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

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2
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0answers
46 views

Intuitive Approach to Sheaf and Cech Cohomology

Sheaf and Cech cohomology $H^*(X,\mathcal{F})$ (which give the same result when applied to good enough topological spaces) are a useful generalisation of the concepts of de Rham and Dolbeault ...
4
votes
1answer
52 views

Proving one version of equivariant formality

Let $G$ be a compact, connected Lie group acting smoothly on a compact, connected and oriented smooth manifold $M$. We denote by $H_G^*(M)$ the corresponding equivariant cohomology. We have a ...
4
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2answers
111 views

Alternate construction of the universal cover of a space

Suppose you have a connected, locally path connected Hausdorff space $Y$ that admits a universal covering (i.e. is semilocally simply connected). It occured to me that maybe one can describe the ...
2
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2answers
73 views

First Cohomology Group

Is it true that the first cohomology group of a differentiable manifold with finite fundamental group is trivial? If so, could you explain why? Thanks very much
0
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1answer
28 views

punctured Mobius band in high dimension

Let $M^{n+1}$ be the non-trivial line bundle over sphere $S^n$. When $n=1$, $M^2$ is Mobius band and the punctured space $M^2\setminus *\simeq S^1$. How about $M^{n+1}\setminus *$ in general? Does ...
4
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1answer
71 views

No Natural Group Structure on Conjugacy Classes

Is it possible to show that there is no natural group structure on conjugacy classes in a group? Alternatively, for a path connected space $ X $, the set $ [S^1,X] $ of free (unpointed) homotopy ...
3
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0answers
31 views

Graph theory application of homology

I am struggling with the idea of local homology groups and would like to see an example of how to go about finding them in general. I'm thinking of the most trivial case to apply the theory of local ...
1
vote
2answers
43 views

Topological group closed path

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$, $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let $f,h$ be closed paths ...
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0answers
12 views

Nested subspace and the fundamental group.

Let $X$ be a topological space, and for each positive integer $n$ let $X_n$ be an arcwise- connected subspace containing the base point $x_0\in X$. Assume that the subspaces $X_n$ are nested, i.e., ...
1
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1answer
18 views

homotopic closed paths in topological group

A topological group $G$ is a group that is also a topological space in which the maps $u:G×G → G$ $v:G→G$ defined by $u(g_1, g_2)=g_1g_2$ and $v(g)=g^{-1}$ are continuous. Let f,h be closed paths in ...
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0answers
27 views

Constructing chain homotopy equivalence related to mapping cones

Problem (Weibel, Introduction to Homological Algebra, Exercise 1.5.8) Given a map $f\colon B\to C$ of complexes, let $v$ denote the inclusion of $C$ into $\operatorname{cone}(f)$. Show that there is ...
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3answers
53 views

Does $\chi(X) = \chi(Y)$ imply $H_p(X) \cong H_p(Y)$?

Does there exist spaces $X$ and $Y$ such that $\chi(X) = \chi(Y)$ such that there is (at least one) $p$ such that $H_p(X) \ncong H_p(Y)$? If so, what can we say about spaces with the same Euler ...
0
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1answer
25 views

Prove that the set of all vector fields $V(S^1)$ is a free $C^{\infty}(S^1)$-module

I need to prove that the set of all vector fields, $X:S^1\to TS^1$ name it: $V(S^1)$, is a free $C^{\infty}(S^1)$-module. So i need a basis $\frac {d}{dx_1},...,\frac {d}{dx_n}$ for $V(S^1)$.It's easy ...
1
vote
1answer
26 views

Tensor product of flat real line bundles and triviality.

Let $L$ be a real line bundle over a manifold $M$. Equivalence classes of real line bundles over $M$ are in correspondence with elements of $H^{1}(M,\mathbb{Z}_{2})$ through the first Stiefel-Whitney ...
1
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2answers
49 views

Show that the plane $\mathbb{R}^2 - \{(-1,0), (1,0)\}$ is homotopy equivalent to $C(1,0) \cup C(-1,0)$

I'm trying to find a deformation retract of the union of the two circles to $\mathbb{R}^2 - \{(-1,0), (1,0)\}$, any help is appreciated.
1
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1answer
60 views

Universal covering space of CW complex has CW complex structure

We know that covering spaces of many low dimensional CW complexes such as graph (CW structures with only 0-cells and 1-cells) and all compact surfaces has CW structure. [The second fact is due to ...
0
votes
1answer
21 views

Reference for a particular case of the classification theorem of covering spaces

Let $X$ be a connected topological space (maybe some other hypothesis should be imposed on $X$). Then I'd like a reference for the following result: The sets: $$A=\{\text{equivalence classes ...
0
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0answers
52 views

Thurston's Geometric structure for 3-manifold

I have an orientable 3-manifold $X$ , such that $$X=\lbrace(x,y,z)\mid x\neq y \neq z \neq x \rbrace\subseteq S^1\times S^1 \times S^1 $$ I would like to know the geometric structure on X. My ...
1
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0answers
46 views

