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

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0
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3answers
47 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
146 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
151 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
140 views

Isomorphism Finite Topological Space

Does there exist a finite topological space with fundamental group isomorphic to $\mathbb{Z_2}$?
1
vote
3answers
414 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
72 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
139 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) = ...
7
votes
3answers
222 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 ...
3
votes
2answers
222 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
58 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
71 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 ...
11
votes
1answer
211 views

$\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$

I have to solve the following: Show that $\mathbb{R}\text{P}^{n-1}$ is not retract of $\mathbb{R}\text{P}^n$ for $n\geq 2$. I have done this with knowledge of homotopy-groups, by showing that ...
1
vote
0answers
42 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
48 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
19 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
68 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
38 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
43 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] $ ...
3
votes
1answer
85 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 ...
4
votes
1answer
415 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 ...
1
vote
0answers
32 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
votes
2answers
121 views

Group acts freely on a closed surface [closed]

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 ...
2
votes
1answer
68 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
104 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
81 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
46 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 ...
5
votes
2answers
304 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
29 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
1answer
62 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?
3
votes
0answers
89 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
253 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 ...
6
votes
1answer
225 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
143 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
90 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
151 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 ...
9
votes
4answers
1k 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 ...
6
votes
1answer
352 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 ...
2
votes
2answers
487 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
1answer
278 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
36 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 ...
3
votes
0answers
61 views

Singular cohomology with compact support

If $X$ is a locally compact Hausdorff space, then for any $n \geq0$ is $H_c^n(X) \cong {\tilde H^n}({X^ + })$? ($H_c^n(X)$ is the Singular cohomology with compact support and $X^+$ is the one-point ...
1
vote
1answer
67 views

Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ? I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point ...
0
votes
0answers
44 views

fundamental group of $\mathbb{C^*}/\{e,a\}$

I'm taking an intro to topology course, and am having trouble with this question. What is the fundamental group of $\mathbb{C^*}/\{e,a\}$, where $e$ is the identity homomorphism and $az=\overline{z}$. ...
6
votes
0answers
307 views

Lefschetz duality for non-compact relative manifolds

I'd like to use the formulation of Lefschetz duality stated here, but I can't seem to find a reference for this particular version of it, and it doesn't seem quite right to me. The exact statement in ...
3
votes
1answer
28 views

The coeffcients of a generator of $H_0(X)$ sum to $\pm 1$?

I'm reading Theorem 4.14 (p. 70) of Rotman's Intro to Algebraic Topology. He proves that if $X$ is a nonempty path connected space, then $H_0(X)\simeq\mathbb{Z}$, and if $x_0,x_1\in X$, then ...
6
votes
0answers
59 views

Surprising constructions in algebraic topology that facilitate one's understanding of underlying theory

I am recently come into the world of algebraic topology and find it a fascinating place with lots of beautiful constructions that challenge one's intuition. Also, understanding these constructions are ...
2
votes
0answers
44 views

Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
1
vote
2answers
299 views

Fundamental Group of Punctured Plane

What is the fundamental group of $(\mathbb{C} \setminus {\{0\}})~/~\{e,a\}$, where $e$ is the identity homeomorphism and $az = -\bar{z}$? Clearly this is homeomorphic to the half cylinder , which is ...
1
vote
0answers
93 views

Real Spectrum of a Commutative Ring

Could anyone explain briefly what is a real spectrum of a commutative ring? If it possible please give me some easy example. Many thanks in advance. $\text{Spec}_{r}(A)$
3
votes
1answer
137 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 ...