Tagged Questions

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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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
votes
1answer
75 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
votes
0answers
97 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 ...
1
vote
3answers
32 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 ...
0
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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 ...
0
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2answers
33 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
votes
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
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3answers
66 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 ...
1
vote
2answers
44 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
votes
1answer
78 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 ...
1
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0answers
62 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 ?
0
votes
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
votes
1answer
35 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
votes
1answer
65 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
votes
1answer
56 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 $X$ ...
0
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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
48 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 ...
0
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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 ...
0
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2answers
52 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
votes
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
vote
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
votes
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
votes
0answers
72 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
vote
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 ...
0
votes
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
vote
2answers
60 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
votes
1answer
69 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
27 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
votes
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
votes
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) ...
0
votes
1answer
34 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 $h \circ i = j$?
0
votes
1answer
34 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 ...
0
votes
0answers
12 views

Continuous mapping from 2-simplex [duplicate]

I am asked to find a map from the 2-simplex to the homotopy square. Such that $$(0,0) \to (1,0,0)$$, $$(0,1) \to (1,0,0)$$, $$(1,1) \to (0,0,1)$$ and $$(1,0) \to (0,0,1)$$ where $ \alpha(t) * ...
1
vote
1answer
28 views

Parameterization problem

I am faced with the very annoying problem of finding a continuous map which will transform the unit square in to the triangle. This is a problem in topology and the map need to obey certain ...
1
vote
1answer
59 views

CW complexes - An algebraic Topology Question

This question regards a particular exercise regarding Algebraic Topology, CW complexes and homotopy. I am trying to prove that the Klein Bottle is homotopic to $S^2 \vee S^1 \vee S^1 $, where $\vee $ ...
0
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1answer
53 views

differential manifolds or algebraic topology [closed]

In our university we must catch a course at least in one of these courses: differential manifolds or algebraic topology . which one is harder to start at first , differential manifolds or algebraic ...
8
votes
1answer
145 views

Hatcher Chapter $0$ Exercise $7$

I am going through Hatcher's Algebraic Topology. But I'm stuck with the question $7$ of chapter $0$. Fill in the details in the following construction from [Edwards 1999] of a compact space $Y ...
1
vote
1answer
32 views

Homology of a 3-manifold with a solid torus attached

Let $M$ be a (connected) compact orientable 3-manifold whose boundary $\partial M$ is homeomorphic to $T^2$ (the torus). Now consider the solid torus $S=S^1\times D^2$ and choose a homeomorphism ...
1
vote
1answer
48 views

invariance of integrals for homotopy equivalent spaces

I just wanted to know whether the integral of a closed n-form is invariant if we integrate it over homotopy equivalent spaces. This seems like a generalization of "Homotopy invariance of line integral ...
1
vote
1answer
19 views

On the relations between rank and torsion of homology and cohomology of a CW pair.

I am reading Massey's book on algebraic topology and on the chapter of universal coefficient theorem of cohomology, there is this exercise 4.1 that I don't know how to solve. Let (X,A)be a pair such ...
1
vote
2answers
68 views

Rigorous Covering Space Construction

Construct a simply connected covering space of the space $X \subset \mathbb{R}^3$ that is the union of a sphere and diameter. Okay, let's pretend for a moment that I've shown, using van Kampen's ...
0
votes
1answer
44 views

Is every finite CW complex is homotopic to simplicial complex?

Is every finite CW complex is homotopy equivalent to a simplicial complex?
0
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0answers
31 views

How is this simplex a subspace?

Let $e_1$, $e_2$, and $e_3$ be the standard basis vectors of $\mathbb{R}^3$. Then the standard 2-simplex, $ \triangle^2$, is of the form $$t_1 e_2 + t_2 e_2 +t_3 e_3$$ where $t_1 + t_2 + t_3=1$. ...