Vector bundle is homotopy equivalence [duplicate]

If $\pi:E\rightarrow B$ is a vector bundle (I allow it to be of nonconstant rank), then I want to prove that it is a homotopy equivalence. As a homotopy inverse I propose the zero section ...
0
votes
1answer
21 views

how to split two tubes intersecting in a closed double curve

Let $T_1$ and $T_2$ be two tubes intersecting in a closed double curve. Let $X$ be an operation which split the two tubes so that the intersection between them becomes empty. The question is what kind ...
4
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0answers
58 views

Homology of $\mathbb{R}\setminus A_+$. [duplicate]

Let $A$ be the unit circle in the $xy$ plane in $3$-dimensional real space and let $A_+$ be a semicircle. I have to compute the homology of $\mathbb{R}^3\setminus A_+$. I was thinking that ...
0
votes
1answer
26 views

Retraction and fundamental groups

If we know $\pi_1(X),\pi_1(Y)$, what we can say about existence of retraction $X$ onto $Y$ and vice versa. I think if $\pi_1(X)$ is 'smaller' than $\pi_1(Y)$ there is no retraction $X$ onto $Y$. For ...
1
vote
1answer
45 views

$\pi_{1}(X)$ abelian criteria

$X$ is a path-connected topological space. Let $\beta_{h}$ be homomorphism $\pi_{1}(X,x_0) \to \pi_{1}(X, x_1), [f] \mapsto [h f h^{-1}]$($h$ is a path from $x_0$ to $x_1$). I want to prove that ...
0
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1answer
36 views

'Elementary' proof of $\tilde{X}$ is contractible iff $\pi_n(X) =0 \forall n \ge 2$.

I have studied algebraic topology, but have not studied $\pi_n(X)$ any further than $n=1$. Is there a proof of $\tilde{X}\cong 1 \Leftrightarrow \pi_n(X) =0 \forall n \ge 2$ that does not require ...
1
vote
0answers
35 views

In what sense is cohomotopy dual to homotopy?

I understand the duality in the case of homology and de Rham cohomology. Through integration a chain can be understood as a linear functional on differential forms and vice versa. Is there a way to ...
2
votes
0answers
70 views

Visualising algebraic topology

I'm new to algebraic topology and although I can follow the arguments it would be nice to be able to visualise important concepts like homology and excision. Can anyone recommend a book or other ...
1
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0answers
18 views

Totalization of cosimplicial map

Let $f^*:X^*\to Y^*$ be a map of cosimplicial spaces such that each map $f^n:X^n\to Y^n$ is homotopically trivial. Is this true that the induced map $Tot(f):Tot(X^*)\to Tot(Y^*)$ is homotopically ...
2
votes
1answer
29 views

If $p:E\to X$ is a covering map ($X$ connected and locally arcwise connected) then is $E$ locally connected?

I recall my definition of a covering map. A continuous and surjective map $p:E\to X$ between topological space, where $X$ is connected and locally arcwise-connected, is called a covering map if for ...
0
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0answers
20 views

Why is a curve bounding a punctured torus nullhomologous?

I read (simplified) the following in a paper by Fintushel and Stern: The curve $\alpha$ bounds a punctured (*) torus and thus is null-homologous. Is this just the consequence of applying the ...
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0answers
22 views

$0$-th homotopy set of $G\times Z_2/H$

For a connected Lie group $G$ and its subgroup $H$, if $\pi_0(G/H) = 1$, is it true $\pi_0(\frac{G\times Z_2}{H}) = \{1,-1\}$ and $\pi_0(\frac{G\times Z_2}{H\times Z_2}) = 1$? I have to understand ...
1
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0answers
19 views

Suppose $S^m$ is a contraction on a complete metric space $(X,d)$. I want to show that this implies $S$ has a unique fix-point. [duplicate]

Let $(X,d)$ be a complete metric space and let $S: X \rightarrow X$ be a mapping. Suppose there exist $m \ge 1$ such that $\underbrace {S^m = S \circ S \circ \dots \circ S}_{\text {m times}}$ is a ...
1
vote
1answer
17 views

Retract and its fundamental group.

Suppose that $X$ is a topological space, and $A$ is a retract with retraction $r: X\rightarrow A$ and $i:A\rightarrow X$ the inclusion map. Prove that if $i_*\pi(A,a)$ is normal then ...
1
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0answers
22 views

A simply connected region $D$ that contains the boundary of $S$ contains $S$

If $D\subseteq X$ is a simply connected subspace of the topological space $X$ and [add assumptions here] and $S\subseteq X$, $\partial(S)\subseteq D$, then $S\subseteq D$. It doesn't seem to be ...
1
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0answers
30 views

Relative singular homology $H(M,\partial M)$ for a manifold $M$?

