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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2answers
58 views

Prove Simply Connected

If $X = U \cup V$ with $U,V$ open and simply connected and $U \cap V$ is path connected, why is $X$ simply connected?
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1answer
26 views

Verify $p _0 : [0,1] \mapsto S^1 , p_0(s)=(\cos(2 \pi s),\sin(2\pi s))$ is a covering map.

I want to verify that the restriction to the interval $[0,1]$ of the map $p : \mathbb{R} \mapsto S^1 $ given by $ p(s)=(\cos(2 \pi s),\sin(2\pi s))$ is a covering map. I tried as follows. Take $s ...
0
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1answer
34 views

Deformation retraction onto the boundary

If I have a square and I remove an open disc from its interior, there exists a deformation retraction onto its boundary. Is this also the case, if I remove a closed disc from its interior? Does the ...
3
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1answer
56 views

Is Orbit Criterion an abstract nonsense? Different induced fundamental groups.

(Orbit Criterion) Let $p:\tilde X \to X$ be a covering map. If $\tilde q, \tilde q' \in \tilde X$ are two points in the same fiber $p^{-1}(q)$, there exists a covering transformation taking $\tilde ...
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1answer
26 views

If f is a path, [f] can be exposed as [f]=$[f_1][f_2]…[f_q]$?

Can you please help me solve this problem? I am completely lost. Thanks in advance. Suppose that X is a space that $X=U \cup V$ with U, V open subsets. Show that if $f$ is a path in X then [f] can ...
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1answer
37 views

Relative singular chains basis

If $(X,A)$ is a pair, then $S_k(X,A):=S_k(X)/S_k(A)$ is free on the singular simplicies of $X$ with image not contained in $A$. Why is this so? I tried to give a proof by checking the mapping property ...
2
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1answer
50 views

Fundamental group: Attaching a disc to a path connected triangulable space

What's the effect on the fundamental group of a path connected triangulable space X if you attach a disc? So far I've figured out that a disc has trivial fundamental group. So the free product of ...
2
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2answers
54 views

On Hatcher's proof that $H_0(X)$ is a direct sum of $\mathbb{Z}$s?

I'm confused about part of the proof of Proposition 2.7 in Hatcher. If $X$ is nonempty and path-connected, then $H_0(X)\simeq\mathbb{Z}$. Hence for any space $X$, $H_0(X)$ is a direct sum of ...
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0answers
34 views

Hatcher Exercise (4.1.20)

I'm having a little trouble doing 4.1.20. It states "Show that $[X,Y]$ is finite if $X$ is a finite connected CW complex and $\pi_i(Y)$ is finite for $i \leq \text{dim }X$", where $[X,Y]$ are the ...
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0answers
19 views

Path-connected Space. Show abelian iff.

Let $x_0$ and $x_1$ be points of a path-connected space $X$. Show that $π_1(X,x_0 )$ is abelian if and only if for every pair $α$ and $β$ of paths from $x_0$ to $x_1$, we have $\bar α = \bar β$ .
2
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1answer
26 views

Universal Coefficient Theorem for Coefficient Group ${\bf R}$

${\rm Ext}\ ({\bf Z}_n,G) =G/nG$ so that if $G={\bf Q},\ {\bf R}$ then ${\rm Ext}\ ({\bf Z}_n,G)=0$ Hence UCT implies $$ H^n(C;G) = {\rm Hom}_{\bf Z} ( H_n(C);G) $$ Hence $$ H^n(C;G) =H_n(C)/{\rm ...
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2answers
45 views

Quotient of surface of genus 2

Let $g$ be the quotient map from $M_2$ to $M_1$ (namely torus). Why $g$ induces isomorphism on $H_2$? Also, Hatcher talks about the degree of $f$ from sphere to sphere in his book. Can it be ...
1
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1answer
47 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
1
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1answer
31 views

Spaces homotopy equivalent to $A_{\infty}$-spaces

I ask this question after reading Peter May's "Geometry of Iterated Loop Spaces", where the problem is definitely hinted at but I couldn't find a definite answer. Recall a symmetric operad ...
2
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0answers
42 views

What does base point by us for algebraic topology?

This may be a vague quesion. I am confusing between base pointed case and non base pointed case in algebraic topology. Is there any convinience in base pointed case? For example, it leads to the ...
3
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0answers
32 views

Why is this Milnor fiber homeomorphic to a cylinder?

Let $f:(\mathbb C^2,0)\to (\mathbb C,0)$ be a holomorphic function with a critical point at the origin. Let us denote by $X_0$ the fiber $f^{-1}(0)$, in which, as we said, the point $0\in X_0$ is a ...
1
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0answers
37 views

what is the covering space of figure eight which is corresponding to commutator subgroup.

