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

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2
votes
1answer
70 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
46 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
583 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
72 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 ...
3
votes
2answers
153 views

Reference Request to Prepare for Hatcher's “Algebraic Topology”

Hatcher himself has an excellent and always generously free set of notes on point- set topology: http://www.math.cornell.edu/~hatcher/Top/TopNotes.pdf It includes up to quotient spaces. It seems ...
1
vote
1answer
134 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
votes
1answer
45 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 ...
3
votes
0answers
83 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 ...
1
vote
3answers
127 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
63 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
71 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 ...
1
vote
2answers
39 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
51 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
74 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
vote
1answer
67 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
342 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
31 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
vote
1answer
55 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
30 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 ...
4
votes
2answers
63 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
42 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
130 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
124 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
13 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
32 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
70 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
votes
1answer
59 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
209 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
76 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
95 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
40 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
155 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
106 views

Is every finite CW complex is homotopic to simplicial complex?

Is every finite CW complex is homotopy equivalent to a simplicial complex?
1
vote
0answers
50 views

Is nerve theorem always true?

Is the nerve theorem true for not paracompact spaces? Background: Nerve theorem states that if $U$ is an open cover of a paracompact space $X$ such that every nonempty intersection of finitely many ...
2
votes
1answer
49 views

Given a group $G$, the existence of a space such that $\pi_1(X)\simeq G$.

I'm having trouble understanding Corollary 1.28 of Hatcher which proves that for every group $G$ there is a space $X_G$ such that $\pi_1(X_G)=G$. Since $G$ is the quotient of a free group $F$, we ...
1
vote
1answer
86 views

Every Group is a Fundamental Group

I am studying elementary Algebraic Topology recently. I have seen that a topological space is identified with a group. We are telling the group as Fundamental Group. So every topological space $X$ and ...
4
votes
1answer
56 views

Is this a valid way to show $\chi(SL_n(\mathbb{R}))=0$?

Why does $\chi(SL_n(R))=0$? I'm going about it like this. Let $X:=SL_n(R)$. Define a map $f:X\to X$ such by $A\mapsto BA$, where $B$ is the identity matrix, except with an extra $1$ in the upper ...
1
vote
1answer
59 views

Hatcher's formula in homotopy equivalence proof

In the proof that two homotopic maps induce the same homomorphism in homology, appears the formula (bottom of p. 112, Hatcher, Algebraic Topology): \begin{gather} P(\partial \sigma) = \sum_{i<j} ...
0
votes
1answer
51 views

What is the map $\Sigma K(G,n) \to K(G,n+1)$?

Since $\Omega K(G, n+1)$ is a $K(G,n)$, we have a CW approximation/homotopy equivalence $K(G,n) \xrightarrow{\sim} \Omega K(G,n+1)$. The adjoint of this map is a map $\Sigma K(G,n) \to K(G,n+1)$. ...
3
votes
1answer
48 views

Map from $\mathbb{R}\mathbb{P}^{2}$ to Klein bottle homotopic to constant map

I'm currently struggling with the following exercise: Show that any continuous map $$f: \mathbb{R}\mathbb{P}^{2} \to K$$ where $K$ is the Klein bottle, is homotopic to a constant map. I know that ...
0
votes
1answer
57 views

Nulhomotopic map from $S^1 \rightarrow \mathbb{C} - \{0\}$

Hullo, I am aware that the inclusion map $i : S^1 \rightarrow \mathbb{C} - \{0\}$ is not nulhomotopic since there is a retraction from $\mathbb{C} - \{0\}$ to $S^1$ making the induced homomorphism ...
1
vote
0answers
67 views

Classification of compact 3-delta-complexes made of a single simplex

With a single 3-simplex (by identifying its faces in couples) it is possible to make 39 compact delta-complexes that can be grouped in 8 classes of complexes having the same homology groups (see ...
3
votes
0answers
67 views

Definition of the triad homotopy groups

Let $ \ n \in \mathbb{N}$, $n \geqslant 2$. Equip $\mathbb{R}^n$ with the usual euclidean norm $ \ \Vert . \Vert : \mathbb{R}^n \to \mathbb{R}$, metric and topology. Consider $ \ p_0 = (1,0,0,...,0,0) ...
0
votes
0answers
40 views

What is the “product rule” for the boundary map of a product of CW-complexes?

I'm reading some notes that compute the action of the boundary maps on the CW-complex $\mathbb{R}P^2\times\mathbb{R}P^2$. I know the chain maps for $RP^2$ itself are $d_0,d_1\equiv 0$, and $d_2=\cdot ...
2
votes
0answers
35 views

Injectivity of a map from a homotopy set to a homology group

Let $\Sigma$ be a closed Riemann surface and $X$ a 1-connected topological space. I would like to prove the following fact. (A) The map $[\Sigma,X] \to H_2(X;\mathbb Z)$ defined by $[f] \mapsto ...
0
votes
1answer
80 views

Proof of FTA from Hatcher

This is the proof of the fundamental theorem of algebra (FTA) given in Hatcher's Algebraic Topology textbook (I have underlined the relevant part): Could someone explain why $r$ needs to be ...
7
votes
2answers
274 views

Understanding Hatcher's proof of Borsuk-Ulam theorem for $n=2$

I am trying to understand the proof of the Borsuk-Ulam theorem for $S^2$ given in Hatcher's "Algebraic Topology" (Th. 1.10), as another person does here, but we are stuck at different places, so I ...
0
votes
1answer
163 views

A question on Kronecker Index

I am reading A book by Milnor (Lectures on characteristic classes) and I can across this section on Stiefel-Whitney numbers (page 16) and he uses the Kronecker index but never defines it (He says to ...
1
vote
1answer
69 views

Homeomorphism - transforming mug into donut

I read that a map is 'visually' a homeomorphism if you don't have to fold or tear the object. Thus, I was wondering what the problem with folding is? I guess that in this statement they don't assume ...