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

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0
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1answer
20 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-conneted, and suppose that $f:\tilde X_1 \rightarrow \tilde X_2$ is an ...
3
votes
1answer
29 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?
1
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0answers
35 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
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2answers
47 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
81 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
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2answers
96 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
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1answer
30 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
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1answer
31 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
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1answer
61 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
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4answers
252 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
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0answers
20 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 ...
2
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2answers
59 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
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1answer
20 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
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1answer
30 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
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3answers
39 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
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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
55 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
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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
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3answers
53 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 ...
14
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0answers
157 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
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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
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0answers
37 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
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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
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0answers
58 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
42 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
218 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
162 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
60 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
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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
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2answers
118 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
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0answers
87 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
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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
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0answers
36 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 ...
1
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0answers
42 views

Projection of a covering map over product set.

Let $p,q$ be a covering maps, $p:\tilde X \rightarrow X$ and $q:\tilde Y \rightarrow X$ and let $Z=\lbrace(\tilde x, \tilde y)| p(\tilde x)=q(\tilde y) \rbrace$, I want to proff that $f:\tilde ...
5
votes
1answer
47 views

Spectral sequence for computing the homotopy fixed points in unstable equivariant homotopy theory

I'm reading Carlsson's "A survey of equivariant homotopy theory" and I have a question. Let $G$ be a topological group and $X$ be a $G$-space (for a nice notion of "space"). He defines ...
1
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1answer
77 views

Homotopy lifting property of $\mathbb{R} \to S^1$ in Hatcher

I am reading Hatcher's proof of the homotopy lifting property of the covering map $p: \mathbb{R}\to S^1$. Starting with a homotopy $F: Y \times I \to S^1$ and a map $\tilde{F}:Y \times \{0\} \to ...
5
votes
1answer
46 views

If the top Stiefel-Whitney class of a compact manifold is nonzer0, must there be another non-vanishing Stiefel-Whitney class?

I was trying to collect some examples of Stiefel-Whitney class computations, just to make myself more familiar with them. It seems from my (relatively short list) that if the top Stiefel-Whitney ...
2
votes
1answer
38 views

$H_n(S^n,A)$ is not trivial

Let $(H_n)_{n\in \Bbb{Z}}, (\partial_n)_{n\in \Bbb{Z}}$ be an ordinary homology theory with values in the category of $R$-modules. Let $A\subset S^n$ be a proper subset. Then $H_n(S^n, A)$ is not ...
1
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0answers
60 views

Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...
1
vote
2answers
61 views

Splitting of Singular Homologies

In Singular homology, let $C_n(X)$ be the free abelian group generated by all the $n$-siimplices of the topological space $X$. Let $U$ be a subspace of $X$, then we have a spliting sequence ...
0
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2answers
67 views

Why does $\overline{\alpha * \beta}=\bar{\beta} * \bar{\alpha}$

I'm working on this question from Munkres' topology: Let $\alpha$ be a path in $X$ from $x_0$ to $x_1$; let $\beta$ be a path in $X$ from $x_1$ to $x_2$. Show that if $\gamma = \alpha * \beta$ , then ...
2
votes
1answer
58 views

Showing two spaces are homotopy equivalent

So I understand the basics about homotopy, I know a punctured disk or $\mathbb{R}- \{ 0 \}$ are homotopy equivalent to $\mathbb{S}^1$. This can be shown using the deformation retract ...