Let $ F$ be the free group on two generators and let $F^{'}$ be its commutator subgroup. Find a set of free generators for $F^{'}$ by considering the covering space of the graph $S^{1} \vee S^{1}$ ...
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0answers
26 views

a problem about surface $M_{g}$.

1.$M_{g}$ has normal universal cover $\widetilde{X}$ with deck transformation $G(\widetilde{X})=\mathbb{Z}^{n}$ if and only if $n \leq 2g$. 2.for $n=3,g \geq 3$ explain such covering. 3.show that ...
5
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0answers
42 views

Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point ...
3
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0answers
44 views

homomorphism of fundamental groups induced by a map of two surfaces

I am trying to find another proof of the following theorem Theorem Let $X$ and $Y$ be two compact surfaces of genus greater than $2$. Then every homomorphism $π_1(X,x_0)→π_1(Y,y_0)$ is induced by a ...
5
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0answers
38 views

A theorem of Kan regarding fibrant replacement

Recall that there is an adjunction $$\mathrm{Sd} \dashv \mathrm{Ex} : \mathbf{sSet} \to \mathbf{sSet}$$ where $\mathrm{Sd} (\Delta^n)$ is the first barycentric subdivision of $\Delta^n$. There is a ...
3
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1answer
35 views

Computing the Euler Characteristic of the $n$-sphere

Let $n\ge 2$. Compute the Euler characteristic of the $n$-sphere $S^n$ using the standard triangulation of the $n+1$-simplex. I know the union of the proper faces of the $(n+1)$-simplex is ...
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0answers
23 views

History of vectorial bundles in articles or papers?

I'm looking for an article or book that gives a thorough and interesting history of bundles and vectorial bundles in algebraic topology. I'm looking for it for my own learning, please help its ...
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0answers
42 views

exercise 17 of hatcher page 80 chapter 1.3

Given a group $G$ and a normal subgroup $ N$, show that there exists a normal covering space $\widetilde{X} \rightarrow X $ with $\pi_{1}(X)\approx G ,\pi_{1}(\widetilde{X})\approx N $, and deck ...
2
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1answer
26 views

Basic question on almost complex structures and Chern classes of homogeneous spaces

Toward the end of "Characteristic Classes and Homogeneous Spaces, III," Borel and Hirzebruch prove that given a compact Lie group $G$ and toral subgroup $T$ (no restriction on rank), one has $w(G/T) = ...
2
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1answer
29 views

f~g iff if $f*\bar{g}$~$\varepsilon_x$

Can you please help me with this question? let f,g:I $\rightarrow$ X be two paths in X from x to y. Prove that f~g iff $f*\bar{g}$~$\varepsilon_x$ (where $\bar{g}$=g(t-1)) Thanks in advance
2
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1answer
36 views

strong deformation retract, of a perforated plane?

Let $x_0 \in R^2$. How do I find a circle in $R^2$ which is a strong deformation retract of $R^2-\{x_0\}$? Is it just a unit circle with the center $x_0$?
1
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1answer
41 views

Why does the intersection change to a union in $r^{-1}(\bigcap r(V_i\cap W_i))=\bigcup V_i\cap W_i$?

Let $q: X\to Y$ and $r:Y\to Z$ be covering maps, $p=r\circ q$. If $r^{-1}(z)$ is finite for each $z$ in $Z$, $p$ is a covering map. There is a proof on ask a topologist, but I can't follow why ...
3
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2answers
70 views

When is $\pi_0$ a group?

In general, $\pi_0(X)$ is the set of path components of $X$ and does not have a group structure. After all, $S_0$ is just two points and the usual way of multiplying using the equation of a sphere ...
0
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1answer
33 views

3 Questions concerning the fundamental group of $S^1\vee S^1\vee S^1$

I have the following questions: 1) What is the fundamental group of $S^1\vee S^1\vee S^1$ and why? 2) Is $S^1\vee S^1\vee S^1$ homeomorphic to the bouquet of 3 circles 3) Are $S^1\vee S^1\vee S^1$ ...
2
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1answer
25 views

Density, Irreducible Topological Space

Let X is a irreducible noetherian topological space, and $U \subseteq X$ is a nonempty open subset. If $B \subseteq U$ is dense, then so is $B \subseteq X$. Is this true or false, and why? Here ...
1
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1answer
18 views

Path components of an $H$-space

From Hatcher, page 291, page 3. Show that if $X$ is an $H$-space such that the set of path-components of $X$ is a group with respect to the multiplication induced by the $H$-space structure, then ...
3
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2answers
54 views

