
votes
2answers
63 views

How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?

I'm in trouble with the following problem: Let $M$ be a manifold with compact boundary $N$ and let $X$ be the double of $M$, that is, the manifold without boundary one gets by glueing $M$ with ...
2
votes
1answer
83 views

Homotopy of spheres $\pi_{n+1}(S^n) \simeq \mathbb{Z}_2$

I have a problem: I have to prove that $$ \pi_{n+1}(S^n) \simeq \mathbb{Z}/2\mathbb{Z} $$ when $n \ge 3$. I know the Freudenthal suspension theorem and the Hopf fibration. Is there an easy method to ...
8
votes
1answer
168 views

Hopf fibration and homotopy of spheres

Let $$ S^3 \to S^7 \to S^4 $$ an the Hopf fibration. We con consider the induced sequence in homotopy $$ \pi_i(S^3) \to \pi_i(S^7) \to \pi_i(S^4) \to \pi_{i-1}(S^3) \to \pi_{i-1}(S^7) \to \cdots $$ ...
2
votes
1answer
93 views

$J$-homomorphism for unitary group

We can define the $J$-homomorphism $$ J: \pi_{r}(SO(n)) \rightarrow \pi_{r+q}(S^q) $$ Can we define $$ J: \pi_{r}(U(n)) \rightarrow \pi_{r+q}(S^q) $$ what information can we deduce in this situation ...
2
votes
0answers
49 views

$\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R}) \simeq S^n \times_{\mathbb{Z}_2} \mathbb{R}$

Let $\operatorname{Hom}_{\mathbb{Z}_2}(S^n,\mathbb{R})$ the set of $\mathbb{Z}_2$-maps from $S^n$ to $\mathbb{R}$ and $S^n \times_{\mathbb{Z}_2} \mathbb{R}$ the fiber product of $S^n$ and ...
1
vote
0answers
67 views

Principal $G$-bundles as pull back bundles.

Let $G$ be a compact Lie group and consider a $G$-universal bundle $\pi: EG \to BG $ where $BG$ is the classifying space for the goup $G$ and the bundle $\pi: EG \to BG $ is defined as the principal ...
1
vote
1answer
74 views

Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $

Identify the Euler characteristic of the edge word $ abc^{-1}b^{-1}da^{-1}d^{-1} c $. The Euler characteristic is $$ X=V-E+F$$ where $V$, $E$ and $F$ are the vertices, edges and faces ...
2
votes
1answer
65 views

Spin group without Clifford algebras

I have to build the spin group $Spin(n)$ without use Clifford algebras. Can I find a complete description of spin group with a topological method? How can I build $Spin(n)$ as the double covering of ...
2
votes
2answers
91 views

Universal property of universal bundles.

A classifying space for a group $G$ is a topological space $BG$ with a principle $G$-bundle $p : EG \to BG$ where $EG$ is contractile, so that $BG = EG/G$. A classifying space is universal in the ...
-1
votes
1answer
62 views

Isomorphism $\left(\mathbb{C}^{n}\setminus\{0\}\right)/\mathbb{Z}$ with $S^{1} \times S^{2n-1}$

I have to prove that there is this isomorphism: $$\frac{\mathbb{C}^{n}{\setminus\{0\}}}{ \mathbb{Z}} \simeq S^{1} \times S^{2n-1},$$ where there is this equivalence relation in the left side: $(w_1, ...
3
votes
1answer
168 views

Tangent bundle of Grassmann manifold

I have to prove that the tangent bundle of Grassmann manifold $G_n(\mathbb{R}^{n+h})$ is isomorphic to $\operatorname{Hom}(\gamma^n(\mathbb{R}^{n+k}),\gamma^\perp)$, with $\gamma^{\perp}$ is the ...
3
votes
0answers
164 views

Imagining four or higher dimensions and the difference to imagining three dimensions

I’m very interested in how people envision four or higher dimensions. And I’m especially interested in how geometers and topologists who actually work in four dimensions do. Now I know of the video ...
5
votes
1answer
105 views

Constructing a odd homeomorphism between $A$ and $S^n$.

Let $A\subset\mathbb{R}^N\setminus\{0\}$ be a closed symmetric set ($x\in A$ then $-x\in A$). Suppose that $A$ is homeomorphic to some sphere $S^n$, $n\leq N$ ($n$ is the dimension of the sphere). Is ...
4
votes
2answers
138 views

Looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex

I am looking for a (nonlinear) map from n-dimensional cube to an n-dimensional simplex; to make it simple, assume the following figure which is showing a sample transformation for the case when $n=2$. ...
2
votes
1answer
80 views

Calculating the Picard group of $\mathbb{C}P^1$ in an elementary way

I have to explain the Picard group to some people that doesn't know the concept of sheaf. So is there a method to calculate $Pic(\mathbb{C}P^1)$ without sheaf theory? Is there a simple and easy proof ...
1
vote
0answers
65 views

Canonical bundle and Möbius bundle

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Möbius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
1answer
37 views

Isomorphism canonical and Moebius bundle.

