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

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3
votes
0answers
121 views

The fundamental group of the union of three convex open subsets of $ \mathbb{R}^n$.

I have to prove that the fundamental group of the union of three open convex subsets of $\mathbb{R}^n$ is trivial or $\mathbb{Z}$. I can show that it has only one generator, but I can't prove that if ...
2
votes
0answers
80 views

The fundamental group of an open set in $\mathbb{R} ^n$ does not have nilpotent elements.

I am studying a little of basic algebraic topology and I thought that this statement could be true. If you have an open connected set $U \subset \mathbb{R}^m$ and a loop $\gamma$ that is not ...
6
votes
1answer
465 views

The Gram-Schmidt process is a deformation retraction

Consider the Gram-Schmidt process $r : GL(n) \rightarrow O(n)$ that sends invertible matrices to orthogonal matrices. I need to show this is a deformation retraction and, by restrictions of $r$, ...
1
vote
0answers
45 views

Classification theorem of the coverings of a given space

I'm trying a lot to find easy examples of classification theorems of covering spaces of a given space. I've already read some examples here at Mathexchange such as Classification of covering spaces ...
2
votes
0answers
62 views

How to find every 4-sheet covering of the wedge sum?

Based on this question 4-sheet covering of the wedge sum of two circles I know how to find one 4-sheet covering of the wedge sum, but how to find every 4-sheet covering of the wedge sum? I really ...
3
votes
1answer
251 views

An open set in $\mathbb{R}^{2}$ is not homeomorphic to an open set in $\mathbb{R}^{n}$ for $n > 2$

I'm taking an introduction to algebraic topology course and this is one of the assigned problems. We have computed the fundamental group of the circle and we have proved that $S^{n}$ is simply ...
4
votes
1answer
193 views

A question on the deformation retract

I want to show that the subspace $A\cup B$ where $A=\{(x,y);(x-1)^{2}+y^{2}=1\}$ and $B=\{(x,y);(x+1)^{2}+y^{2}=1\}$ is a deformation retract of $X=\{(x,y);x^{2}+y^{2}\leq 4\}-\{(1,0),(-1,0)\}$. For ...
3
votes
0answers
87 views

Construct a space with free involution and homological restriction

I'm looking for a space $X$ which satisfies the following conditions: $X$ is a compact manifold. $H_\ast (X;\mathbb Z)$, the integral homology groups $X$, are torsion free. There is a free ...
5
votes
3answers
148 views

Homeomorphism exchanging two homotopic paths

Let $X$ be a compact simply connected space and $\gamma_1, \gamma_2 : [0,1] \to X$ be two (homotopic) simple paths between two different points $x,y \in X$. Does there exist a homeomorphism $\varphi : ...
0
votes
1answer
82 views

Abelianization of the $\mathbb R \mathbb P^2$#$\mathbb R \mathbb P^2$

I know that the fundamental group of $\mathbb R \mathbb P^2$#$\mathbb R \mathbb P^2$ is $<a,b|a^2b^2>$. When we abelianize this we have this presentation: $Ab(\pi_1(\mathbb R \mathbb ...
1
vote
1answer
133 views

A covering space of “8” defines the commutator subgroup of F2

Let M be the squares space $(R\times Z) \cup (Z\times R)$ covering the 8 space (pointwise union of two circles) by calling one circle of the 8 (path starting and ending at the dot of intersection of ...
12
votes
1answer
1k views

fundamental group of the Klein bottle minus a point

I'm trying to see the fundamental group of the Klein bottle minus a point without success. I know how to solve the torus minus a point giving a deformation retraction to the wedge sum of two circles. ...
8
votes
1answer
263 views

The complement of Jordan arc

If $A$ is the image of a Jordan arc in $S^2$, that is, $A$ is the image of an injective continuous map from $[0,1]$ to $S^2$, is $S^2-A$ necessarily a simply-connected set?
12
votes
2answers
381 views

Torsion on $\pi_1(X)$, $X$ connected and open in $\mathbb{R}^n$

Can the fundamental group of an open connected subset $X$ of $\mathbb{R}^n$ have a torsion element?
2
votes
1answer
106 views

Definition of cochain algebra

In Felix, Halperin, Thomas "Rational homotopy theory" on page 46 we find the following definition : A cochain algebra is a dga $(R, d)$ with $R = \{R^n\}_{n\ge o}$. I can't get the ...
2
votes
0answers
66 views

fundamental group of complement in the thickened torus

What is the fundamental group of $(T \times I ) \backslash J$ where $J$ is a closed loop going around the vertical $\mathbb{S}^{1}$ of the thickened torus twice.
8
votes
0answers
205 views

How did Chern pictured the first Chern number?

