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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2
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Applications of Topological Complexity of configuration space

I'm starting to work on Topological Complexity of configuration spaces . Articles say that it has applications in robotic and control theory . My questions are : 1) How Topological complexity can help ...
0
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2answers
41 views

Strong Topology is the strongest topology?

In his article Construction of universal bundles. II (1956), John Milnor defines the strong topology in a join of spaces, but his definition is By a strong topology in $A_1\circ A_2\circ \dots \...
4
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0answers
39 views

Is there a general way to tell whether two topological spaces are homeomorphic?

We know that if two topological spaces $X$ and $Y$ are homeomorphic, then they have the same fundamental groups, and the same homology. In other words, we have functors $$\pi_1 : \mathsf{Top} \to \...
4
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0answers
63 views

example of toric varieties with nontrivial first cohomology group

If I remember correctly, when a toric variety is smooth or simplical (the moment polytope is simplicial rational), then there is no odd dimensional cohomology group and the dimension of the even ...
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1answer
33 views

Topological Join of Unit Balls

I have seen that apparently one has for spheres that $S^n*S^m=S^{n+m+1}$. Is there a similar result for unit balls? Thank you.
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2answers
138 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, $T_{...
3
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1answer
29 views

Construct smooth mapping $f: B^{n + 1} \to S^n$ with two singularities at which $f$ has degree $+/- 1$.

I'm currently working through a paper by Pjotr Hajlasz who wants to show that For smooth manifolds $M,N$, if $\pi_{[p]}(N) \neq 0$ and $1 \leq p < n = \dim M$, then the smooth mappings $C^\...
2
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1answer
72 views

Show that the homology induced homomorphism $f_*:H_3(RP^3)\rightarrow H_3(S^2\times S^1)$ is a zero map.

Let $f:\mathbb{RP}^3\rightarrow S^2\times S^1$ be a continuous map. Prove that induced map $f_*:H_3(\mathbb{RP}^3)\rightarrow H_3(S^2\times S^1)$ is a zero map. I found that the third homology of ...
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1answer
30 views

Interpretation of points in covering spaces as homotopy classes of paths [on hold]

If $p:\widetilde{X} \to X$ is a covering map, $y \in \widetilde{X}$ determines a homotopy class of paths in $X$ joining the base point $x_0$ to the point $p(y)$. But a homotopy class of paths in $X$ ...
2
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2answers
65 views

Fundamental group contains $\mathbb{R}$ or $\mathbb{Q}$

Is there any topological space whose fundamental group contains $\mathbb{Q}$ or $\mathbb{R}$? In case of (singular) homology or cohomology, we can change its coefficients to any abelian groups (with ...
6
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1answer
61 views

For $n\geq 2$, any continuous map $f:\mathbb{C}P^n\rightarrow S^2$ induces the zero map on $H_2(*)$

I am working through an old qualifying exam from another university. My course did not cover as much material as what is on this test (e.g. we did not cover cohomology). So I am just working through ...
7
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1answer
87 views

Why the attachment to simplices in (co)homology?

I've been thinking a bit about why we define the singular homology and cohomology groups with simplices rather than, say, cubes, and it seems to me that the elementary aspects of the theory would all ...
2
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1answer
62 views

Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
3
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1answer
59 views

If two maps are homotopic, are the images homotopy equivalent?

My question is; If two continuous maps $f,g:X\rightarrow Y$ are homotopic, are the images $Im(f),Im(g)$ homotopy equivalent? Clearly, the converse is false. If it is false, is there any condition ...
4
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2answers
168 views

Broken line is NOT diffeomorphic to the real line

This is from Bredon's Topology and Geometry, page 71. This comes right after the very definition of differentiable manifold, so I think no use of tangent space or 'differential' is permitted. (Bredon ...
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1answer
20 views

Question About Covering Space Classification Theorem

I'm a bit confused by Hatchers choice of words here. He says "The main classification theorem for covering spaces says that by associating the subgroup $p_{*}(\pi_{1}(\tilde{X},\tilde{x_{0}}))$ we ...
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0answers
25 views

A doubt in Whitehead's proof about cohomology with local coefficients [on hold]

In the proof of Theorem 4.9 says that $p^*:H^n(X_n,X_{n-1};G|X_n) \to Hom(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$ has image $Hom^{\pi}(H_n(\widetilde{X}_n, \widetilde{X}_{n-1}),G_0)$. ...
2
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1answer
28 views

