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

learn more… | top users | synonyms (1)

1
vote
1answer
85 views

computing fundamental group of $S^1 \times S^1$ [duplicate]

How to find $\pi_{1}(S^{1}\times S^{1})$ ? I know $\pi_1(S^1)$ but ho to do this ?
1
vote
1answer
58 views

How to show that a locally exact form is globally closed

Every closed differential form is locally exact. That's quite obvious, as it follows (Poincare lemma) from the vanishing $k$-th cohomology groups ($k>0$) of contractible open subsets of $\mathbb ...
2
votes
1answer
62 views

Cohomology of $\mathbb{R}P^2$

Note: this is from some homework exercises which my lecturer has left us which $\textbf{don't}$ count towards our grade in any way. We have $$0\xleftarrow{0}\mathbb{Z}^2 \xleftarrow{ ...
2
votes
1answer
42 views

How can I complete this proof of a algebraic topology theorem?

The theorem is: Let be f, g : X→Y two continuous maps between topogolical spaces and H : X$\times$[0, 1] a homotopy such that H(x,0)=f(x) and H(x,1)=g(x). Given $x_0\in X$ let be $y_0=f(x_0)$, ...
3
votes
1answer
97 views

Non-existence of embedded incompressible surfaces

I want to prove the following assumption: Let $g,h$ be natural numbers with $g > h$ and let $S_g$ be the closed, orientable surface of genus $g$. Then, there is no (smooth) map $f: S_g \to S_h ...
0
votes
1answer
60 views

Natural versus Unatural splitting, the difference is…

I am working on an assignment currently and come across the Universa Coefficient Theorem for Homology/Cohomology. I am still getting somewhat used to the terminology and concepts in questions but I ...
5
votes
1answer
61 views

Where have we used that $G$ is a compact Lie group?

Let $G$ be a compact Lie group acting on a topological space $X$. I need to show that the orbit map $p:X\rightarrow X/G$ has the path lifting property. Here is the proof - Let ...
1
vote
0answers
42 views

Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map.

Let $q: X\to Y$ and $r: Y\to Z$ be covering maps; let $p=r\circ q$. Show that if $r^{-1}(z)$ is finite for each $z\in Z$, then $p$ is a covering map. I've almost completed solving this problem, but ...
5
votes
1answer
86 views

Example of a cogroup in $\mathsf{hTop}_{\bullet}$ which is not a suspension

Let $\mathsf{hTop}_{\bullet}$ denote the homotopy category of pointed topological spaces. More precisely, the objects are pointed topological spaces and for two objects $X$ and $Y$, the morphisms from ...
1
vote
1answer
44 views

Can every CW complex be presented as a countable collection of integer valued matrices?

I had this epiphany today and i'd really like to verify it, since i'm starting to seriously doubt myself... Let $X$ be a connected CW complex. It can be written as a colimit of it's sequence ...
1
vote
1answer
35 views

Degree of the map $S^1\to S^1:z\to z^n$

I want to compute the (topological/differential) degree of the map $f:S^1\to S^1:z\mapsto z^n$. I've have shown that the degree of the map $g:\mathbb{C}^*\to \mathbb{C}^*:z\mapsto z^n$ is $n$. Is ...
2
votes
1answer
61 views

How can I prove that the horn torus and $\mathbb{T} \cup D_1$ have the same homotopy type?

I'm trying to prove that the horn torus ($W$) defined by rotating the circumference $(x-1)^2+z^2=1, y=0$ around the z axis and $A=A_1 \cup A_2$ where $A_1$ is the torus obtained rotating ...
3
votes
1answer
78 views

If X is simply-connected then any two paths are homotopic via a homotopy relative to the points where they agree

Let $X$ be a simply-connected space and $f,g:I\to X$ two paths with the same endpoints and $A=\{s\in I:f(s)=g(s)\}$. Since $X$ is simply-connected there is a homotopy $F:I\times I\to X$ relative ...
1
vote
1answer
29 views

Is the projection of two homotopic maps path homotopic?

Let $\alpha$ and $\beta$ be two homotopic paths in a path connected topological space $X$. Let $\alpha(0)=x_0$ $\alpha(1)=x_1$ $\beta(0)=x_2$ and $\beta(1)=x_3$. Let $p:X\rightarrow Y$ be a continous ...
1
vote
1answer
30 views

Is $\gamma$ homotopic to $g\circ\gamma$?

Let $X$ be a simply connected topological space. Let $x_0,x_1\in X$ and let $\gamma$ be a path in $X$ from $x_0$ to $x_1$. Let $g$ be a homeomorphism of $X$ with itself. Then $g\circ\gamma$ is a path ...
2
votes
1answer
44 views

Uniqueness of (co)homology theories on category of finite CW pairs

By a (co)homology theory on the category of (finite) CW pairs I mean a functor from the category of (finite) CW pairs to the category of graded $R$-modules, satisfying the Eilenberg-Steenrod ...
0
votes
0answers
40 views

What is dual to relative homology?