The first Chern number $\cal C$ is known to be related to various physical objects. Gauge fields are known as connections of some principle bundles. In particular, principle $U(1)$ bundle is said to ...
0
votes
2answers
416 views

Fundamental group of this space

Based on this question: What is the homology groups of the torus with a sphere inside? I'm trying to find the fundamental group of this space using the Seifert–van Kampen theorem. If $U$ is the torus ...
4
votes
0answers
153 views

Calculating H_0 directly from Eilenberg-Steenrod axioms

It's well-known that every homology theory satisfying Eilenberg-Steenrod axioms is isomorphic to singular homology. I tried to perform some homology calculations directly from axioms but couldn't do ...
2
votes
1answer
278 views

What is the homology groups of the torus with a sphere inside?

I'm trying to find the homology groups of this space X below. The easiest part is $H_0(X)=\mathbb Z$, because $X$ is connected. In order to find $H_1(X)$ I think the easiest method is finding its ...
1
vote
1answer
38 views

maximal tori and principal $N(T)$-bundles.

Let $U(n)$ be the unitary group and $T^{n}= S^{1} \times \cdots \times S^{1}$ a maximal torus in $U(n)$. Let $N(T^{n})$ be the normalizer in $U(n)$ of $T^{n}$. How can i prove that $U(n) \rightarrow ...
3
votes
2answers
173 views

What is the homology group of the sphere with an annular ring?

I'm trying to compute the homology groups of $\mathbb S^2$ with an annular ring whose inner circle is a great circle of the $\mathbb S^2$. space X Calling this space $X$, the $H_0(X)$ is easy, ...
1
vote
1answer
82 views

Submanifolds of a space of functions?

Let $f:\mathbb{R}\rightarrow(\mathbb{R}\rightarrow\mathbb{R})$ be a function mapping a real number uniquely into the set $\mathbb{F}$ of total functions from $\mathbb{R}$ to $\mathbb{R}$. $\mathbb{F}$ ...
3
votes
0answers
134 views

Cellular inclusion is a cofibration

I have to show that if $X$ is a CW-complex, $A$ is a subcomplex and $i:A\hookrightarrow X$ is a cellular inclusion, then $i$ is a cofibration. My attempt is as follows. I think my proof, which first ...
5
votes
1answer
95 views

help-need to determine that this induced map is the zero map

Let $f : S^3 \rightarrow S^3$ have the property $f(x) = f(-x)$ for every $x \in S^3$. Show that $f_{*} : H_{3}S^3 \rightarrow H_{3}S^3$ is the zero map.
6
votes
3answers
302 views

Does the total space of a fibre bundle have the homotopy type of a CW complex if the base and the fibers have?

Let $$ F\to E\xrightarrow{\pi} B $$ be a fibre bundle over a connected and compact base $B$. Is it true that the total space $E$ has the homotopy type of a CW complex, if the fibre $F$ and the base ...
4
votes
1answer
151 views

Fiber bundle on Stiefel manifold

Let $V_{n}(\mathbb{C}^{k})$ the Stiefel manifold of $n$-frame in $\mathbb{C}^{k}$. We can see $V_{n}(\mathbb{C}^{k})$ as a subset of $n$ copies of the cartesian product $S^{2k-1} \times \cdots \times ...
7
votes
1answer
368 views

why the K3 surfaces are minimal surfaces

I need to prove that all K3 surfaces are minimal surfaces, so that every birational map between K3 surfaces is an isomorphism. I've started to read beauville's book on complex algebraic surfaces: ...
3
votes
2answers
150 views

why the orthogonal group $O(k,l)$ is homotopy equivalent to $SO(K)\times SO(l)$

I want to prove that the orthogonal group $O(k,l)$ (http://en.wikipedia.org/wiki/Indefinite_orthogonal_group)is homotopy equivalent to $SO(k)\times SO(l)$, so that ...
5
votes
1answer
167 views

Relative CW approximation, Hurewicz fibrations and Serre fibrations

Let $B$ be a finite CW complex and $p:E\to B$ a Hurewicz fibration. I know that every topological space $E$ has the weak(!) homotopy type of a CW complex $D$ realized by a weak equivalence $f:D\to ...
6
votes
1answer
164 views

Example of a pair of non-cobordant manifolds

So far, any source I consult will gladly talk about cobordism classes of closed (compact and without boundary) oriented manifolds, but I have yet to see an example of a pair of manifolds which are not ...
4
votes
1answer
346 views

Connection between Euler characteristic and degree of the Gauss map

Let $M\subset \mathbb{R}^n$ be a closed compact smooth oriantable manifold. Let $E\to M$ be the normal bundle of the embedding and $N\subset \mathbb{R}^n$ a (closed) tubular neighbourhood. $N$ is an ...
2
votes
0answers
108 views

Is this map the Gauss map?

