Questions about algebraic methods and invariants to study and classify topological spaces: homotopy groups, (co)-homology groups, and beyond.

learn more… | top users | synonyms (1)

0
votes
0answers
16 views

$H_n(X)\to H_{n-1}(X_1\cap X_2)\stackrel{g_*}{\to}H_{n-1}(Y_1\cap Y_2)$ same as $H_n(X)\stackrel{f_*}{\to}H_n(Y)\to H_{n-1}(Y_1\cap Y_2)$

I've been reading about the Mayer-Vietoris sequence, but I don't follow a certain naturality condition. Suppose two spaces can be written as $X=X_1^\circ\cup X_2^\circ$ and $Y=Y_1^\circ\cup ...
0
votes
2answers
14 views

Why is the product of path homotopy classes not defined sometimes?

Munkres says on pg. 346 that the set of path homotopy classes does not aways form a grop uder the operation $*$ because the product of two path homotopy classes is not always defined. What does this ...
2
votes
1answer
36 views

Is there a cayley graph for the Klein bottle?

When studying algebraic topology we learned about the fundamental group of the $2$-torus $T^2$ which is isomorphic to $$\langle a, b \mid aba^{-1}b^{-1} \rangle$$ (the free abelian group on two ...
1
vote
0answers
14 views

Cohomology after Dehn surgery

A $0/1$-type Dehn surgery on a 3-manifold $M$ is to remove a solid torus from $M$ and thereafter to sew it back in $M$ such that the meridian disc goes once time along the longitude and no times along ...
0
votes
0answers
10 views

How to prove homomorphisms with 'lifting'

for topology i just started a chapter called lifting and i'm having trouble using this concept to prove a homomorphism of a lifting correspondance. This is my following question: let $m \in ...
2
votes
1answer
33 views

Is the Hopf fibration the only fibration with total space $S^{3}$?

Let $S^{1}$ bundle over $S^{2}$ is homeomorphic (diffeomorphic) to $S^{3}$. Is the Chern class of the fibration 1, i.e. is the Hopf fibration up to isomorphism the only fibre bundle with total space ...
1
vote
0answers
48 views

A doubt in Munkres' Topology.

On pg. 343 of Munkres' Topology (Second Edition), there is a diagram given that I am having problems understanding. This diagram is Figure 51.7. Why are the paths $i$ and $e_0$ defined from $I$ to ...
1
vote
3answers
39 views

A problem regarding $k\circ (f*g)=(k\circ f)*(k\circ g)$.

My Algebraic Topology book states the following: Let $k:X\to Y$ be continuous path. If $f$ and $g$ are two paths in $X$ with $f(1)=g(0)$, then $$k\circ(f*g)=(k\circ f)*(k\circ g)$$ I'm trying ...
2
votes
0answers
25 views

Where have I gone wrong in understanding of CW complex and Cell homology?

I seem to have wrong understanding of CW complex and it would be nice if someone could help me out. The definition I have for Cell complex is the usual one I think. We define glueing of cells and ...
4
votes
2answers
85 views

Homeomorphism Compact Subsets

Are there compact subsets $A,B \subset \mathbb{R^2}$ with $A$ not homeomorphic to $B$ but $A \times [0,1]$ homeomorphic to $B \times [0,1]$?
7
votes
1answer
101 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
1
vote
3answers
36 views

Cohomology groups of real projective space

My question concerns the cohomology groups $H^k(RP^n,\mathbb{Z}_2)$. We know that $H_k(RP^n,\mathbb{Z}_2) = \mathbb{Z}_2$ if $0 \leq k \leq n$ and is trivial otherwise. I looked up the solution and it ...
0
votes
1answer
24 views

Question on Hamming distance

Let V_n be n-dimentional vector space over GF(q). E is k-dimentional vector subspace which is a linear q-ary (n,m,d) code and also consider the radius e = [(d-1)/2]. Assume that E is not a perfect ...
3
votes
2answers
27 views

the mapping class group of the disk is trivial proof

Proof : Identify $D^2$ with the closed unit disk in $\mathbb{R}^2$. Let $\phi : D^2 \rightarrow D^2$ be a homeomorphism with $\phi_{\partial D^2}$ equal to the identity. We define, $F(x,t) = ...
1
vote
0answers
28 views

Computing fundamental groups.

How do I compute the fundamental groups of these spaces: (a) $\{(x,y)\in\mathbb{R}^2|x^2+y^2>1\}$; (b) $\mathbb{R}^2$ with two points deleted; (c) $\mathbb{C}$P$^n$, the complex projective ...
6
votes
3answers
88 views

let $\xi$ be an arbitrary vector bundle. Is $\xi\otimes\xi$ always orientable?