Continuous mapping from n-sphere to (n+1)-sphere

Are there any "nice" functions that can take a point from the surface of an n-sphere and map it to a the surface of an (n+1)-sphere? By "nice", I mean it should be continuous, one-to-one (but not ...
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0answers
11 views

Orthogonal Array Planes

In an OA-plane (Orthogonal Array-Plane) how we define the touching of two lines? If we consider OA of strength of 3 with unit index , that is the plane OA-plane of rank 3 (we call this Laguerre ...
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0answers
79 views

nice space with wild fundamental group

I would like to know an example of nice space with very strange fundamental group. With simplices and similar things I only get finitely presented groups. Edit. I know from comments that Hawaiian ...
1
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1answer
28 views

Computing homology group of product of spheres

I am having trouble computing $\tilde{H}_n(S^3\times S^1)$. I am supposed to use Mayer-Vietoris sequence. I know $\tilde{H}_n(A\vee B)\cong\tilde{H}_nA\oplus\tilde{H}_nB$ if there is a contractible ...
0
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0answers
26 views

is there a specific way to find deck transformation and its related group?

is there a specific way to find deck transformation and its related group? this question came to my mind at the first time I studied deck transformation and related topics of covering space. it ...
2
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0answers
38 views

Special representations for morphisms of spectra from a smash product [migrated]

I follow the definitions of spectra, function, morphism, found on Switzer, chapter 8. After definition 8.15 where he defines homotopies of spectra, he says: In terms of cofinal subspectra we can ...
1
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1answer
36 views

Constructing a ring in Top by wedging and smashing pointed spaces

I'll list some things I believe I can do, any of which might I might not actually be able to, and then, if I'm right, I'll ask if there's any point to what I've done. The category of pointed locally ...
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1answer
42 views

A certain homotopy equivalence…

A few friends and I have been stuck on this old qualifying question for quite some time now... Let $D$ be the diagonal subspace of $\Bbb S^2 \times \Bbb S^2$. Show that the projection onto the first ...
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1answer
61 views

Induced nontrivial homomorphism?

I am caught up in a minor detail on a qualifying exam problem I am doing: Show that there are no injective continuous functions $f: \Bbb R^n \to \Bbb R^2$ , $n>2$ with $f(0)=0$. So far I have ...
1
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1answer
27 views

Is there a name for the result that homotopy groups are insensitive to removing submanifolds of sufficiently large codimension?

First some motivation. Consider $\mathbb{R}^n-\{0\}$. This is simply connected iff $n > 2$, since it deformation retracts to $S^{n-1}$. If instead we consider $\mathbb{R}^n - L$ where $L$ is a ...
2
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3answers
99 views

Is it possible a trivial fiber bundle with nonzero holonomy?

Let $P\rightarrow M$ be a principal bundle with structure group $G$. Suppose that the bundle is trivial $M\times G$; is it possible to have a nonzero holonomy along some closed trajectory on $M$ for ...
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1answer
79 views

A basic question on Relative Homology

And so this week, our algebraic topology class starts with relative homology groups. But there are some (REALLY) basic parts of the definition of the relative homology group that I don't understand ...
2
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3answers
67 views

Why is the fibre of each point compact?

For a compact covering space, the fibres of the covering map are finite. I am working on the same question as the one posed in this link, but there was an unanswered question at the end, namely, why ...
0
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1answer
12 views

Degree of the map and path components

I am trying to solve the following question: for each $a\in \mathbb{C}\setminus S^1$ calculate the degree of the map $$ \phi_a: S^1\to S^1; z\mapsto \frac{z-a}{|z-a|} $$ and deduce that $a,b\in ...
2
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1answer
77 views

Question about the definition of homology

i have this paragraphe: Can someone explaine me what it means ? if i understand $H_n$ measure the numbers of holes with dimension $n$ but what about $H_0$ what is the relation between the holes of ...
3
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2answers
95 views

$G$ is Topological $\implies$ $\pi_1(G,e)$ is Abelian

Hypothesis: Let $G$ be a topological group with identity element $e$. Let $\mu$ denote the multiplication mapping in $G$. Goal: Show that $\pi_1(G,e) = \pi(G)$ is an abelian group via the hint ...
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0answers
18 views

branched cover along a closed curve in the $3$-sphere

Let $c$ be a closed embedded smooth curve in the $3$-sphere $\mathbb S^3$. I was told that $\mathbb S^3$ admits a two fold branched cover $X(c)$, branched along $c$, which corresponds to the ...
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0answers
30 views

all line bundles over $S^1$

can anyone help me with this problem show that all line bundles over $S^1$ are equivalent to either the trivial bundle or the standard Mobius strip? thanx.