Let $M\subset \mathbb{R}^n$ be a closed compact smooth oriantable manifold. Let $E\to M$ be the normal bundle of the embedding and $N\subset \mathbb{R}^n$ a (closed) tubular neighbourhood. $N$ is an ...
2
votes
1answer
346 views

Visualized definition of cohomology

I cannot imagine how cohomology is related to graph theory, actually I read solid definition from wiki, and to be honest, I cannot understand it. e.g I know what is homology (in simple term), group ...
5
votes
1answer
511 views

Cohomology of Grassmannian

Let $G_r$ the infinite complex Grassmannian manifold. We know that $H^{*}(G_r)=\mathbb{C}[x_{1}, \cdots, x_{n}]$ where $x_i$ are the Chern classes of tautological bundle. But $H^{*}(G_r)$ is also ...
2
votes
3answers
275 views

$\pi_1(S^n)=0$ for $n\geq2$

Hello friends of math :D I want to prove the result named in the heading. I have some hints but i can't imagine how wo work with this to conclude the result: Consider $S^n\subset\Bbb{R^{n+1}}$ (as ...
6
votes
2answers
442 views

If two CW complexes have isomorphic homotopy groups, are they homotopy equivalent? [duplicate]

I denote with $\pi_{i}$ i-homotopy group. If I have $X,Y$ CW-complex and $\pi_{i}(X)=\pi_{i}(Y)$ for all $i$. Can I say that $X$ and $Y$ are homotopic equivalent? What type of equivalence is it?
1
vote
1answer
93 views

Fixed-point problem (Weyl group)

Let $G=U(n)$ a compact Lie group and $T$ a maximal torus in $G$ (subgroup of diagonal matrix). We define $W=N(T)/T$ the Weyl group where $N(T)$ is the normalizer of $T \in G$. I have to prove that $W ...
2
votes
1answer
100 views

Prove the isomorphism

I am trying to prove the following: $$\left[S^1 \vee S^1 \vee ... \vee S^1; S^1\right] \cong \mathbb{Z}^n$$ where $\vee$ is a wedge sum and $[X,Y]$ is homotopy class between $X$ and $Y$. I would ...
2
votes
1answer
114 views

Algebraic Topology-Explanation required for the following definition

I am currently reading the book A combinatorial introduction to topology by Michael Henle. Under "Compactness and Connectedness" there is the following definition which I didn't understand at all. I ...
8
votes
2answers
1k views

Unit sphere in $\mathbb{R}^\infty$ is contractible?

Let $\mathcal{T}_{\infty}= \left\{ U \subset \mathbb{R}^{\infty}: \ U \cap \mathbb{R}^n \in \mathcal{T}_n, \text{ for } n=1,2,... \right\} $. Of course $\mathcal{T}_{\infty}$ is topology in ...
1
vote
2answers
288 views

The homotopy equivalence classes of the sentence “I love the algebraic topology”

Determine the homotopy equivalence classes of the sentence "I love the algebraic topology". I want to learn that how we can define a homotopy on the set of letters of a sentence. Please with an ...
11
votes
1answer
572 views

When is a homology class a fundamental class?

Let $X$ be a real connected orientable closed $n$-dimensional compact differentiable manifold. A connected oriented closed $d$-dimensional submanifold $i:M\to X$ (i.e. $M$ is a real connected ...
2
votes
0answers
73 views

Equivariant Mayer Vietoris and Borel localization

We have this theorem: Let $U$, $V$ two open sets of manifold $M$, ($U \cup V = M$). If they are $G$-stable the induced sequence in cohomology $$ \cdots \rightarrow H^{k}_{G}(U \cup V) \rightarrow ...
2
votes
0answers
74 views

Global sections of covering spaces

Let $p:C\to X$ be a covering space having a global section $s:X\to C$. I can show that this implies that $s(X)$ is disconnected from the rest of $C$. Is there any reference where this is explicitly ...
1
vote
1answer
277 views

Prove that the spaces have the same homotopy type

This exercise was taken from the book "Fundamental Groups and Covering Spaces", from Elon Lages Lima. "Let $X=C_1\cup\cdots\cup C_k$ be a finite union of convex open sets in the Euclidean space ...
4
votes
1answer
904 views

Natural examples of deformation retracts that are not strong deformation retracts

The question here asks for examples of deformation retracts which are not strong deformation retracts. A comment is given refering the asker to Exercise 0.6 in Hatcher. (The description of this space ...
4
votes
2answers
107 views

help me find counterexample

for two comparable topologies on a set X, compactness of the bigger one (bigger in the sense of containment) implies the compactness of the smaller one, can anyone help me find example of two ...
18
votes
6answers
2k views

What is the difference between homotopy and homeomorphism?

What is the difference between homotopy and homeomorphism? Let X and Y be two spaces, Supposed X and Y are homotopy equivalent and have the same dimension, can it be proved that they are homeomorphic? ...
2
votes
0answers
122 views

Question On Cech Cohomology

In section 16 of the topology book of Bott and Tu, there is a path fibration $\Omega S^2 \to PS^2 \to S^2$. The $E_2$ page of the spectral sequence of this fibration is $$E_2^{p,q}=H^p(S^2,H^q(\Omega ...