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

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1answer
102 views

Use of Low dimensional Paths vs High dimensional Cubes

The universal covering manifold has a construction as follows: Fix a base point $p$ of $M$. Two paths $c_i\colon I \rightarrow M$ ($i=0,1$) on $M$ with terminal point $p$ and the same starting point ...
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0answers
285 views

Fundamental group of a space of infinite genus and an accumulation point

The fundamental group of $\mathbb{R}^2$ with a point removed is $\mathbb{Z}$. The fundamental group of $\mathbb{R}^2$ with $n$ points removed is the free group of $n$ generators. Is the fundamental ...
12
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1answer
93 views

If the connected sum of a manifold $M$ with itself gives back $M$, does it imply $M$ is a sphere?

Let $M$ be a compact, connected, oriented $n$-dimensional manifold without boundary. Suppose that $M\#M\cong M$. Does it imply that $M \cong S^n$? Sorry if this is a naive question. This is not my ...
9
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1answer
418 views

Is the quotient map a homotopy equivalence?

It is well known that, if $A \subset X$ is a reasonable contractible subspace, then the quotient map $X \to X/A$ is a homotopy equivalence ("reasonable" means that the pair $(X,A)$ has the homotopy ...
2
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1answer
29 views

$g : Z \rightarrow W$ a regular covering map where $Z$ and $W$ are path-connected, and $\pi_1(Z)$

Suppose that $g : Z \rightarrow W$ is a regular covering map where $Z$ and $W$ are path-connected, and $\pi_1(Z) = Z_2.$ Is it true that $g_* : H_1(Z) \rightarrow H_1(W )$ is injective? Prove or give ...
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1answer
43 views

Homology with coefficients

i am starting to study homology groups and i have a question, maybe is a dumb question but if someone could help me would be great. The question is $H_i(X,A)=H_i(X,A;\mathbb{Z})?$ I mean equal or ...
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1answer
83 views

what's wrong with this categorical proof that maps between two covering spaces are unique?

Let $\mathcal{C}$ be the category of finite covers of a fixed base space $S$ (say, connected, locally path connected, locally simply connected. Hell, we can even assume $S$ is a manifold). Morphisms ...
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1answer
688 views

Klein bottle covered by the torus

Maybe this is an idiot question and I'm missing something very trivial. This question question was asked here before, but the answer (which apparently is equal to the one that I created) seems ...
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0answers
165 views

Is such an infinite dimensional metric space, weakly contractible?

We counteract this answer by adding the rigidity assumption: Is there still a counterexample? Let $(X_{n},d_{n})_{n \in \mathbb{N}}$ be a sequence of complete geodesic metric spaces such that: ...
2
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1answer
110 views

Hatcher P151 Example 2.47.

Hatcher P151 Example 2.47. We can decompose the Klein bottle K as the union of two Mobius bands $A$ and $B$ glued together by a homeomorphism between their boundary circles. Then $A$, $B$ , ...
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1answer
143 views

Cech cohomology and cohomology of a category : a cluster of questions.

I apologize in advance : what follows is a bit of a mess. Also, I think it might be a big tautology, but i don't see it yet. My question is about the rapport of Cech cohomology and cohomology of a ...
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2answers
135 views

The complex tangent bundle of $\mathbb{C}P^1$ is not isomorphic to its conjugate bundle.

In " Characteristic Classes" by Milnor and Stasheff on pages 167-168, the authors give a brief argument about why: The complex tangent bundle of $\mathbb{C}P^1$ is not isomorphic to its conjugate ...
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0answers
127 views

Klein Bottle considered to be two Möbius band $A,B$ glued together

In Hatcher, a Klein Bottle $K$ is considered to be two Möbius band $A,B$ glued together. I see the map $\phi$ is $\mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$, but why $1 \mapsto (2,-2)$? I other ...
3
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1answer
192 views

Induced isomorphism of homology implies isomorphism with coefficients in any group?

If $\alpha\colon C \to C'$ is a map of chain complexes (of free abelian groups) that induces an isomorphism on homology $a_{*} \colon H_n(C) \simeq H_n(C')$, then I know that $\alpha$ induces an ...
2
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1answer
71 views

Basic constructions for graded algebras.