Map inducing zero on first cohomology is nullhomotopic (plus assumptions on fundamental group and universal cover)

Currently I am studying for a topology exam next week and came across an exercise where I could need some hints (cf. here): Let $X$ be a path-connected space with $\pi := \pi_1(X,*)$ abelian and ...
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0answers
18 views

Explicit isomorphism $H (E, E^0) \cong H_{cv} (E)$

Yo! I've been trying to understand better why $$H (D, D-\{x\}) \cong H_c (D)$$ for a disk $D$ and, more generally, why $$H (E, E^0) \cong H_{cv} (E)$$. In general, the first isomorphism can be seen as ...
4
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0answers
95 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
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0answers
15 views

Product and Join of $G$-CW-Complexes

Given a topological group $G$ and two $G$-CW-Complexes $X$ and $Y$ I want to understand the natural CW-structure on $X\times Y$ and $X*Y$. I understand that the concepts are very similar, so I want to ...
1
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1answer
36 views

If 2 loops with equal base points are homotopic, must they be homotopic relative to the base point?

Let $X $ be a topological space and $\mathbb {S}^1$ be the set of complex numbers with magnitude 1 equipped with the inherited topology from $\mathbb {C} $. If we have 2 loops $f,g:\mathbb {S}^1\...
10
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1answer
436 views

covering space of $2$-genus surface

I'm trying to build $2:1$ covering space for $2$- genus surface by $3$-genus surface. I can see that if I take a cut of $3$-genus surface in the middle (along the mid hole) I get $2$ surfaces each one ...
0
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1answer
31 views

Difficulties with the description of $p*$ in the serre spectral sequence(Bullet 7 of Mosher Tangora Page 76):

For the first difficulty, let $E^r$ be the rth page of a first quadrant spectral sequence with elements $E^r_{p,q}$ , where $p$ is the filtering degree. Difficulty 1: On bullet 7 of Mosher and ...
2
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2answers
44 views

Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
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2answers
23 views

A relation between interior and closed sets

A topological space $X$ is said to be completely regular provided that it is a Hausdorff space such that, whenever $F$ is a closed set and $x$ is a point in its complement, there exists a function $f\...
3
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0answers
48 views

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...
2
votes
1answer
47 views

Fixed-point free map of the 2-sphere which has order 4

The antipodal involution of $\mathbb{S}^2$ clearly has no fixed points. However I cannot think up an example of homeomorphism of order 4 which has no fixed points. Could you help me?
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1answer
22 views

Steenrod Operations an algebraic Approach

Assume that $q=p^{r}$, where $p$ is a prime either 2 or odd and $\mathbb{F}_{q}$ is a Galois field and $V$ a finite dimensional $\mathbb{F}_{q}$-vector space. Then due to Larry Smith in this http://...
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2answers
57 views

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$?

How to show that $\pi_1(\mathbb{RP}^2) = \mathbb{Z}_2$? In general is well known that $\pi_1(\mathbb{RP}^n) = \mathbb{Z}_2, ~ n \ge 2.$ But how to show this assertion? I have a few knowledge about ...
0
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0answers
44 views

Prove that the fundamental group of $P^2$ the real projective plane is the group with two elements

Presumably the group will just contain $e$ and another $g$. My idea is to use the proof that $\pi_1(S^1)=\mathbb{Z}$ where the covering map from $S^2$ to $P^2$ maps to the line in $\mathbb{R}^3$ ...
3
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2answers
62 views

Topology of complex projective plane

It is well known there are two ways to construct topology of $\mathbb{C}P^n$: quotient space of $S^{2n+1}$ by identifying $x$ with $\lambda x$, where $\lambda$ is complex nonzero constant. According ...
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1answer
39 views

Can we lift paths of a Lie group quotient $G\to G/H$?

Question: Let $G$ be a Lie group and $H\subseteq G$ a closed normal subgroup. Let $$\pi:G\to G/H$$ be the quotient map. If $\gamma:[0,1]\to G/H$ is a smooth path, can we find a smooth path $\tilde{...
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0answers
28 views

Universal Abelian Covering Space of genus two surface [closed]

Let M be a surface M, i am concerned with Abelian covers. These are the covering spaces for which the deck group is Abelian. The largest such cover corresponds to the commutator subgroup of the ...
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0answers
24 views

The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$

**Note:I was extremely confused when I wrote this post. Please see the linked one. I left this one as it is, because what I understand now is so radically different then what I wrote below ** Let $...
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0answers
21 views

basepoint problem: Is there an action of $\pi_1(B)$ on $\pi_1(F)$ for $F$ path connected

I am doing this to try to figure out The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$ . Let the fibration $F \hookrightarrow E \xrightarrow{p} B$ be ...
0
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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 ...
2
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0answers
25 views

If $p: E \rightarrow X$ is a covering map with $E$ connected and $|p^{-1}(x_{0})|=k$ for some $x_{o}$ then $|p^{-1}(x)|=k$ for all $x \in E$.

