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
1answer
2k views

Fundamental group is abelian iff the fundamental group isomorphisms (a-hat) coincide

I want to show that if $X$ is a non-empty path connected space, then the fundamental group is abelian if and only if given any points $y, z\in X$ and paths $\alpha, \beta$ from $y$ to $z$, ...
5
votes
2answers
350 views

Homotopy fiber of inclusion of projective spaces equivalent to sphere $S^3$

Consider the inclusion map $S^2=\mathbb{C}P^1 \overset{f}{\to} \mathbb{C}P^\infty$ ($\mathbb{C}P^\infty$ is the sum [direct limit] of $\mathbb{C}P^n$s) and the mapping space $E_f\subseteq ...
2
votes
1answer
302 views

Fundamental group of multiplicative group in Zariski topology

What is the fundamental group of the multiplicative group of the complex numbers $\mathbb{G}_m(\mathbb{C})$ with respect to the Zariski topology. More precisely, what are the homotopy classes of ...
1
vote
2answers
157 views

Short Five Lemma for Fibrations

Is there a short five lemma for fibrations in algebraic topology (in whatever category where it would be suitable -- the topological category, the homotopy category, whatever). By short five lemma I ...
0
votes
3answers
65 views

Is every fiber-preserving map between coverings again a covering?

Suppose we have two coverings $$p_1:Y\rightarrow X$$ and $$p_2:Y^\prime\rightarrow X$$ and further a continuous map $$\pi\colon Y\rightarrow Y^\prime,$$ such that $$p_2\circ\pi=p_1.$$ Ist $\pi$ ...
5
votes
2answers
371 views

Fundamental group of shrinking wedge of spheres.

Let $X$ be the subspace of $\mathbb R^3$ which is union of the spheres of radius $1/n$ and centered at $(1/n,0,0)$. Then $X$ is simply connected. I had thought for it in this way to attach $2$-cells ...
0
votes
2answers
248 views

Retraction from torus to linked circle [duplicate]

Possible Duplicate: Why is this entangled circle not a retract of the solid torus? I am stuck with exercise 16 (c), pag.39 of Hatcher's Algebraic Topology: prove that there is no retraction ...
3
votes
1answer
88 views

Prove that such homeomorphisms have fix points

If a homeomorphism $f:R\rightarrow R$ satisfies $f^2=1$, prove that it has at least one fix point. What if we set $f^n=1$ instead of $f^2=1$?
5
votes
0answers
129 views

$\operatorname{Spin}^c(n)$ is a Lie group?

Assume that $n\geq 3$. (This implies $\pi_1(\operatorname{SO}(n))=\mathbb{Z}/2$.) Let $\rho\colon \operatorname{Spin}(n)\to \operatorname{SO}(n)$ be the 2-fold universal cover. Identify $\ker(\rho)$ ...
8
votes
2answers
1k views

This quotient space is homeomorphic to the Möbius strip?

Let $G:\mathbb R \times [-1,1]\to \mathbb R \times [-1,1]$ be a map defined by $G(x,y)=(x+1,-y)$ This space $Q=\mathbb R\times [-1,1]/\sim$, where $(x_1,y_1)\sim (x_2,y_2)$ if and only if there is ...
4
votes
1answer
369 views

Rotman Introduction to Algebraic Topology Question 0.5

I've been stuck on a problem from Rotman's Introduction to Algebraic Topology for a while. I'm doing the exercises outside of class right now so it's difficult to ask for help. I'm hoping someone here ...
6
votes
1answer
838 views

contractible and simply connected

Every contractible space X is simply connected because X is homotopy equivalent to a point. Is there a direct proof of this fact? There obviously is a (free) homotopy between any loop and the trivial ...
83
votes
3answers
8k views

Topology: The Board Game

Edit: I've drawn up some different rules, a map and some cards for playing an actual version of the game. They're available at my personal website with a Creative Commons Attribution 4.0 license. ...
7
votes
1answer
303 views

Fundamental group calculations

I'm a student taking my first course in algebraic topology. I've stumbled across this exercise: calculate the fundamental group of $S^3-\gamma$, where $\gamma$ is a circumference in $\mathbb{R}^3$ ...
5
votes
0answers
161 views

Symmetric product of genus 2-surface

Let $\Sigma$ be the genus 2-surface. Denote $\operatorname{Sym}^2(\Sigma)=\Sigma\times \Sigma/\mathbb{Z}/2$, where $\mathbb{Z}/2$ acts on $\Sigma\times \Sigma$ by $(x,y)\mapsto (y,x)$. In the very ...
6
votes
1answer
391 views

A question about the contractibility of the Sierpinski space

The two-point Sierpinski space is usually defined as follows: Let $X =\{x,y\}$ be the two-point space where the only open sets are $X, \varnothing, \{x\}$. I think from this it can be inferred that ...
3
votes
1answer
310 views

Homologies of the pairs are same but they are not homotopy equivalent as pairs.

