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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3
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2answers
43 views

The Incluson Map $S_1\to S_1\times S_1$ Induces an Injection in First Homology.

Let $T=S^1\times S^1$ be the torus and $A=S^1\times \{x_0\}$ be the "vertical" circle in the usual depiction of the torus as a tyre tube sitting "horizontally". Let $i:A\to T$ be the inclusion map....
0
votes
1answer
22 views

Proving weak homotopy equivalence of a map

This is a modification of the question I previously asked here. Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting ...
8
votes
1answer
123 views

What is this manifold?

As picture below ,it is a Mobius band with a cylinder crossing it .Let it be $\Omega$ . Obviously , $\partial \Omega$ is a circle. Now , what is $\Omega/\partial \Omega$ ( I mean glue the boundary to ...
0
votes
0answers
71 views

Homology Groups of Tangent 2-Spheres

I have been trying to compute the Homology Groups $ H_n $ of two tangent 2-Spheres (we will call this space, X). By previous results, I am able to easily determine that $ H_0(X) $ is isomorphic to the ...
0
votes
1answer
73 views

Cayley graph of the fundamental group of the 2-torus

Does anybody knows some description or some good picture on the internet for the Cayley graph of the fundamental group of the 2-torus, when this is constructed by connected sum of two torus. I know, ...
5
votes
1answer
74 views

The Euler Characteristic of $\mathbf RP^2$ is a Fraction.

Problem 22 in Section 2.2 in Hatcher's Algebraic Topology reads For $X$ a finite CW complex and $p:\tilde X\to X$ an $n$-sheeted covering space, show that $\chi(\tilde X)=n\chi(X)$. Here $\chi$ ...
3
votes
1answer
73 views

Non contractible space with contractible suspension

I know that there exist non-contractible spaces $X \not\simeq \ast$ with contractible suspension $\Sigma X \simeq \ast$. For instance the 2-skeleton of the Poincaré homology 3-sphere is such a space. ...
3
votes
1answer
67 views

Correspondence $\{$principal $G$-bundles on $M\}\leftrightarrow\{$conjugacy classes of homomorphisms $\pi_1(M)\to G\}$

Context. I'm reading Qiaochu's short note Surfaces and the representation theory of finite groups which aims to prove Mednykh's formula inspired by ideas from topological quantum field theory. On page ...
2
votes
2answers
76 views

In the quotient topology $D^2/{S^1} \cong S^2 $

Let $X = D^2 = \{(x, y) \in R^2\ : x^2 + y^2 \le 1\}$ be the closed unit disc (in the standard topology). Identify $S^1$ with the boundary of $D^2$. Now I have to prove that $$D^2/S^1\cong S^2$$ I ...
3
votes
2answers
80 views

Is there any 'nice' space with fundamental group $\mathbb{Z}_3$?

I'm trying to build up intuition for the fundamental group, as it occurs in physics. In the simplest examples, the fundamental group is trivial, or $\mathbb{Z}$, or $\mathbb{Z}^n$. We can also get $\...
1
vote
1answer
58 views

Arc connectedness of Telophase topology [duplicate]

In Counterexamples in Topology Book by Lynn Steen i found that Telophase topology is arc connected. How can we build an arc in this topology ? And also could you give me an idea of how to prove that ...
2
votes
2answers
33 views

Compositions of homotopic maps are homotopic

I'm reading some lecture notes on homotopy, and the author has just proved the theorem: If $f_{0} \simeq f_{1}$ and $g_{0} \simeq g_{1}$, then $g_{0} \circ f_{0} \simeq g_{1} \circ f_{1}$ ...
4
votes
0answers
72 views

what is a (co)homology theory?

There are different axioms for a (co)homology theory: Homotopy invariance is always there, but for the rest there is or isn't: long exact sequence for pairs of topological spaces exact sequence for ...
3
votes
2answers
155 views

projective space, constant function, homotopy

Show, that every continious function $f:\mathbb{R}P^2\to S^1$ is a homotopy to a constant function. (Hint: Has $f$ a lift $\tilde{f}:\mathbb{R}P^2\to\mathbb{R}$) Hello, I want to solve this ...
2
votes
1answer
98 views

Generators of the fundamental groups of the 8-figure and the torus

I have two doubts strictly related to each other. 1) Firstly, consider the $8$-figure, namely the union of two circles in a point $x_1$. Using the Seifert-Van Kampen's theorem I proved that its ...
2
votes
0answers
95 views

“admissible” maps from context

I have been reading Massey's Algebraic Topology and on page 158 came across the following "semi-mystical principle" which he says guides much mathematical research: Whenever we wish to gain ...
1
vote
0answers
211 views

Deformation retract of wedge sum

Let $(X_\gamma)_{\gamma \in \Gamma}$ be a collection of topological spaces, and let $x_\gamma \in X_\gamma$ be a fixed point for each $\gamma$. Fix some $\alpha \in \Gamma$, and suppose that for $\...
5
votes
1answer
53 views

