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

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-6
votes
0answers
23 views

quotient space obtained by S^2 modula equator S^1 with equivalent relation x~-x [on hold]

This is Problem 10 in Chapter 2 of Hatcher's book, but I do not know how to prove it.
0
votes
1answer
23 views

Covering spaces of $S^1$

Put $\tilde X=\lbrace (exp(2\pi if(t),t)| t\in \mathbb{R} \rbrace$ where $f:\mathbb{R}\rightarrow \mathbb{R}$ is any continuous function and let $\pi_1$ be the projecction on the first coordinate. If ...
2
votes
1answer
35 views

A morphism of principal bundles is an isomorphism.

Let $p_1:E_1\to B$ and $p_2:E_2\to B$ be two principal $G$-bundles. Let $f:E_1\to E_2$ be a principal $G$-bundle morphism. I want to show that $f$ is an isomorphism. I read somewhere that it is ...
1
vote
1answer
41 views

$S^1 \times S^2$ vs $S^1 \vee S^2 \vee S^3$

This is a multi-part problem. Let $X = S^1 \times S^2$ and $Y = S^1 ­\vee S^2 \vee S^3.$ Compute $\pi_1$ of those spaces. Do there exist $\phi:S^3 \to X$ and $\psi:X \to S^3$ such that $\psi \phi ...
1
vote
0answers
18 views

Do the paths of a deformation retraction cover the boundary?

Let $A$ be a compact set in $R^n$, $U$ its open neighbourhood, and $H:U\times I \to U$ a strong deformation retraction of $U$ onto $A$. It seems plausible that for any point $a\in\partial A$ there ...
2
votes
1answer
38 views

Groups $\pi$ with $K(\pi, 1)$ a finite CW-complex

For what (say, finitely generated) groups $\pi$ is $K(\pi, 1)$ a finite CW-complex? Such a group must necessarily be torsion-free, since otherwise $H^*K(\pi, 1) = H^* \pi$ would be nontrivial in ...
2
votes
0answers
41 views

Homology of mapping cone

Let $f:X\to Y$ be a map, and $\text{cone}(f) = CX \sqcup_f Y$ its mapping cone. Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be a homology theory with values in the category of $R$-modules. ...
1
vote
0answers
22 views

Simplicial function space and homotopy colimits

I am currently reading the book by Bousfield and Kan, in particular Ch. XII, par. 2, and would like to understand why the functor $hocolim: Top_{+}^{I} \rightarrow Top_{+}$ is left adjoint to $hom(I ...
6
votes
0answers
65 views

Fundamental class of the connected sum of two closed orientable manifolds

I need to find a representation of $ [M \mathbin\sharp N] \in H_n(M \mathbin\sharp N) $ in terms of the fundamental classes $[M]$ and $[N]$. My idea is that $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
1
vote
1answer
37 views

An exhaustive continuous map is a covering map.

$p_1:\tilde X_1 \rightarrow X \, ; \, p_2:\tilde X_2 \rightarrow X$ two coverings maps, where $X$ connected and locally path-connected, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
3
votes
1answer
38 views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
2
votes
0answers
45 views

When the free loop space fibration splits?

Let $X$ be a (nice) connected topological space. Let $LX=Map(S^1,X)$ be the free loop space and $\Omega X = Map_*(S^1,X)$ the subspace of based loops (with some choice of base point for X). Now, there ...
2
votes
2answers
54 views

constructing a CW Complex

I am looking at an example of constructing a CW complex for a space X. The example i am looking at is that for The quotient of $S^2$ obtained by identifying north and south poles. The solution is as ...
2
votes
2answers
85 views

How to prove $\deg( f\circ g) = \deg(f) \deg(g) $?

If $ f,g:S^1 \rightarrow S^1$ continuous maps then \begin{equation*} \deg( f\circ g)= \deg(f)\deg(g). \end{equation*} Unfortunately, i haven't made any progress in solving it. I've tried considering ...
4
votes
2answers
100 views

Classify sphere bundles over a sphere

Problem (1) Classify all $S^1$ bundles over the base manifold $S^2$. (2) Do the same question for $S^2$ bundles. Moreover, does there exist a universal method to solve this kind of ...
1
vote
1answer
33 views

Multiplicative spectral sequence

I have a simple question regarding the definition of a multiplicative spectral sequence, which I couldn't answer myself by looking at the definitions in various texts: Is the product assumed to be ...
0
votes
1answer
34 views

Why is $Z*Z/({xyx^{-1}y} )= 1$?

I was reading notes on the Fundamental Group and came across the statement $Z*Z/({xyx^{-1}y}) = 1$. Why is this true? If a quotient group is trivial doesn't it mean that the two factors are ...
1
vote
1answer
69 views

Homology functors defined on $\mathsf{Top} \times \mathsf{Top}$ in Eilenberg-Steenrod axioms?