Let $(X,A)$ be a pair of manifolds, where $A \subseteq X$. Then we can define the relative cohomology of the pair $H_{\bullet}(X,A)$ to be the homology of the chain complex ...
2
votes
0answers
41 views

Structures created from cell complexes

I've recently started stydying algebraic topology and am now learning about cell complexes. I understand the iterative construction of such spaces, but I lack any intuition concerning the limitations ...
4
votes
1answer
102 views

Isomorphism involving Eilenberg-Maclane space, Tors.

Let $\pi$ be a group and let $K(\pi, 1)$ be a connected CW complex such that $\pi_1(K(\pi, 1)) = \pi$ and $\pi_q(K(\pi, 1)) = 0$ for $q \neq 1$. Does there exist an isomorphism between $H_*(K(\pi, 1); ...
1
vote
1answer
128 views

Does having the same fundamental group imply that two spaces have the same homotopy type?

I have to prove that two different topological spaces $X,Y$ have the same homotopy type. I've been able to prove so far that $\pi_1(X)=\pi_1(Y)$ but I don't know if this is enough to say that $X$ and ...
1
vote
1answer
52 views

Relations between covering map and (co-)homology groups

This occurs to me when considering the homomorphism $p_*$ induced by the covering map/attaching map: $$p:S^n\to \mathbb{R}P^n$$ on its homology group: $H_n(S^n)\to H_n(\mathbb{R}P^n)$ which sends the ...
1
vote
1answer
28 views

Equivalent statements to simply connectedness

Show that the following are equivalent for a path connected space $X$. (a) $X$ is simply connected. (with the definition that the fundamental group is trivial) (b) If two paths $\alpha$ and $\beta: ...
0
votes
1answer
49 views

How to show that positive definite symmetric $nxn$ matrices are simply connected? [closed]

I was told that they are "star-shaped" as a hint but would like to see how this is proven.
0
votes
0answers
17 views

Set of beginning part and ending part of all loops based at a point is an open neighborhood

Problem: Suppose topological space $X$ is connected and locally path-connected, and the function $f:X\rightarrow S^1$ is continuous. Prove that if the induced homomorphism ...
0
votes
0answers
14 views

Find and sketch a lift for the covering $p:\mathbb{R}\to S^1$ given by $t \to e^{2\pi i t}$

Actually my question is is it the same as the path $f: I\to S^1$ beginning at $(1,0)$ given by $f(s)=cos\pi s, sin\pi s)$ lifted to $f'(s)=s/2$ beginning at $0$ and ending at $1/2$?
2
votes
1answer
64 views

What is a Covering Space (intuitively) and is it related to the concept of open cover?

I do know the definition of a covering space (e.g. Wikipedia: https://en.wikipedia.org/wiki/Covering_space), but am a bit confused what exactly is it. Is it related to the concept of Cover in point ...
2
votes
1answer
35 views

Definition of symmetric product in Milnor's paper

I am currently reading Milnor's paper which discusses the group action on spheres without fixed point. At the second page of the paper, he denotes $$M^n*M^n$$ to be a symmetric product of a manifold. ...
2
votes
0answers
69 views

Not every path connected space is contractible.

I wrote a proof that any path connected space is contractible which is completely wrong but i was not able to see what goes wrong in my proof: Let $X$ be a path connected space. Let $P$ be point in ...
1
vote
1answer
53 views

Recommended Book for Open Book Exam (Algebraic Topology)

This is a very soft question --- I have an upcoming open book exam on Algebraic Topology (includes Fundamental Groups/Homology/ Covering Spaces). Any recommendations on what books/notes are suitable ...
6
votes
1answer
64 views

Are all instances of torsion special cases of the same concept?

The concept of 'torsion' pervades mathematics. As far as I know the origin of the word is in algebraic topology where it was used to describe chains $\gamma$ which are not boundaries but such that ...
2
votes
2answers
66 views

A surface with Euler characteristic of $-1$ [closed]

Is it possible to a have a surface that has an Euler Characteristic of $-1$ and what would that surface be homeomorphic to? $\displaystyle \chi \left({M}\right) = -1$
0
votes
0answers
42 views

Riemann - Hurwitz Formula for topology.