Characterizing spaces with no nontrivial covers

I know that simply connected locally path-connected spaces have no nontrivial covers. Is there a characterization of spaces with this property?
0
votes
2answers
47 views

Hatcher basic terminology/phrasing

I'm trying to self-study some algebraic topology, reading Hatcher. His questions seem much less straightforwardly worded than Munkres - with Munkres it was always clear that you weren't expected to ...
1
vote
1answer
40 views

Proof of More Flexible Mayer-Vietoris for Calculating Homology Groups

On pg. 150 of Hatcher's Algebraic Topology, the author writes: Let $X$ be a topological space and $A$ and $B$ be subspaces of $X$ such that $X=A\cup B$. Suppose there are open subspaces $U$ and $V$ ...
3
votes
1answer
37 views

Preimage of a simply closed curve under the two-dimensional antipodal map

Suppose $p:S^2\to P^2$ is the quotient antipodal map, and $J$ is a simply closed curve in $P^2$, then $p^{-1}(J)$ is either a simply closed curve in $S^2$, or two disjoint simply closed curves in $S^2$...
4
votes
1answer
41 views

Cohomology class of zero set of a section

Say we have a rank $r$ smooth vector bundle $E\to X$ and a smooth section $s:X\to E$ of it. Shortly after the 30 min mark in this video Joe Harris defines the $r$th Chern class of this bundle to be ...
1
vote
1answer
19 views

Relative Homology of the Mapping Cylinder w.r.t a Subspace

Given a continuous map $f:X\to Y$, the mapping cylinder of $f$ is defined as the space obtained from $(X\times I)\sqcup Y$ by identifying $(x, 1)$ with $f(x)$ for all $x$. Let $f:S^n\to S^n$ be a ...
1
vote
0answers
41 views

Bott's topological obstruction to integrability

Let $M$ be a smooth manifold, $E\subseteq TM$ be an integrable distribution and $\pi:TM\to Q=\frac{TM}{E}$ the quotient bundle with $\dim Q=q$. Bott showed that \begin{equation} Pont^k(Q)=0 \qquad \...
5
votes
1answer
44 views

Concluding $\Bbb Z$-cohomology from $\Bbb Z_2$-cohomology using Bocksteins

According to a theorem of Serre, the cohomology algebra $H^*(K(\Bbb Z,3); \Bbb Z_2)$ is a polynomial ring on elements $\iota_3, \,\operatorname{Sq}^2(\iota_3), \,\operatorname{Sq}^4\operatorname{Sq}^2(...
0
votes
1answer
36 views

Proof of the cellular boundary formula

I'm trying to understand the proof in Hatcher (p. 141) of the cellular boundary formula. Now there's one thing that Hatcher does several times in his book and that I don't understand very well: he ...
1
vote
0answers
60 views

Proving a function isn't homotopic to a map to the boundary

Let $X$ be a compact Hausdorff space and $S$ a finite dimensional, convex, Hausdorff space. Moreover, let $f:X \to S$ be a closed, surjective map with $\tilde{H}^q(f^{-1}(s))=0$ for all $q\ge 0$ and $...
2
votes
1answer
44 views

Proving that a map is a weak homotopy equivalence

Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting diagrams where all the morphisms are serre fibrations $$\begin{...
6
votes
1answer
184 views

How to know if you are “tough enough” to study Algebraic Topology [closed]

I am graduating with a BA this summer and I am very interested in topology. I admit it, I never went that deep into topology and all I know is about point-set topology (metric spaces etc.) but from ...
0
votes
1answer
46 views

Projective space, $S^n$

We observe the projective space $\mathbb{R}P^n$ for $n>1$. Let $e\in S^n$ be random. a) The quotient map $p:S^n\to\mathbb{R}P^n$ is an overlapping and $U_i:=\{p(x):x\in S^n, x_i\neq 0\}\...
6
votes
3answers
97 views

Intuitive reason why the Euler characteristic is an alternating sum?

The Euler characteristic of a topological space is the alternating sum of the ranks of the space's homology groups. Since homeomorphic spaces have isomorphic homology groups, however, even the non-...
0
votes
1answer
48 views

A modern approach to homotopy theory in $\mathbf{SSet}$

I'm currently trying to understand the basics of homotopy theory for simplicial sets. However, my current sources (Peter Mays "Simplicial objects in algebraic topology" and Kans original "on c.s.s. ...
2
votes
1answer
54 views

Hodge theory on semi-Riemannian manifolds [reference request]

I need to learn a bit of Hodge theory on manifolds and I am looking for a reference which covers the case where the metric has arbitrary signature $(p,q)$. Most books I have found seem to focus on the ...
0
votes
0answers
25 views

Let U be a simple connected open set in R×R. If C is a simple closed curve lying in U, then each bounded component of R×R - C also lies in U.