I have to prove that the canonical bundle over $\mathbb{R}P^1$ is isomorphic to Moebius bundle. We define the caninical bundle as $\xi=\{E^{\perp},p,S^1 \}$ where $E^\perp:=\{(l,v) \in \mathbb{R}P^{1} ...
0
votes
0answers
55 views

Compactness of covering space

If we have space $X$ with and $n$ sheeted covering space $Y$ is $Y$ compact iff $X$ is? Torus or sphere, make me believe the answer is yes.
0
votes
1answer
75 views

Universal bundles and classificant maps.

We have this theorem: Let $\xi=(E,p,X)$ be a vector fiber bundle with rank($\xi)=n$. We can find a map $f: X \rightarrow Gr_n(\mathbb{C}^{\infty})$ such that $\xi=f^*(\gamma_n)$. I denote with ...
2
votes
0answers
47 views

Linkage of two circles in $\mathcal{R}^3$

Is there an easy analytic expression to determine whether two circles two circles (of equal radius) in $\mathcal{R}^3$ are linked? I guess one could look at their supporting planes, compute the ...
1
vote
0answers
81 views

Reference for Topology and Geometry

While trying to read the book "Three-Dimensional Geometry and Topology" by William P. Thurston and Silvio Levy I just realized that my knowledge of Topology is still very scarce...Is there any other ...
3
votes
1answer
96 views

From (algebraic) topology to geometry

I am thinking about a "correct" didactic way of linking topology (algebraic topology) to geometry. Usually, we are taught introducing geometry first, then topology, almost as an abstraction of ...
5
votes
1answer
180 views

Isotopy and homeomorphism

Let $X$ and $Y$ be topological spaces. Suppose we have an isotopy between maps $f, g: X\to Y$. The question is that is there a homeomorphism $h: Y\to Y$ such that $h\circ f =g$? I am especially ...
4
votes
1answer
81 views

Fuchsian Group without fixed points

I'm searching for a Fuchsian Group without fixed points. (because i need an example for a group $\Gamma$, so that $\mathbb{H}/\Gamma$ is a Riemannian surface, and therefore $\Gamma$ has to be a ...
0
votes
1answer
231 views

De Rham cohomology question

I'm trying to compute a certain DeRham cohomology. Consider $M = S^n-C$, where $C$ is the disjoint union of closed disks $C = \cup_{i=1}^m D_i$, and $m,n \geq 1$. How can we compute the cohomology ...
7
votes
4answers
206 views

How do you imagine the shape of a manifold $S^2 \times S^1$?

In 3-dimensional manifold theory, I have encountered the manifold $S^2 \times S^1$ many times. (The following story can be applied not only this manifold but also for any 3-dimensional manifold.) But ...
8
votes
1answer
443 views

Into how many parts do $n$ ellipsoids divide $\mathbb{R}^{3}$?

What is the maximum number of regions into which $\mathbb{R}^{3}$ can be divided by $n$ ellipsoids? (Each ellipsoid has the same size). Let´s denote this number by $r_{n}$. Clearly $r_{1}=2$. But ...
1
vote
1answer
81 views

How do I find the number of vertices on a planar diagram of a surface?

Cheers, I have a question which I just do not seem to see the answer for: I am proving the classification theorem for compact surfaces and use planar diagrams as representation of the surfaces. I ...
1
vote
0answers
103 views

Question on essential map

Suppose $M$ is a closed smooth n-manifold. a)Does there always exist a smooth map $f:M\to S^n$ from $M$ into the n-sphere, such that $f$ is essential(i.e. $f$ is not homotopic to a ...
5
votes
0answers
270 views

Definition of Reshetikhin-Turaev TQFT

I am studying Reshetikhin-Turaev TQFT. In their paper or in the book " Quantum invariants of knots and 3-manifolds", they first define an invariant $\tau(M)$ for a closed orientable 3-manifold $M$ and ...
4
votes
1answer
244 views

Surgery link for lens spaces

Let $p$ and $ q$ be a relatively prime integers. I want to know how to prove that a Hopf link with framing $-p$ and $-q$ is a surgery link for a lens space $L(p,q)$. The lens space is first a result ...
0
votes
1answer
203 views

orientability of the möbius strip using homology

I read in Hatcher's "Algebraic topology" book about orientability of topological maifolds using homology. now I would like to know how one can apply this to show that the möbius strip is not ...
2
votes
0answers
94 views

A Heegaard splitting of $S^2\times S^1 \# S^2\times S^1$.

For a Heegaard splitting of $S^2 \times S^1$, we can take two copies of genus 1 handlebodies and glue boundaries with the identity map. I want to generalize this a little bit. In the case of ...
4
votes
2answers
216 views

Heegaard splitting of a 3-manifold with boundary

A Heegaard splitting of a closed orientable 3-manifold $M$ is $M=H \cup H'$, where $H$ and $H'$ are handlebodies. Is there any similar concept for orientable 3-manifolds with boundaries?
3
votes
0answers
76 views

How to do this surgery?

Let $L$ be a $0$-framed trivial knot in $S^3 \subset B^4$. Take $B^3 \subset B^4$ such that $B^3$ splits $B^4$ into two and $\partial B^3$ intersects $L$ only two points. Take a neighborhood $U$ of ...