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

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0answers
23 views

Geometric Realization of Finite Dimensional Abstract Simplicial Complex

I am learning the theory of complex. And there are two theorems presented by our teacher: Every abstract complex $K$ has its geometric realization. Every $n$-dimensional abstract complex $K$ has its ...
8
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1answer
70 views

Is a space with no nontrival vector bundles contractible?

Let $X$ be a "nice" space, say having the homotopy type of a CW complex. Suppose also that $X$ is connected. Suppose that all real vector bundles on $X$ are trivial. Does it follow that $X$ is ...
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5answers
159 views

Algebraic topology in high school?

This winter I am planning on teaching a small seminar (20 lectures 45 minutes each) for high school students. I was was given the freedom to choose the topic of the seminar, but it is supposed to be ...
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0answers
38 views

Roots of monic complex polynomial lie on a circle of radius $R$.

A problem in my topology course asks to show that there exists a large enough $R$ such that $f(x)=z_n x^n + \dots + z_0$ has no roots on $ \mid z \mid =R$. I am not sure how to approach this problem ...
2
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1answer
55 views

Single $\Delta$-complex structure on $S^3$

Is it possible to identify pairs of faces of $\Delta^3$ to produce a $\Delta$-complex structure on $S^3$ having a single simplex? I'm thinking to identify left with right and behind with bottom. But ...
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1answer
17 views

rigorously defining homotopy inverses between the wedge sum and union of growing circles

I'm trying to solve the exercise 1.2.20 of Hatcher algebraic topology and stuck with the homotopy equivalence part. I can't construct explicit homotopy inverses. Could anyone show me what the ...
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1answer
39 views

constructing an explicit homotopy

I can see that the paths $(\cos(\pi s), \sin(\pi s))$ and $(\cos(\pi s), -\sin(\pi s))$ in $\mathbb{R}^2 \setminus \{0\}$ are 'homotopic' But can't construct an explicit homotopy between them. Could ...
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1answer
51 views

fundamental group of $\Bbb R^3\setminus S^1$ with explicit deformation retract

İ am studying Tamma tom Dieck algebraic topology book and i see question 2.8.1 Question : Let $ D = \{(0,0,t) \mbox{ } | -2 \leq t \leq 2 \} $ and $S^2(2) = \{ x \in \mathbb{R}^3 \mbox{ } | ...
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1answer
34 views

Derivation of the Maurer-Cartan formula

The left-invariant Maurer-Cartan forms are given by $$g^{-1}dg, $$ wher $g$ a Lie group $G$ to $M_n(\mathbb{R})$. My question is why is $$d(g^{-1}dg)=(g^{-1}dg)\wedge(g^{-1}dg)\quad ? $$ How come one ...
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0answers
43 views

Additivity for Relative Homology

If $(X_\alpha,A_\alpha)$ are disjoint topological pairs, is the following statement true? $$ \bigoplus_\alpha H_n(X_\alpha,A_\alpha)=H_n\left(\bigcup_\alpha X_\alpha,\bigcup_\alpha A_\alpha\right) $$ ...
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1answer
45 views

Fundamental group of a genus-2 surface

I want to calculate the fundamental group of a genus-2 surface, i.e. a double torus. Using Van-Kampen I obtain ( with the notation generators- relations) $$\Pi_1(X,p) = < \alpha, \beta, \alpha_1, ...
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0answers
21 views

Constructing orientable surfaces

In Hatcher's "Algebraic Topology" (http://www.math.cornell.edu/~hatcher/AT/AT.pdf), page 5: an orientable surface $M_g$ of genus $g$ can be constructed from a polygon with $4g$ sides by ...
3
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1answer
74 views

A question about contractibility of a Lie group

In Lawson's book Spin Geometry, chapter II, before Proposition 1.11, it mentions a fact that if the 1st, 2nd and 3rd homotopy groups of a Lie group $G$ vanish (although $\pi_2(G)=0$ automatically), ...
4
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0answers
96 views

De Rham Cohomology is Sheaf Cohomology

I'm trying to prove that de Rham cohomology computes the cohomology of the constant sheaf $\mathbb{R}$ of real valued smooth functions on a manifold. I would like to do this without using Čech ...
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3answers
31 views

unreduced suspension

Is the definition $SX=\frac{(X\times [a,b])}{(X\times\{a\}\cup X\times \{b\})}$ of the unreduced suspension the standartdefininition? If I consider X=point, the suspension of X is a circle. But I saw ...
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3answers
37 views

Amalgamated Product example.

I have been reading about amalgamated products online where they are defined as the quotient group formed by the free product and a normal subgroup of the free product. I am having a hard time ...
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2answers
31 views

Normal subgroup of a free product : how does $f_1(h)f_2(h)^{-1}$ generate a normal subgroup of $G_1 * G_2$?