Suppose U = B(0,2), C is a simple closed curve lying in U, then theorem obviously true competition. Let U be a simple connected set in R×R?
1
vote
0answers
62 views

Borsuk lemma doesn't hold if f is not injective. [closed]

Give an example to show that the conclusion of the Borsuk lemma need not hold if the map is not injective. Statement of the aforementioned lemma: Let $A$ be a compact topological space and $f:...
5
votes
1answer
80 views

Why does Van Kampen Theorem fail for the Hawaiian earring space?

The Hawaiian earring space has notoriously complicated fundamental group, and is essentially not as simple as the wedge sum of countably many circles whose fundamental group is straightforwardly given ...
2
votes
1answer
54 views

Degree formula for smash product

Let $f : S^n \to S^n$ and $g : S^m \to S^m$ be two maps with degrees $d_f$ and $d_g$ respectively. These two map gives rise to a map $f \wedge g : S^{n+m} \to S^{n+m}$. My question is how the degree ...
0
votes
2answers
59 views

submanifold with same homology

Suppose $M$ is a manifold without boundary, and $N\subseteq M$ is any submanifold, possibly with boundary. If $H_*(N)\cong H_*(M)$, is it necessarily true that $N\cong M$?
0
votes
0answers
40 views

Calculating The Fundamental Group of the Hopf Fibration

I want to calculate the fundamental group of the Hopf fibration (or rather, the fundamental group of the total space of the fibre bundle that is the hopf fibration). I know that $S^3$ is simply ...
0
votes
0answers
15 views

winding number, “function degree”

For every continuous function $f: S^1\to S^1$ with $f(1)=1$ we define the "function degree" $deg(f)$ as winding number of the path $f\circ w_1$ Show, that: a) $deg(f)=n\Leftrightarrow f\circ ...
2
votes
0answers
35 views

Proving that Emb($D^m,N$) is homotopy equivalent to $V_m(TN)$

I am reading online lecture notes by John Francis on h-principle. I want to prove that Emb($D^m,N$) is homotopy equivalent to $V_m(TN)$ where $V_m(TN)$ is stiefel bundle of the tangent bundle on $N$....
0
votes
1answer
50 views

$\pi_1(S^n) = 0$ for $n \geq 2$

I have few of questions for the following proof in hatcher's book. (1)Why f being continous imply that for each $s \in I$ has an open neighborhood $V_s$ in I mapped by f to some $A_{\alpha}$. (2)Why ...
1
vote
0answers
63 views

Generalized Euler Characteristic

I was asking myself what kind of generalizations of the euler characteristic are there? I've heard about the homotopy cardinality, but I was rather interested in a construction involving generalized ...
4
votes
0answers
84 views

Topology on $\mathcal{C}(X,Y)$ to work with homotopy.

We know that the compact open topology on $\mathcal{C}(X,Y)$ is a good choice for topology on the set of continuous maps, but this seems really efficient, both naively and with respect to existence of ...
2
votes
1answer
47 views

Cohomological AHSS for projective space $\mathbb{C}P^n$ ALWAYS collapses at the second page

While I was computing the cohomology ring $E^*(\mathbb{C}P^n)$ for $E$ an oriented ring spectrum, via AHSS I realised that orientation is not a necessary hypothesis in order to compute the cohomology ...
1
vote
1answer
50 views

Is the rank of this group always finite? Why?

Take $U$ an open subset of the plane. Consider $C_0(U)$ the free abelian group over the points of $U$, $C_1(U)$ the free abelian group over continuous paths (i.e. continuous maps $[0,1]\to U$), and $...
1
vote
1answer
30 views

Proving that a map is Null Homotopic

Suppose $X$ is a manifold of dimension $n$ and $f:Y \to Z$ is an $n-$connected map. Then I want to show that given any map from $g:X \to Y$, the composite map $f \circ g$ is nullhomotopic. Definition ...
2
votes
1answer
54 views

How to conceptually visualize the homotopy map?

I hope to be clear in my question, I've been meditating on the definition of Homotopy of two continuous maps and I've come to the following thought: This is the definition I'm adopting: let $f_0, f_1:...
1
vote
0answers
47 views

Relation between homology class and homology groups and betti numbers

I am reading about algebraic topology from various different sources. I found a lot of material on calculating homology groups using chain complexes and computing their betti numbers. I think I have ...
6
votes
2answers
91 views

Infinite sequence of distinct spaces, all with same homology

Using the following fact, we get infinitely many non-homotopic maps $f_k:S^{2n-1}\to S^n\vee S^n$. Fact: $\pi_{2n-1}(S^n\vee S^n)$ contains a $\Bbb Z$-summand. So we can consider the spaces $X_k=...