Let $\xi=(E,p,B)$ be a line bundle (not nec. orientable). Then the tensor product $\xi\otimes \xi$ is orientable. I obtain this by choosing $b\in U\cap V$, $U,V$ open in $B$ such that ...
1
vote
2answers
82 views

tensor product of two vector bundles

Let $\xi$ and $\eta$ be two vector bundles over the same base space $B$. Then $\xi\otimes \eta$ is orientable if and only if $\xi$ and $\eta$ are both orientable. How to prove this true or not true? ...
1
vote
1answer
44 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 ...
2
votes
1answer
48 views

Covering map of $\mathbb R \mathbb P^2$

The question I am trying to answer is: Does the quotient map $ q:[0,1] \times [0,1] \to \mathbb R \mathbb P^2$ extend to a covering map $\mathbb R^2 \to \mathbb R \mathbb P^2$ I know that the ...
4
votes
0answers
43 views

$RP^{(n-1)}$ is not retract of $RP^n$

I have to solve the following: Show that $RP^{(n-1)}$ is not retract of $RP^n$ for $n\geq 2$. I have done this with knowledge of homotopy-groups, by showing that $\mathbb{Z}$ cannot factor through ...
0
votes
0answers
17 views

Euler characteristic and free action

If $K$ is a finite simplicial complex, and $G$ acts simplicialy on $K$ with no fixed points, show $\chi(K) = |G|\cdot\chi(K^2/G)$. Could I have a hint for how to start this question? I was told to ...
0
votes
1answer
24 views

Show that $\tilde{X} \rightarrow X$ is a covering map.

Let $\tilde{X}=\{(x.y)\in \mathcal{R}^2; \text{x or y is an integer}\}$ Let X=$\{(z_1, z_2) \in S^1\times S^1; z_1=1$ or $z_2=1\}$ and let $p:\tilde{X}\rightarrow X$ be defined by $p(x,y)=(exp(2\pi ...
0
votes
1answer
11 views

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups.

Show that a finite graph product of finitely generated groups is finitely generated, and similarly for finitely presented groups. I think when we have a finitely generated groups,the graph product of ...
1
vote
1answer
35 views

What is the second homotopy group of $R^3 \setminus \{ (0,0,0) \}$

I was told that it was $\mathbb{Z}$, and I can imagine a subgroup isomorphic to $\mathbb{Z}$ of 'wrappings' of the sphere around the point, but I am still convinced there are more homotopy classes. I ...
0
votes
0answers
32 views

Equivalence of Cohomology groups

Suppose $n=i+j,$ with $n, i,j$ positive integers. Let $I^k$ denote the $k$-dimensional unit square. It is claimed (in Hatcher's Algebraic Topology text) that $H^i(\mathbb{R}^n, \mathbb{R}^n \setminus ...
1
vote
1answer
18 views

In homology, when we operate the boundary twice we get zero, that is, $\partial^2=0$. Need help understanding proof.

Proof for $S=\Delta_n=(v_0 ... \hat{v_i} ...v_n)=d_i$ $\partial=\displaystyle\sum_i^n(-1)^id_i.$ Thus, $\partial^2=[\displaystyle\sum_i^n(-1)^id_i][\displaystyle\sum_j^n(-1)^jd_j] $ ...
0
votes
1answer
51 views

Open and Closed Covering [on hold]

Suppose $p:\widetilde{X} \mapsto X$ is a covering with $f,g:Y \mapsto \widetilde{X}$ continuous such that $pf = pg$. Why is the set of points in $Y$ for which $f=g$ open and closed in $Y$?
3
votes
0answers
31 views

Finite index embedding of $F_{4}$ in $F_{2}$

In this question $F_{n}$ is the free group with $n$ generators. Is there a subgroup of $F_{2}$, isomorphic to $F_{4}$, which index is finite but not in the form of $3k$(not multiple of $3$)? The ...
2
votes
0answers
20 views

Cross product of cohomology classes: intuition

Let $X$ and $Y$ be topological spaces and consider cohomology over a ring $R.$ Hatcher (in his standard Algebraic Topology text) defines the cross product of cohomology classes $$H^k(X) \times ...
0
votes
0answers
11 views

is simplicial approximation of $gof$ equal to the combination of simplicial approximation of $g$ with simplicial approximation of $f$?

suppose $M,L,K$ are complex and $f:|K| \rightarrow |L|$ and $g:|L| \rightarrow |M|$ are continues maps,can we consider combination of simplicial approximation of $f$ and $g$ as a simplicial ...
1
vote
0answers
17 views

Explain the terms k-simplex and simplical complex geometrically?

I m new to algebraic topology .so confused with these terms pls suggest simple books
1
vote
2answers
51 views

Group acts freely on a closed surface

My question is as follows: Let G be a finite group which acts freely as a group of homeomorphisms of a closed surface S (so the only element with fixed points is the identity) Then: Show the orbit ...
1
vote
1answer
26 views

Let $p$ be a covering space and $X, Y$ be path connected. Show there exists a map $q$ such that $q\circ p=1_{X}$ iff $p$ is a homeomorphism.

Let $p\colon X\rightarrow Y$ be a covering map where $X$ and $Y$ are path connected. Show that there exists a map $q\colon Y\rightarrow X$, such that $q\circ p=1_{X}$ if and only if $p$ is a ...
0
votes
1answer
31 views

Mayer-Vietoris sequence confusion

In the Mayer-Vietoris exact sequence $$... \rightarrow H_n(A) \oplus H_n(B) \rightarrow H_n(X) \rightarrow H_{n-1}(A \cap B) \rightarrow H_{n-1}(A) \oplus H_{n-1}(B) \rightarrow...$$ I am confused ...
3
votes
0answers
47 views

does the pullback of a covering space correspond to the pullback of the corresponding representations of $\pi_1$?