Suppose we have $G_1$ and $G_2$ groups and let $$f_1: H \to G_1$$ $$f_2: H \to G_2$$ be group homomorphisms. The amalgamated product $G_1 *_{H} G_2$ is defined as follows: Let $N$ be the normal ...
3
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1answer
46 views

Van-Kampen and Covers

Is there a Van-Kampen-style theorem for universal covers? I was looking for a reference. I was looking for something like: given two topological spaces $X,Y$, the universal cover of $X\cup Y$ follows ...
0
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0answers
60 views

Tubular neighborhoods in the proof of the Morse homology theorem

I have a question regarding the proof of the Morse homology theorem given by D. Salamon in "Morse theory, the Conley index and Floer homology". The full text can be found here: ...
2
votes
3answers
64 views

What is free product?

I have searched for it, but I found there are several many different definitions. Even wikipedia states just free product of $2$ sets, not an infinite product. I know what exactly free group of a ...
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2answers
41 views

Cover of a finitely punctured plane

Let $X_n$ be the plane with a finite number $n$ of punctures, and let $p : Y \rightarrow X_n$ be a covering map (it may have infinite degree). Can we say anything about the topology of $Y$? (I know ...
3
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1answer
71 views

A valid proof for the invariance of domain theorem?

The invariance of domain theorem states that, given an open subset $U\subseteq \mathbb{R}^n$ and an injective and continuous function $f:U\rightarrow\mathbb{R}^n$ then $f$ is a homeomorphism between ...
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0answers
60 views

Finite graph products of finite groups have free subgroup of finite index

This is a problem in Hatcher's Algebraic topology. Show that a finite graph product of finite groups has a free subgroup of finite index, by constructing a finite-sheeted covering space of $K\Gamma$ ...
2
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1answer
48 views

Fundamental group of $GL(n, \mathbb{C}) $

I want to prove that the fundamental group of $GL(n, \mathbb{C}) $ is infinite. I don't know how to proceed, any hint ?
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1answer
33 views

$\mathbb{R}P^2$ and its fundamental group by identification of edges of unity square

Suppose we identify edges of the the unity square $[0, 1] \times [0, 1]$, as in the picture: http://de.wikipedia.org/wiki/Datei:ProjectivePlaneAsSquare.svg Now to compute the fundamental group, for ...
2
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1answer
33 views

Triangulation of Torus

I was asked to find out the simplicial homology groups of the torus $T=S^1\times{}S^1$ embedded in $R^3$. I triangulated the torus like this : Here the $0$-simplices are $\{v_0\}$. $1$-simplices ...
3
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1answer
62 views

Equivalence of categories $\Pi_1:\mathbf{K^1_{CW,*}}\to \mathbf{Grp}$

Let $K(G,1)$ denote the Eilenberg-Maclane spaces with fundamental groups isomorphic to $G$. Consider the category $\mathbf{K^1_{CW,*}}$ where objects are pointed $K(G,1)$ CW complex, and morphisms ...
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0answers
30 views

Why does an inclusion $N \subset M$ imply that $H_k(N) \rightarrow H_k(M)$ is surjective?

In my particular case, $N \subset M$ is $\mathbb{S}^2$ and I am interested in $H_2(M)$, where $M$ is closed and simply connected. We have $H_2(N;\mathbb{Z}) = \mathbb{Z}$. Why must $H_2(N; \mathbb{Z}) ...
0
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1answer
54 views

Comb space is contractible but not base point preserving

For each positive integer $n$, let $I_n=\{1/n\}\times I$ as a subset of $I\times I$. Let $X=(I\times0)\cup(0\times I)\cup(\bigcup_{n\geq 1} I_n).$ Let $x_0=(0,1)\in X$ be the base point. Show that $X$ ...
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1answer
34 views

Free abelian groups in Algebraic Topolgogy

In the context of algebraic topology, free abelian groups are frequently used like the parameter of functions $\textbf{X}$ in $f(\textbf{X})$ Since i am lack of knowledge about abstract algebra, i'd ...
2
votes
0answers
46 views

automorphisms of the torus bundle over the circle

Let us consider a $\mathbb{T}^2$-bundle $\xi$ over circle $S^1$. These bundles are completely determined by their monodromy matrix. A fiber-preserving homeomorphism of $\xi$ is called automorphism of ...
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0answers
10 views

difference between $n$-simplex, singular $n$- simplex

From studying the difference between $n$-simplex, singular $n$- simplex, i have a question. Reading Hatcher's textbook, i noticed for singular $n$-simplex, includes that $\sigma$ need not be a nice ...
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2answers
51 views

Does the Seifert-van Kampen Theorem applied to loop spaces say anything about higher homotopy groups?

