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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20 views

Specific topological example of Nielsen-Schreier theorem

I'm assuming that the following question should be basically trivial, and that I'm just misunderstanding something basic, but some clarification would be much appreciated. There is a section in my ...
0
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0answers
17 views

Infinite Concatenation of Homotopies

In Chapter 0 of Hatcher's Algebraic Topology book, it is proven that CW pairs $(X,A)$ have the homotopy extension property (pg 15- I would include an image, but I don't have enough reputation to do ...
1
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1answer
30 views

Projection map of a vector bundle induce isomorphism on top cohomology.

I'm reading a passage in Milnor-Stasheff about Euler class, and I noticed that he states that the projection map $$\pi \colon E \to B $$ where $(E,\pi,B)$ is a n-dim vector bundle, induces a canonical ...
2
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0answers
31 views

There is no smooth submersion from $S^2$ to $S^1$.

Show that there is no smooth submersion from $S^2$ to $S^1$. I know of one algebraic topology proof which I think is not the shortest one. That submersion is an open map should be a useful fact in ...
8
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1answer
49 views

Can (singular) homology classes always be represented by images of closed manifolds?

My intuition tells me that if $A \in H_2(M;\mathbf Z)$, then $A$ can be represented by a map $\Sigma \to M$, where $\Sigma$ is a closed (= compact boundaryless) surface, i.e., the connected sum of ...
1
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0answers
15 views

Why universal G-bundles are contractible?

Let $G$ be a nice topological group and $E\to B$ a universal $G$-bundle. I'm interested to a proof of contractibility of $E$ using only the universal property of it. I also know that if there is a ...
1
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1answer
18 views

Mayer-Vietoris sequence in reduced homology.

By using the Mayer-Vietoris sequence in reduced homology : $...\overset{\Delta_{n+1}}{\longrightarrow} \tilde{H_n}(A)\overset{E_{n}}{\longrightarrow} \tilde{H_n}(X_1)\times ...
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2answers
30 views

Homotopy equivalence between circles

I'm wondering about one thing: let's consider a plane with one hole $ \mathbb{R}^2 \setminus \{0\} $. I'm wondering whether the two subsets: $$ S^1 = \{(x,y) \in \mathbb{R}^2 \setminus \{0\}: x^2 + ...
2
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1answer
47 views

If a polynomial maps a region onto a neighborhood of zero, does it follow that it has a zero in some “robust” sense?

Let $B^n\subseteq\Bbb R^n$ be a unit ball, $P: B^n\to\Bbb R^m$ is polynomial in each component, and assume that the image of $P$ contains $0$ in its interior. Does it follow that for some $\epsilon$, ...
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0answers
32 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.
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1answer
61 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. ...
2
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1answer
41 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 ...
2
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1answer
48 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 ...
2
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0answers
45 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
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1answer
40 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 ...
3
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1answer
61 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
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0answers
25 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
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0answers
79 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
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1answer
39 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
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1answer
40 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
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0answers
47 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
57 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
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2answers
87 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
101 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
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
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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
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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
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4answers
272 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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1answer
34 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
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3answers
86 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
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
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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
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3answers
44 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
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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
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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
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1answer
44 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
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 ...
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0answers
189 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 ...
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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 ...
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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
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0answers
56 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
63 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
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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 ...
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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
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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
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2answers
244 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
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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
164 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, ...