Let $M$ be an orientable manifold. What can be said about the relative homology $H(M,\partial M)$? Perhaps one can calculate the homology using excision?
1
vote
1answer
11 views

Geometric interpretation of the evaluation of Poincaré dual with a fundamental class

Given oriented, closed submanifolds $X^k$ and $Y^{n-k}$ in an oriented, closed $n$-manifold, is there a nice geometric interpretation of the evaluation $\langle \operatorname{PD}([X]),[Y]\rangle$? I ...
0
votes
0answers
18 views

Suspension of a cup product

Suppose we have a multiplicative cohomology theory $E$. Hence we have suspension isomorphisms $E^n(X, o) \to E^{n+1}(\Sigma X, o)$. Take two elements $x, y \in E^*(X, o)$ of degree's $i,j$ with ...
1
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1answer
36 views

connected manifolds are path connected

prove every connected manifold is path connected manifold . my thought: connected space : Let $ X$ be a topological space. A separation of $ X $ is a pair $U, V$ of disjoint nonempty ...
-1
votes
1answer
39 views

Homotopy between two paths in a path connected space

I'm trying to show that any two paths in a path connected space,are homotopic ( homotopic not path homotopic) ,any help?
1
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1answer
36 views

Exemplification in special function

this question have 3 sections that are together. $ a)$ Give an example of a map that is open but not closed, and an example of a map that is closed but not open. $b)$Determine whether the ...
1
vote
1answer
26 views

Homotopically equivalence for a 2 dimensional manifold

I have the following problem in my homework for algebraic topology: Does there exist a compact 2-dimensional manifold $M$ without boundary such that $M\times M$ is homotopically equivalent to $M$? I ...
0
votes
0answers
14 views

given a specific vector bundle how to see whether the first Pontryagin class is zero

Let $Z_2$ be the group with $2$ elements. Let $a\in Z_2$ be the nontrivial element. Let $S^n$ be the $n$-sphere. Let $C(S^n,2)=\{(x,y)\in S^n\times S^n\mid x\neq y\}$. Let $a$ act on $C(S^n,2)$ by ...
2
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0answers
7 views

easy question involving definition of bousfield localization

Easy question: If $X$ is a space/spectrum such that $H_*(X)=0$ (where $H_*$ is ordinary homology), then why does this imply the Bousfield localization $L_HX$ is contractible? My attempt: $L_HX$ is ...
2
votes
1answer
16 views

Representation of the fundamental class of a closed orientable $n-$manifold

Let $M$ be a closed orientable $n$-manifold with a ∆-complex structure. Let ${σ_1 . . . , σ_k}$ be the set of all $n$-simplices. How does one prove that the fundamental class $[M]$ can be represented ...
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0answers
19 views

Existence of a degree 1 map from a connected closed orientable $n-$manifold to $S^n$

For a map $f : M → N$ between connected closed orientable $n$-manifolds with fundamental classes $[M]$ and $[N]$, we define the degree of $f$ to be the integer $d$ such that $f∗([M]) = d[N]$. How do I ...
1
vote
1answer
46 views

Give a direct simple proof that comb space cannot be strong deformation retracted to $(0,1)$?

Give a direct simple proof that comb space cannot be strong deformation retracted to $(0,1)$ where comb space is $\bigl(\bigcup_{n\in \mathbb{N}} \{\frac{1}{n}\}\times[0,1]\bigr)\cup ...
0
votes
0answers
20 views

Understanding the morphisms for an $H$-cogroup

I'm reviewing my notes for an Algebraic topology course and am having trouble with the notion of an $H$-cogroup. I know that an $H$-group $X$ should have morphisms $$\mu:X\times X\to X$$ $$i:X\to ...
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1answer
69 views

A statement in tom Dieck's Algebraic Topology

In the cone construction (section 9.3.1) of tom Dieck's book Algebraic Topology, the author defines the following map $$q:\Delta ^{n-1}\times I\rightarrow \Delta ^n,\;\;\;((\mu_0,\dots ...
4
votes
0answers
50 views

representatives of $\pi_n(U(n))$

I know that $\pi_n(U(n)) \cong \mathbb{Z}$ for $n$ odd. I am looking for a description of an explicit map $S^n \rightarrow U(n)$ that represents the generator in this group. By precomposing with maps ...
1
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0answers
23 views

Flat vector bundles and constant transition functions

Let $E\to M$ be a vector bundle endowed with a flat connection. Then, does $E$ admit a bundle atlas with constant transition functions? For a vector bundle with constant transition functions, are ...
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0answers
25 views

finding homology of disjoint closed balls

If $U \subset \mathbb{R}^m$ and $B$ is a closed ball contained in $\mathring{U}$ (interior of $U$) with $B_1,B_2,\ldots,B_n$ are disjoint closed balls contained in $\mathring{B}$ (interior of $B$). ...