The Seifert-van Kampen tells us that $\pi_1$ preserves pushouts of the form $U \cap V \subseteq U, V$ (for $U \cap V$, $U$, $V$ open, path-connected). Can this information be used to say anything ...
0
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1answer
54 views

Showing that there is no base-point preserving homotopy

I'm working on this problem and showed that X is contractible. In fact I showed that X has the origin (0,0) as its deformation retract. However, I'm stuck at the second part. It seems intuitively ...
1
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1answer
48 views

Borsuk - Ulam Theorem for $n=2$

Show that Borsuk -Ulam Theorem for $n=2$ is equivalent to the following statement : For any cover $A_1, A_2, $ and $A_3$ of $S^2$ with each $A_i$ closed, there is at least one $A_i$ containing a pair ...
0
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0answers
38 views

showing that no retract from solid torus to the entangled circle

This is the exercise 1.1.16 (c) of Hatcher algebraic topology. I've looked through other questions and the answers keep saying that the entangled circle inside the solid torus can be retracted to a ...
0
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0answers
71 views

Showing whether the torus can retract onto the diagonal circle

I got an assignment requiring me to show whether the torus $S^1\times S^1$ can retract upon the diagonal circle which is the set of all $(x,x)$ where $x$ belongs to $S^1$. I searched on the internet ...
1
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2answers
27 views

About homotopy groups of pairs

Suppose $A$ is a deformation retract of $X$, for $n\ge 2$ and for any $x_0\in A$, how to show $$\pi_n(X,x_0)=\pi_n(A,x_0)\oplus\pi_n(X,A,x_0)$$ I am not clear why the homotopy groups are not ...
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0answers
35 views

Identifying Objects with Polygons

I can't seem to find anything regarding how one identifies something like a torus with am oriented square. I would like to know the significance of: How does the rectangle depict the torus? Why are ...
1
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2answers
57 views

Maps that induces identity on fundamental groups are homotopic to identity?

Suppose $X$ is path connected, let $F:(X,x_0)\to (X,x_0)$ be a map such that $F_*: \Pi_1(X,x_0)\to \Pi_1(X,x_0)$ is identity, does it imply that $F$ is homotopic to identity? Let $y_0$ be arbitrary, ...
1
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1answer
34 views

Explicit calculation of simplicial homology

Is it possible to calculate simplicial homology of $n$-dimensional simplex just by definition, without using homotopy invariance of homology(or it's equality to singular or cellular ones)? I've done ...
2
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1answer
64 views

Showing that Möbius band can't retract onto the boundary circle

Trying to prove that Möbius band can't retract onto the boundary circle, I got an idea that I must show the below thing. If $\alpha\in\pi_1 (\partial M)$ is a generator, its image $i^*(\alpha) \in ...
2
votes
1answer
26 views

Continuity, Smash product, etc.

Let $X,Y,K$ be pointed spaces and $K$ locally compact Hausdorff. Let $f:X\rightarrow Y^K$ and define $g:X\wedge K\rightarrow Y$ by $g(x\wedge k)=f(x)(k)$. I want to prove that $f$ is continuous iff ...
0
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1answer
33 views

Is there any way to show that an equation of this form splits?

If I have this exact sequence $\mathbb{Z}^2 \rightarrow B \rightarrow \mathbb{Z}^n\rightarrow \mathbb{Z}$, does it split? If so how do I know.
1
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1answer
40 views

classifying space of permutation groups

Given a group $G$, how to compute the classifying space of $G$? In particular, how to compute the classifying space $B\Sigma_n$ for a permutation group $\Sigma_n$?
2
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0answers
25 views

How can I prove that if for all $X \in \Bbb{Top_*}$ , $[X,W]_*$ has a natural group structure, then $W$ is an $H$-group?

I know that if $\enspace[X,W]_*$ has a natural group structure, in particular $\enspace[W \times W, W]_*$ has it. If $p_1,p_2:W \times W \to W$ are the projections, it seems that defining $ \mu : W ...
2
votes
2answers
30 views

Maps to Sn homotopic

At the moment I'm taking an algebraic topology course. In our current exercise we have to prove the following: Consider any topological space $X$ and any two maps $f, g : X \to S^n$, such that $f(x) ...
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1answer
33 views

Are all embeddings of $D^n$ into $R^n$ ambiently homeomorphic? [closed]

To make the question more precise: let $i, j : D^n \to R^n$ be topological embeddings. Does there exist a topological automorphism $h : R^n \to R^n$ such that $h \circ i = j$?
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1answer
31 views

Mapping Cylinder.

I don't understand the following fact I've read: Any map $f:X \rightarrow Y$ can be written as composition $X \stackrel{i}{\hookrightarrow} M_f \stackrel{j}{\rightarrow} Y$, where $i$ is the ...
2
votes
2answers
108 views

Elementary way to show the exact sequence $0 \to M \to \mathbf Z^2 \to \mathbf Z \to 0$ implies $M = \mathbf Z$

I am computing the singular homology of spheres by induction. In the process, I have come across the following short exact sequence $$0 \to H_1(S^1) \to \mathbf Z^2 \to \mathbf Z \to 0.$$ I wonder ...