I'm reading about the Weil algebra of a Lie group and it involves some constructions I'm not very familiar with, for instance the "free graded-commutative graded algebra on $a_1...a_n$ with degrees ...
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1answer
208 views

interior boundaries and closures

Let $X = \{a,b,c,d,e\}$ where the $5$ elements are all different. Let $T = \bigl\{X, \emptyset,\{a\},\{a,b\},\{a,c,d\},\{a,b,c,d\},\{a,b,e\}\bigr\}$. Is $T$ a topology on $X$? List all the closed ...
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1answer
86 views

Fix point problem: a continuous mapping from $S^{2n-1}$ to $S^{2n-1}$

Is somebody able to prove that: If $f$ is a continuous mapping from $S^{2n-1}$ to $S^{2n-1}$ (the surface of a unit ball in $2n$ dimensional Euclidean space) and $f$ is not homotopic to the identity ...
2
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1answer
66 views

Spaces sharing all higher homotopy groups

Is it possible that two topological spaces share all higher homotopy groups, but are not homeomorphic? I should note that I have not studied much in the way of the theory of higher homotopy groups; I ...
4
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1answer
227 views

induced maps in homology

Let $S^1=I^1/ \partial I^1$, where $I^1=[0,1]$ with base point ${0}$ and $I^n=I^1\times I^1 \times \dots \times I^1$ (n times). $S^p=S^1 \wedge \dots \wedge S^1 $ where $\wedge$ is the smash product. ...
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1answer
224 views

Example on relative homology

I am trying to prove that $$H_p(B_{n+1},S_n;\mathbb{A}) \cong \left\{\begin{array}{ll} H_{p-1}(S_n,\mathbb{A}) & \text{if } p\geq2\\\ 0&\text{if } p=1, n\geq 1\\ \mathbb{A} &\text{if } ...
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2answers
111 views

Calculate the Euler Characteristic of M

Let $M$ be the following subsets of $\mathbb R^4$:$$M= \{(x,y,z,w), 2x^2+2=z^2+w^2, 3x^2+y^2=z^2+w^2 \}$$ we know $M$ is a submanifold of $\mathbb R^4$,what is the Euler Characteristic of M?
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1answer
48 views

Question on relative homology [duplicate]

i have that $H_p(X,Y)$ is isomorphic to $Z_p(X,Y)/(B_p(X)+C_p(Y))$, where $Z_p(X,Y)=\lbrace \sigma\in C_p(X), \partial\sigma\in C_{p-1}(Y)\rbrace$ and i want to deduce that $H_0(X,Y)$ is the free ...
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1answer
184 views

Is there anything to be learned from the spectrum of a cohomology ring?

Given some topological space, $X$, is there any benefit to studying $Spec(H^*(X))$, or is everything we care about already available "in the algebra"? As $H^*$ is a graded ring, does this question ...
0
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1answer
45 views

symmetry in the homotopy relation

Suppose $\alpha, \beta : I \to X$ are paths and suppose $\alpha $ is homotopic to $\beta$, $\alpha \cong \beta$. So, can find a continuous function $F(s,t) = f_t(s)$ such that $$ f_t(0) = \alpha(0) ...
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1answer
151 views

Homology of a torus

I am trying to compute the homology of a torus by its chain map, rather than its equivalence to $\oplus H_n (S^1)$. The post Homology groups of torus has been really helpful, but my question is, how ...
0
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1answer
57 views

Definition of degree

By Hatcher P134, degree is defined from a map $f: S^n \to S^n$ - but degree must be able to applied to all maps. Can I arbitrarily generalize the definition of degree to a map between any two spaces? ...
7
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1answer
186 views

What exactly is duality?

In general, I am familiar with this notion of duality (i.e. in category theory, a statement is dualized simply by "reversing all arrows" and leaving objects unchanged). There are a couple of questions ...
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1answer
239 views

Relative homology and path connected space

I want to prove that if $X$ is a path connected space and if $Y$ is nonempty then $$H_0(X,Y)\simeq 0$$ it is sayed that we have this chain: $H_0(Y)\rightarrow H_0(X)\rightarrow H_0(X,Y)\rightarrow 0$ ...
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1answer
44 views

Is there a compact complex manifold with trivial $H_2$?

I don't believe that every complex manifold should have nontrivial $H_2$, otherwise we would easily prove the Chern's conjecture... But the problem is I don't have any counterexample. The Kähler ...
2
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1answer
117 views

Homology and cohomology are basically the same

Is my following understanding correct: A chain complex $(C,\partial)$ is a family $\{C_i\}_{i\geq 0}$ of $R$-modules ($R$ is a given ring) together with a family of $R$-module homomorphisms ...
4
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1answer
93 views

Any two paths in $X = \mathbb{R}^n$ having same initial and end point are homotopic

Suppose $X = \mathbb{R}^n$. Let $\gamma, \alpha : [0,1] \to X $ be to paths such that $\gamma(0) = \alpha(0) = x_0 , \; \; \gamma(1) = \alpha(1) = x_1$. We want to show $\gamma$ and $\alpha$ are ...
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0answers
42 views

What is the name of $C(A)/A$

Given a topological space $A$, $C(A)$ is the cone of $A$. The space $C(A)/A$ is clearly homotopic to the suspension. My question is if it has a widely known name?
3
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1answer
84 views

What's the colimit of the n-sheet covering spaces over the circle?