I am quite confused about the notion of branch points at infinity? and even in general the idea of branch points? I know branch points to be where points diverges to infinity. Could someone please ...
0
votes
2answers
53 views

Suppose $p:E\to B$ is a covering map and $B$ is connected. Prove that if $p^{-1}(\{b\})$ has n points $p^{-1}(\{b\})$ has n points for every $b\in B$

My idea is to somehow show that the group $O_n$ is both open closed which will imply $O_n=B$. Then assign to each $n$ the set of points $O_n\subseteq B$ such that $p^{-1}(b)$ has exactly $n$ points. ...
2
votes
1answer
32 views

Calculation kernel of boundary operator. (Easy example in Hatcher)

Consider the graph as given in the third page (pg.99) in the link below. https://www.math.cornell.edu/~hatcher/AT/ATch2.pdf We define a homomorphism $\partial: C_1 \to C_0$ by sending each basis ...
9
votes
0answers
161 views

Prerequisites for studying Perelman's proof of the Geometrization Conjecture

I want to set a course toward understanding Perelman's proof of the Geometrization Conjecture. I realize this will be a lengthy undertaking, but hopefully only on the order of one to two years. I am ...
2
votes
3answers
119 views

Torus cannot be embedded in $\mathbb R^2$

I've shown that $T^2$ can be embedded in $\mathbb R^3$. I just can't see why it can not be embedded in $\mathbb R^2$. Ideas: suppose $F: \mathbb S^1\times \mathbb S^1 \to \mathbb R^2$ is continuous ...
1
vote
1answer
58 views

Motivations for Homology or Cohomology Theory

In some standard books on Algebraic Topology (except Hatcher's), the motivation for homology or cohomology theory is stated with the help of Cauchy's theorem, Green's theorem, Stoke's theorem etc (for ...
2
votes
1answer
71 views

The Klein bottle is homeomorphic to the boundary of the product of the Möbius band with a disk

Can someone please give me a hint or the intuition in how to prove that the Klein bottle $\cong \partial$(Möbius strip $\times D^1 $ ) where $\cong$ means homeomorphic.
5
votes
2answers
117 views

Applications of Hodge theory to topology and analysis

I am going to give a talk for the PhD students' seminar at my university. The audience is composed mainly by algebraic topologists, algebraic geometers and analysts. I have decided that I'm going to ...
2
votes
0answers
88 views

the join of nonempty path connected X and any space Y has a trivial fundamental group

I am doing the problem 21, section 1.2 in Hatcher's book. Show that the join $X*Y$ of two nonempty spaces X and Y is simply connected if $X$ is path conn'd. I proved that $X*Y$ is path connected ...
11
votes
2answers
294 views

Laymans explanation of the relation between QFT and knot theory

Could someone give an laymans explanation of the relation between QFT and knot theory? What are the central ideas in Wittens work on the Jones polynomial?
2
votes
1answer
85 views

$H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$ are not isomorphic as rings

I'm stuck to prove that the singular cohomology groups with coefficients in $\mathbb{Z}$, $H^*(S^1\times S^3;\mathbb{Z})$ and $H^*(S^1\vee S^3 \vee S^4;\mathbb{Z})$, are not isomorphic as rings. What ...
0
votes
0answers
35 views

Help with formalisation of a reasoning about simplicial sets

We briefly mention during lectures that the homotopy relation between two maps of simplicial sets is not necessarily symmetric. So I thought for a counterexample but I don't know how to formalise it ...
0
votes
0answers
37 views

Given $(X,A)$,$(Y,B)$ such that $X/A$ and $Y/B$ are homotopy equivalent,are their relative homology groups isomorphic?

Suppose that $(X,A)$,$(Y,B)$ are pairs of topological spaces. If $X/A$ and $Y/B$ are homotopy equivalent, are $H_*(X,A;\mathbb{Z})$ isomorphic to $H_*(Y,B;\mathbb{Z})$ ?
3
votes
0answers
52 views

Simplicial homology for infinite complexes

Simplicial homology can be viewed as a covariant functor from the category of finite simplicial complexes with continuous maps over support polyhedra, to the category of sequences of abelian groups. A ...
2
votes
1answer
58 views

Is there a long exact cofiber sequence for a homotopy pushout?

Let $f:Z \to X$, $g: Z \to Y$ and let $M(f,g)$ be the corresponding mapping cylinder. Does the homotopy pushout diagram induce a long cofiber sequence? If so what does it look like?
1
vote
1answer
49 views

How does singular homology work?: $H_1(S^1)$.

This is a fundamental inquiry into the nature of singular homology. Let $\gamma$ and $\sigma$ be the following singular 1-chains on the circle: $$\gamma(t)=2\pi t,~~~~~\sigma(t)=4\pi t,$$ Now, based ...
1
vote
2answers
66 views

triviality of vector bundles with the reduced homology of base space entirely torsion

Let $\xi$ be an $n$-dimensional vector bundle over a manifold $M$ such that the reduced cohomology $\tilde H^*(M;\mathbb{Z})$ is entirely torsion (every element has finite order under addition). ...
2
votes
1answer
26 views

Hatcher example 3H.3. Local coefficients via Modules

I'm trying to understand the following example made by Hatcher at page 329 in the section "Local Coefficients via Modules" The problem arises when I start to prove that a basis for $C_n^+(X')$ is ...
2
votes
1answer
30 views

Action induces action of group ring on singular chain complex. [duplicate]

See here for a related question. Let $X$ be a space that satisfies the hypotheses used to construct a universal cover $\overline{X}$. Let $\pi = \pi_1(X)$ and consider the action of the group $\pi$ ...