Consider the pairs $(\mathbb D^{n},S^{n-1})$ and $(\mathbb D^{n},\mathbb D^{n}-\{0\})$ ,clearly their homologies are same in each dimensions but these pairs are not homotopy equivalent. Any homotopy ...
2
votes
1answer
188 views

Topology: homotopy of product of paths

I'm trying to prove that the product operation on paths induces a well defined operation on path homotopy classes defined by the equation $[f]*[g]=[f*g]$ Let $F$ be a path homotopy between $f$ and ...
15
votes
1answer
306 views

Geometric interpretation of the map $SO(4) \to SO(3)$

Let me first explain the background of my question. As is well known, the group $SO(n+1)$ acts transitively on the sphere $S^n$, and the stabilizer is the group $SO(n)$, so that we get a fibration ...
5
votes
2answers
321 views

Finitely generated singular homology

Let G be a finitely generated abelian group and M a compact manifold, I want to prove that $H_r(M,G)$ is finitely generated for $r\ge 0 $. First I was thinking if I could do induction over $r$ ...
8
votes
2answers
516 views

What is the point of a lift in topology?

I've just covered 'lifts' in topology and also homotopy lifting to a covering map but I'm struggling to understand the intuition behind lifts and essential the 'point' of them. Could someone please ...
7
votes
1answer
98 views

Why is a smooth algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ of degree 5 simply connected?

The title says it all. In fact, i am only trying to prove that if $S$ is an irreducible smooth algebraic surface of degree 5 in $\mathbb{P}_{\mathbb{C}}^3$ (hence a four dimensional manifold over the ...
0
votes
0answers
252 views

Induced map on homology

If I'm given a map from $\mathbb{C}P^1\times\mathbb{C}P^1$ to $\mathbb{C}P^3$ which sends $([z_{0},z_{1}],[w_{0},w_{1}])$ to $[z_{0}w_{0},z_{0}w_{1},z_{1}w_{0},z_{1}w_{1}]$, how do I compute the ...
10
votes
2answers
909 views

Computing the homology groups.

Let $X$ be space obtained by first removing the the interior of two disjoint closed disks from the unit closed disk in $\mathbb R^{2}$ and then identifying their boundaries clockwise. Compute the ...
1
vote
1answer
83 views

Importance of Sheafs in Analysis and what type of tricks we study there?

There is a concept of Sheafs in topology and Algebraic geometry that i dont know what things we prove using them which are useful in topology or in analysis (eg. Harmonic Analysis). Can any body ...
3
votes
2answers
77 views

Isogenies of Riemann surfaces

Fix $n$ natural. I want to characterize all compact Riemann surfaces $M$ such that $M$ is an unramified covering of degree $n$ over itself. How do I construct this covering map? This map is called ...
1
vote
1answer
629 views

$4$-sheet covering of the wedge sum of two circles

I'm trying to find the $4$-sheet covering of the wedge sum of two circles I don't know even how to begin, I know just the definitions of coverings and simple examples, I really need help here. ...
2
votes
1answer
128 views

Generator of the fundamental group of $ \mathbb RP^{2}$.

Take a closed hemisphere and identify the antipodal points on the equator ,we get $\mathbb RP^{2}$ and inside $ \mathbb RP^{2}$ we have copy of $ \mathbb RP^{1}$.So, what will be the induced map on ...
3
votes
0answers
303 views

Orientation of manifold in topological sense

What do we mean by orientability of a topological manifold? How do we orient two dimensional Euclidean space and why is Moebius band non-orientable? And it would be a great favour to me if you can ...
3
votes
1answer
1k views

Finite covering is compact Hausdorff iff base space is

I am in need of solution or tip for this question. I thank you. Let $ p: \widetilde X \to X $ be a covering space with $ p^{-1}(x) $ finite and nonempty for all $ x \in X$. Show that $ \widetilde X$ ...
3
votes
1answer
99 views

Monodromy Representations

Let, be $V$ a connected smooth manifold and $q_1,q_2\in V$ and $F:U\to V$ a connected covering of degree $d$. This covering induces two monodromy representations $\rho_1:\pi_1(V,q_1)\to S_d $ ...
2
votes
1answer
85 views

Finding a covering space for $P \times P$

Let $P$ be a real projective plane. Since the fundamental group of $P \times P$ is $Z_2 \times Z_2$ (abelian group with 5 subgroups), there exists five covering spaces. What is the explicit covering ...
1
vote
0answers
84 views

Postnikov invariants & Cohomology of EM spaces

I'm in trouble in understanding these two statements in Morita's Geometry of Characteristic Classes book: First: what's up with the "twisted" product $K(\pi_2(X),2)\times_{k^4} ...
2
votes
2answers
280 views

Does every lift of a constant path is constant?