The Eilenberg-Steenrod axioms state that a homology theory is a sequence of functors $H_n : \mathsf{Top} \times \mathsf{Top} \to \mathsf{Ab}$ satisfying some additional properties. What I don't ...
6
votes
4answers
271 views

An introduction to algebraic topology from the categorical point of view

I'm looking for a modern algebraic topology textbook from a categorical point of view. Basically, I'd like a textbook that uses the language of functors, natural transformations, adjunctions, etc. ...
2
votes
1answer
32 views

Homology of homotopy fiber of degree map between spheres

From Hatcher's Spectral Sequences: Compute the homology of the homotopy fiber of a map $S^k → S^k$ of degree $n$, for $k,n > 1$. Here's where I am: For $k > 1$, the sphere $S^k$ is ...
3
votes
3answers
83 views

Compute the arithmetic genus of the union of the three coordinate axes $Z(x_1x_2,x_1x_3,x_2x_3) \subseteq \mathbb{P}^3$

Compute the arithmetic genus of the union of the three coordinate axes $Z(x_1x_2,x_1x_3,x_2x_3) \subseteq \mathbb{P}^3$ The arithmetic genus of $X \subseteq \mathbb{P}^n$ is ...
0
votes
1answer
21 views

The action of the orthogonal group $O_n(\mathbb{R})$ on the Stiefel manifold $V_{k,n}(\mathbb{R}) $ .

I'm trying to prove that $O_n(\mathbb{R}) / O_{(n-k)}(\mathbb{R}) \cong V_{k,n}(\mathbb{R})$ where $ O_n(\mathbb{R}) = \left\lbrace A \in M_n(\mathbb{R}) / A A^t = I_{n}\right\rbrace $ is the ...
0
votes
1answer
33 views

$\mathbb{R}^{2}$ and $\mathbb{R} \times [0, +\infty]$ are homotopy equivalent, but not homeomorphic

So, let's consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ - two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ ...
1
vote
3answers
43 views

Homology of $P^n$ minus a point

Denote real projective space by $P^n.$ I want to compute $H_*(P^n - p)$ (for some $p \in P^n$) given that I already know $H_*(P^n).$ Now we know that, as for any $n$-manifold, the local homology ...
1
vote
1answer
45 views

Show $H_{dR}^1(S^n)=0$ for $n>1$ without de-Rham Theorem, and some similar questions.

Question Without using de Rham's theorem, prove: (1) Show $H_{dR}^1(S^n)=0$ for $n>1$. (2) Use (1) to show $H_{dR}^1(RP^n)=0$ (3) a n-form $\Omega$ is exact on $S^n$ if and only if $$ ...
0
votes
2answers
38 views

Group homomorphism on unit circle

For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$. How would one compute the induced group ...
6
votes
2answers
58 views

Homotopical perspective on the long exact sequence in homology and Mayer-Vietoris

For a while I have been wondering whether the long exact sequence in homology and the Mayer-Vietoris sequence can be phrased in homotopical terms. Recently, I heard that both may be reformulated via ...
2
votes
3answers
39 views

What are the subgroups of $(S^1)^n$ isomorphic to the standard copy of $(S^1)^k$?

Let $H$ be the set of all subgroups of the $n$ dimensional torus $(S^1)^n$ that are isomorphic by an element of $Aut ((S^1)^n)$, the set of continuous automorphisms of $(S^1)^n$, to the standard copy ...
0
votes
1answer
43 views

does the closure of interior of a set equal to closure of this set?

Does the $\text{Cl}(\text{Int} A)=\text{Cl}(A)$? Here "Cl" denotes closure, "Int" denotes interior. That is not a duplicate of the question of "does the closure of interior of a set equal t the ...
2
votes
3answers
55 views

Does there exist a surjective continuous map $D^2 \to S^1$?

By considering the induced homomorphism on the fundamental groups, we know that there is no retract $D^2 \to S^1$. But is there any continuous surjection from $D^2$ to its boundary? It seems unlikely ...
15
votes
0answers
181 views

When is there a submersion from a sphere into a sphere?

(Edit: Now posted to MO.) That is: For which positive integers $n, k \ge 1$ does there exist a submersion $S^{n+k} \to S^k$? The discussion at this math.SE question has narrowed it down to the ...
1
vote
1answer
31 views

Problem about covering space

Let $p:\tilde{X}\to X$ be a covering space, $\tilde{X}$ and $X$ are both path-connected and locally path-connected, if $p(x_1)=p(x_2)=x$, is ...
1
vote
0answers
44 views

Why is the Hopf link the only link with knot group $\mathbb{Z} \oplus \mathbb{Z}$?