Prove that if $p:E \rightarrow X$ is a covering map with $E$ connected and $p^{-1}(x_{0})$ has $k$ elements for some $x_{0} \in X$, then $p^{-1}(x)$ has $k$ elements for every $x \in X$. Is my proof ...
3
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1answer
73 views

Action of $\mathbb{Z}/3\mathbb{Z}$ on $P^{1}$

I am reading from the book Topics in Galois theory by Serre. I have the following question , take $G=\mathbb{Z}/3\mathbb{Z}$. The group $G$ acts on $P^1$ by $$\sigma x\;=\;1/(1-x)$$ where $\sigma$ ...
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1answer
31 views

verifying homeomorphism of orbit space and suggestions for further study

Define an action of $\mathbb{Z}_2$ on $S^1$ by $(0,z)\mapsto z$ and $(1,z)\mapsto \bar{z}$. An orbit of $z$ is then the set $\{z,\bar{z}\}$. I claim the orbit space $S^1/\mathbb{Z}_2$ is homemorphic ...
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1answer
99 views

Is the nth homotopy group isomorphic to $[T^n, X]$

Following Spanier's book on algebraic topology chapter $1$, section $6$ about suspensions, I'm wondering about the following questions: 1) We know that $S^n$ is an $H$-cogroup for all $n\geq1$ ...
9
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1answer
166 views

When can we recover a manifold when we attach a $2n$-cell to $S^n$?

I have a question related to this one. In my answer I was going to try and say something about the possible manifolds that might arise in this way, i.e. as mapping cones of elements of $\pi_{2n-1}(S^...
6
votes
3answers
532 views

Cellular Boundary Formula

In Hatcher's book we find, when computing the boundary maps of cellular homology, Cellular Boundary Formula: $d_n(e^{n}_\alpha)=\sum_\beta d_{\alpha\beta}e^{n-1}_{\beta} $ where $d_{\alpha\beta}$ is ...
8
votes
2answers
85 views

Homology and cohomology of 7-manifold

I have the following problem: Let $M$ be a connected closed $7$-manifold such that $H_1(M,\mathbb{Z}) = 0$, $H_2(M,\mathbb{Z}) = \mathbb{Z}$, $H_3(M,\mathbb{Z}) = \mathbb{Z}/2$. Compute $H_i(M,\...
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0answers
24 views

Two sheeted covering projection

In Hatcher on page 144, example 2.42, I see $RP^n$ described as a CW structure with one cell $e^k$ in each dimension $k\leq n$, and the attaching map for $e^k$ is the 2-sheeted covering ...
2
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0answers
49 views

Cohomology Group basis

I'm reading a text on Complex Torus and Abelian Variety and at a time is written as follows: The cohomology group $H^{1}(T,\mathcal O_{T})$ has a basis $w_{j}=d\overline{z}_{j}, j=1,2,...,g,$ as ...
2
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1answer
43 views

Restriction of a $C^{\infty}$ vector bundle over a regular submanifold.

This question is about the content of page 134 of Tu's An Introduction to Manifolds. A $C^{\infty}$ vector bundle or rank r is a triple $(E,M,\pi)$ consisting of manifolds $E$ and $M$ and a ...
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1answer
44 views

If $M \rightarrow N$ and $N \rightarrow V$ are maps of connected and locally path connected spaces, show that $M \rightarrow N$ is a covering space

I'm working on some exam prep questions, and I am having a bit of trouble with this one: If $f:M \rightarrow N$ and $g:N \rightarrow V$ are maps of connected and locally path connected spaces, ...
0
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1answer
56 views

Every map $S^1 \rightarrow X$, extends to a map $D^2 \rightarrow X$

I am reviewing some solutions for past papers for an upcoming exam, and I am a bit confused with the solution for this question. I am hoping someone can shed some light for me: I want to prove that: ...