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

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2
votes
0answers
42 views

The fundamental group of $S^n$

I want to prove that $\pi_1(S^n,x_0)$ is trivial if $2\leq n,$ BUT using universal covering. So let $p:\tilde S^n \rightarrow S^n$ the universal covering. Define $f:D^n\rightarrow S^n$ such that ...
6
votes
1answer
385 views

A question about the contractibility of the Sierpinski space

The two-point Sierpinski space is usually defined as follows: Let $X =\{x,y\}$ be the two-point space where the only open sets are $X, \varnothing, \{x\}$. I think from this it can be inferred that ...
2
votes
0answers
25 views

Does the product functor preserve quotient maps?

In Hatcher's Albebraic Topology, he presents a proof that if $(X,A)$ satisfies the homotopy extension property, and $A$ is contractible, then $X \simeq X/A$. Part of Hatcher's proof goes: Suppose ...
0
votes
0answers
30 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 ...
1
vote
1answer
25 views

Does every n-chain have a homology class?

I was under the impression that not every (singular) $n$-chain has a homology class, since $H_n(X) = Z_n(X)/B_n(X)$, and not every $n$-chain is an $n$-cycle. But I came across the following in ...
8
votes
1answer
52 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 ...
0
votes
0answers
20 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
vote
1answer
65 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. ...
1
vote
1answer
31 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
votes
0answers
36 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 ...
1
vote
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 + ...
6
votes
1answer
146 views
+300

Intersection theory proof of the poincare hopf theorem.

Suppose that $M$ is a connected compact oriented smooth manifold, and $X:M\rightarrow TM$ a vector field with isolated zeros. Then if $Z$ is the zero set of $X$ (a $0$-dimensional oriented manifold), ...
1
vote
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 ...
3
votes
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
vote
1answer
20 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 ...
6
votes
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 $$ [M \mathbin\sharp N] = [M]+[N]$$ ...
3
votes
1answer
50 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 $\epsilon$, ...
2
votes
0answers
48 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
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
votes
2answers
123 views

$S^2$ with countably many points removed is path-connected

Prove that after removing a countably infinite number of points from $S^2$, it remains path-connected. This was a question that arose in the algebraic topology course I have this term. I thought ...
-6
votes
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.
2
votes
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
votes
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
votes
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 ...
2
votes
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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
votes
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 ...
2
votes
0answers
48 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 ...
3
votes
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?
15
votes
0answers
191 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 ...
4
votes
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
vote
1answer
34 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 ...
1
vote
0answers
170 views

Proving that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$

I'm trying to prove that $\deg(fg) = \deg (f) \deg (g)$ for $f:S^1 \to S^1$. The intermediate step is proving that: if $a$ is a lift of $f \circ \exp$ and $b$ is a lift of $g \circ \exp$ then $a+b$ ...
0
votes
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 ...
4
votes
0answers
65 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 ...
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
273 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. ...
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
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 ...
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
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
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 ...
6
votes
2answers
59 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 ...
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 ...
7
votes
1answer
58 views

Boundary of boundary of singular cube is zero (Spivak)

At the bottom of page 99 of M. Spivak's Calculus on Manifolds he arrives at the formula $$\partial (\partial c)=\sum_{i=1}^n \sum_{\alpha=0,1} \sum_{j=1}^{n-1} \sum_{\beta=0,1} ...
15
votes
1answer
375 views

Can we generalize the regular value theorem even beyond the Ehresmann's theorem?

The formulation is complicated, but the answer may be some clever usage of the partition of unity, because locally the answer is given by the regular value theorem and the whole problem is to glue it ...
1
vote
1answer
41 views

“Cohomology classes correspond to homotopy classes of maps to Eilenberg Maclane spaces” and cup product?

I read this in Hatcher. I am especially interested in knowing if the cup product can be understood from this perspective? I would appreciate a reference.