We can use the Loop Theorem to show that if $\Sigma$ is a minimal-genus Seifert surface for a link $L$, then $\pi_1(\Sigma)$ injects into the knot group $\pi_1(S^3 \setminus L)$. An orientable ...
2
votes
0answers
55 views

Examples of vector bundles

I know three examples of interesting vector bundles (not taking into consideration what one gets from them using algebraic operations): Tangent bundle of a smooth manifold Tautological bundle over ...
3
votes
0answers
61 views

Hatcher 2.1.10…

Hatcher asked a question in chapter 2 (a) Show the quotient space of a finite collection of disjoint 2-simplices obtained by identifying pairs of edges is always a surface,locally homeomorphic with ...
3
votes
3answers
98 views

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map?

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map? I get the picture that if the identity map $id$ is homotopic to the constant map then as the circle transforms ...
1
vote
1answer
43 views

simplicial homology definition

I am revising for my algebraic topology exam based on Hatchers 2nd chapter of algebraic topology and I have this question regarding the definition of simplicial homology on pages 104-105: Hatcher ...
2
votes
1answer
60 views

Eilenberg–Maclane space: nonvanishing cohomology

I would like to prove (self study) that $H^{np}(K(\mathbb{Z},n),\mathbb{Z})\neq 0$, where $n$ is even , $p\geq1$ and $K(\mathbb{Z},n)$ is the Eilenberg–Maclane space. I used the adjunction between $[ ...
17
votes
2answers
236 views

Existence of submersions from spheres into spheres

I would like to know if there exist submersions $f\colon \mathbb{S}^4\to \mathbb{S}^2$ and $g\colon\mathbb{S}^6\to \mathbb{S}^2.$ It is simply a question of curiosity. Any suggestion or ideas are ...
2
votes
0answers
31 views

Brouwer fixed-point theorem on non-convex sets [duplicate]

I read about the Brouwer fixed-point theorem in Wikipedia, and got confused about whether it holds when the domain of the function is non convex. On one hand, convexity is explicitly mentioned as a ...
2
votes
3answers
163 views

Winding maps of spheres?

I know that the second homotopy group of $S^2$ is $\mathbb{Z}$. I also know that a simple representative of the class of maps that generates this group is the identity map on the sphere, ...
0
votes
1answer
42 views

How many connected components does the punctured cone of isotropic vectors have?

Consider a real vector space $T$ of dimension $p+q$ with a non-degenerate symmetric bilinear form, $B:T\times T\to\mathbb{R}$, with signature $(p,q)$. Define the cone $$ ...
5
votes
1answer
96 views

If the action is free, is it necessarily a covering space action?

Suppose a group $G$ acts simplicially on a $\Delta$-complex $X$, where "simplicially" means that each element of $G$ takes each simplex of $X$ onto another simplex by a linear homeomorphism. If the ...
6
votes
1answer
61 views

When does covering preserve rational cohomology?

Let $X$ and $Y$ be compact manifolds. $p:X \rightarrow Y$ is a covering. Generally it is not true that $$ H^* (X , \mathbb{Q} ) = H^* (Y , \mathbb{Q} )$$ For instance, if $Y$ is a sphere with $y$ ...
5
votes
2answers
61 views

Prob. 5 (a) in Supplementary Exercises in Munkres' TOPOLOGY, 2nd ed: How to show that this map is a homeomorphism?

Let $G$ be a topplogical group, and let $H$ be a subgroup of $G$. Let $G / H $ denote the collection of all left cosets of $H$ in $G$, and let $a$ be a fixed element of $G$. Let the map $f \colon G / ...
3
votes
2answers
119 views

First book on algebraic topology

Is there a good book on introduction to Algebraic Topology which is self-contained and does not assume any background in topology, only standard calculus and linear algebra courses?
2
votes
0answers
89 views
+50

The map $\lambda: H^*(\tilde{G}_n)\to H^*(\tilde{G}_{n-1})$ maps Pontryagin classes to Pontryagin classes; why?

In Milnor/Stasheff Characteristic classes on page 180 there is a statement (inside a proof) that I don't fully understand. Let $\tilde{G}_k$ be the oriented real grassmanian, $e$ its Euler class: ...
0
votes
0answers
28 views

An introduction to monodromy in topology and algebra

I'm looking for some basic references on monodromy and monodromy groups, in particular I'd like something which describes the interplay between the topological definition (in the theory of covering ...
6
votes
1answer
81 views

What is the class of topological spaces $X$ such that the functors $\times X:\mathbf{Top}\to\mathbf{Top}$ have right adjoints?

For any topological space $X$, define a functor $\times X:\mathbf{Top}\to\mathbf{Top}$ by $Y\mapsto Y\times X$ (and acting on the hom-sets in the natural way). I know that if $X$ is locally compact, ...
0
votes
0answers
37 views

Cone of projective space?

Is the cone of $\mathbb{C}\mathbb{P}^2$ a familiar topological space? What about $\mathbb{C}\mathbb{P}^3$? I'm having a lot of trouble visualizing it. I just learned the notion of the cone of a ...