Say you have a covering space $C \rightarrow X$ corresponding to some homomorphism $\pi_1(X)\rightarrow S_n$. Suppose you have an arbitrary (continuous) map $f : Y\rightarrow X$. Then we may pull back ...
0
votes
1answer
19 views

Is the tangent bundle of an oriented surface plus a trivial bundle trivial?

Let $\Sigma$ be an oriented closed surface and $E$ be the direct sum of $T\Sigma$ with a trivial line bundle. Is $E$ a trivial rank $3$ vector bundle? For genus $0$ and $1$ the answer is yes since ...
4
votes
2answers
81 views

Fundamental group of the Poincaré Homology Sphere

I'm working on the Poincaré Homology Sphere $P_3$ and would like to compute it's Homology $H_1$ and fundamental group. I would like to identify it's fundamental group with the binary icosahedral group ...
0
votes
1answer
11 views

Determining images of points in a path homotopy.

Say the two paths $f_0$ and $f_1$ ae homotopic. Then $(1-t)f_0+tf_1$ is the homotopy between the two paths. Say $f_0,f_1\in\Bbb{R^2}$, and there is a point $(a,b)$ in $f_0$. How can we find which ...
0
votes
0answers
25 views

What is the complement of a loop?

My Algebraic Topology book says $A$ is a loop in the complement of another loop $B$ What does "in the complement of" mean here?
2
votes
0answers
29 views

Hopf Invariant Definitions

I have seen two definitions of the Hopf invariant given: (1) Cohomological Definition: Let $S^{n}$ denote the oriented $n$-sphere, where $n \geq 2$. Let there be given a map $f:S^{2n-1} \rightarrow ...
2
votes
2answers
53 views

Proving that the fundamental group of Klein bottle is generated by two elements without using covering spaces and van Kampen theorem, etc.

This is a problem in 'Topology and Geometry' by Bredon. I tried hard on this problem, but have no idea what to do. This question was onced asked by someone, but there was no satisfactory answer. Let ...
3
votes
1answer
70 views

Topological invariance of chern classes

Are Chern classes topological invariants? To be more precise: Given two complex manifolds $M$ and $N$. Does a homeomophism $f:M\to N$ map Chern classes to Chern classes?
1
vote
1answer
44 views

Fundamental Group of Klein Bottle?

Let $C^{*}=C \setminus \{ 0 \}$. What is the fundamental group of $C^{*}/H$, here $H=\{\psi^n;n \in \mathbb{Z}\}$ with $\psi(z) = 2 \bar{z}$?
0
votes
1answer
36 views

Question about degrees of maps from $S^1 \rightarrow S^1$

Note: Since the degree of a map is independent of the base-point I'll speak loosely and just say $\pi_1(S^1)$. One definition of the degree of a map $f$ from the circle to itself is the number $k$ ...
1
vote
1answer
29 views

Show that if $h,k: S^1\rightarrow S^1$ are homotopic, they have the same degree.

We define the degree of a continuous map $h: S^1 \rightarrow S^1$ as follows: Let $b_0$ be the point $(1,0)$ of $S^1$; choose a generator $\gamma$ for the infinite cyclic group $\pi_1(S^1,b_0)$. If ...
5
votes
4answers
525 views

Topological groups, why need them?

I'm reading through Munkres and Armstrong's books on topology. However, I find topological groups to be really complicated objects! I feel they are twice as hard to deal with then just groups and ...
4
votes
1answer
49 views

Show that if B is simply-connected, then p is a homeomorphism.

Let $p: E \rightarrow B$ be a covering map with $E$ path-connected. Show that if $B$ is simply-connected, then $p$ is a homeomorphism. I'm checking to see if my solution is flawed. Since $p$ is a ...
1
vote
2answers
102 views

Let $p: E\to B$ be a covering map. If $B$ is compact and $p^{-1}(b)$ is finite, then $E$ is compact. [duplicate]

So I start off and assume that some $\{U_\alpha\}$ is a cover of $E$. I want to reduce this cover to a finite subcover of $E$. Since $p$ is a covering map it is also an open map, therefore ...
1
vote
0answers
123 views

Help Understanding/Completing Proof of Prop 3.18/3.19 in Hatcher's Algebraic Topology

I apologize right away for the wall-o-text. I'm participating in a cohomology reading course, and I'll be leading the class through the following proposition later this week, but I'm having a hard ...
1
vote
1answer
26 views

A question about subexponential growth about group

Recall a group $\Gamma$ is said to have subexponential growth if lim sup$|E^{n}|^{1/n}=1$ for every finite subset $E\subset \Gamma. (E^{n}=\{g_{1}g_{2}...g_{n}: g_{i}\in E\}.)$ My question is: Can we ...