I was thinking in computing the filtered colimit of the n-sheet covering spaces $f_n: \mathbb{S}^1 \longrightarrow \mathbb{S}^1$ ($f(z) = z^n$) in the comma category of topological spaces with the ...
2
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1answer
564 views

Homology of quotient of 3-sphere by identifying antipodal points on equator

I'm trying to solve exercise 2.2.10 in Hatcher's Algebraic Topology: Let $X$ be the quotient space of $S^{3}$ under the identifications $x\sim-x$ for $x$ in equator $S^2$. Compute the homology ...
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1answer
205 views

universal cover of a finite cw complex?

Is it true that the universal cover of a finite cw complex $X$ with finite fundamental group is also a finite cw complex?
2
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1answer
134 views

The image of homomorphism of fundamental group of closed surface

$\phi: \pi_1(S)\to \pi_1(S)$ is a homomorphism of fundamental group of closed orientable surface $S$ of genus $\geq 2$. If $\phi$ is not an epimorphism, can we find a non-surjective self map $f: S\to ...
2
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0answers
42 views

Reference request for an explicit description of the group of deck transformations acting on the universal cover of a Riemann Surface

Let $\Sigma_g$ be a geuns $g$ Riemann surface with $g \geq 2$. It can be thought of in the following way: it is the quotient space $$\mathbb{H}/\pi_1(\Sigma_g)$$ where an element of ...
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2answers
66 views

Understanding attaching space

I have some problem understanding the attaching space when learning topology. I cannot understand these two examples. Example 1 Let $X=A=S^1$,$Y=I\times S^1$ and $B=\{0\}\times S^1$. Let $h:B\to ...
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3answers
93 views

Symmetry of “is homotopic to” detail in the proof

Let $f,g:X\rightarrow Y$. If $f$ is homotopic to $g$ then $g$ is homotopic to $f$. Let $F:X\times I\rightarrow Y$ be a homotopy from $f$ to $g$ so $F(x,0)=f(x)$ and $F(x,1)=g(x)$ for all $x \in ...
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0answers
108 views

Different profinite topologies on a group?

I have some general questions around the profinite topology on a group $G$. On the page http://groupprops.subwiki.org/wiki/Profinite_topology one can read, that The profinite topology on a group is ...
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3answers
275 views

Cell complex of torus

Suppose we have a plane model for torus. We want to decompose the plane model of torus into cell complex which satisfies 1)every face of the cell complex has exactly 6 boundary edges 2)at each ...
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3answers
105 views

Number of homotopy classes

For topological spaces $X$ and $Y$, let $[X,Y]$ denote the set of homotopy classes of continuous maps $X\to Y$. If $I=[0,1]$ is the unit interval, then $[X,I]$ has only one element. If $X$ is path ...
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1answer
54 views

How to show that homotopy is preserved after composition?

I have two homotopies: $f\simeq f'$ and $y\simeq y'$. How can I show that $fy\simeq f'y'$ is again a homotopy?
3
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2answers
155 views

How to compute the fundamental group of a necklace of $\mathbb{S}^1$' s?

I was trying to compute $\pi_1 (X)$ where $X =$ "necklace of $n$ $\mathbb{S}^1$'s". At first, I tried using Van Kampen theorem however I could not find open sets $U$ and $V$ such that $U \cap V$ is ...
3
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1answer
61 views

Understanding Quillens Theorem A

Let me restate the theorem: Let $F\colon\mathcal{C}\to\mathcal{D}$ be a functor. If $F\downarrow x$ is contractible for every $x\in\operatorname{Ob}(\mathcal{D})$, then $F$ is a homotopy ...
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0answers
85 views

Subgroup Separability translated in Profinite Topology

The normal definition of subgroup separability is: A group $G$ is said to be subgroup separable if for every finitely generated subgroup $H\leq G$ and $g\in G\setminus H$ there exists a subgroup of ...
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1answer
162 views

How to determine space with a given fundamental group.

I would like to give examples of topological spaces such that their fundamental groups are, respectively, $\mathbb{Z} \oplus \mathbb{Z}_{n}$ and $\mathbb{Z} \ast \mathbb{Z}_{n}$. For the latter, I ...
4
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1answer
190 views

Functors that are the homology of a chain complex

Is there an a priori reason why the singular homology and cohomology groups of a space should be computable as the homology of chain complexes? Certainly you can express any family of functors (say, ...
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2answers
98 views

Is Map($T^4$,$S^2$) connected?

Consider the set $Map(T^4,S^2)$ of continuous maps from the 4 dimensonal torus $T^4$ to the 2 dimensional sphere $S^2$, endowed with compact-open topology, can we show it is not connected? How can we ...
2
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2answers
144 views

Question on relative homology

I have this question and I'd like an idea to solve it: If $Z_p(X,Y)=\lbrace \sigma\in C_p(X), \partial\sigma\in C_{p-1}(Y)\rbrace$, $1)$prove that $H_p(X,Y)$ is isomorphic to ...