I'm trying to prove that every lift of a constant path is constant using the path lifting property which says that for each path $f:I\to X$ and each lift $\tilde x_0$ of the starting point $f(0)=x_0$ ...
4
votes
4answers
1k views

Why this map is a covering map?

I'm trying to find the universal covering space of the Klein bottle. I know that $\mathbb R^2$ covers the Klein bottle , but I don't know how to prove, I found this proof on internet: Someone knows ...
6
votes
2answers
313 views

Why the Picard group of a K3 surface is torsion free

Let $X$ be a K3 surface. I want to prove that $Pic(X)\simeq H^1(X,\mathcal{O}^*_X)$ is torsion free. From D.Huybrechts' lectures on K3 surfaces i read that if $L$ is torsion then the Riemann Roch ...
12
votes
1answer
300 views

Is it possible for a closed manifold to deformation retract onto a proper subset of itself?

Let $M$ be a closed (compact, without boundary) topological manifold. Is it possible for there to exist a subset $A$ of $M$ such that $M$ deformation retracts onto $A$?
2
votes
0answers
84 views

4D TQFT construction from a modular tensor category

I know the construction of 3D topological quantum field theory (TQFT) from a modular tensor category. I heard that we can even (mathematically) construct 4D TQFT from a modular tensor category. I ...
3
votes
0answers
102 views

How to prove that a lie group is simply connected

I need to prove that $O(3,19)/SO(2)\times O(1,19)$ is simply connected. In particular $O(n_{+},n_{-})$ denotes the orthogonal group of $\mathbb{R}^{n_{+}+n_{-}}$ endowed with the diagonal quadratic ...
1
vote
1answer
140 views

confusion about cup product in cohomology ring

I have some confusion that i would like to clarify. The product in cohomology is not the cup product $\smile$ but it is another product $*$ that is constructed from cup product. Indeed wrinte ...
2
votes
1answer
70 views

Real projection spaces

It seems that both $RP^2$ and $RP^3$ have the same fundamental group $Z_2$, but Why no map from $RP^3 \to RP^2$ induces an isomorphism between their fundamental groups?
4
votes
0answers
65 views

Computing number of path components.

Let $H \subset \mathbb{R}^n$ be a closed subset homeomorphic to $\mathbb{R}^{n - 1}$. Could you give me an idea of how to prove that $\mathbb{R}^n - H$ has exactly two path components. I am expecting ...
2
votes
1answer
107 views

What is the topology on $[0,1]^2$ in the defintion of a homotopy?

I have a question about the definition of a homotopy between loops: Let $\alpha$ and $\beta$ be loops with base point $x$ in a topological space $X$. A homotopy from $\alpha$ to $\beta$ is a ...
3
votes
1answer
172 views

Building a space with given homology groups

Let $m \in \mathbb{N}$. Can we have a CW complex $X$ of dimension at most $n+1$ such that $\tilde{H_i}(X)$ is $\mathbb{Z}/m\mathbb{Z}$ for $i =n$ and zero otherwise?
5
votes
1answer
139 views

Proving that $O(n,m)$ is simply connected.

My question is the following: Under which conditions on given integers $n\le m$ is $$O(n,m) = \{A \in \mathbb R^{m\times n} : A^TA = \mathbf 1\}$$ simply connected? Does anyone know a reference for ...
0
votes
1answer
286 views

Co-homology Groups of the Torus

I wanted to explain to me, or give me a reference of how to calculate the cohomology groups of the complex and real, torus $\mathbb{T}^2$. I want to use this as an example in a seminar that I will ...
4
votes
2answers
420 views

If a covering map has a section, is it a $1$-fold cover?

If $q: E\rightarrow X$ is a covering map that has a section $(i.e. f: X\rightarrow E, q\circ f=Id_X)$ does that imply that $E$ is a 1-fold cover?
1
vote
0answers
177 views

homotopy groups of mapping space

I got this homework problem: $X,Y$ finite CW-complexes with $\dim X=m$ and $Y$ is $n$-connected. Prove that $\pi_k(map(X,Y))=0$ for all $k \le n-m$. Thanks for the help!
2
votes
0answers
210 views

Transition functions for the tautological bundle

Define the tautological bundle over $CP^1$ to be $\tau = \{[a_1, a_2], (z_1, z_2) \in CP^1\times\mathbb{C}^2 | \exists \lambda \in \mathbb{C} \;\text{such that} \;\lambda (z_1,z_2) = (a_1, a_2) \}.$ ...
4
votes
1answer
92 views

Fuchsian Group without fixed points

I'm searching for a Fuchsian Group without fixed points. (because i need an example for a group $\Gamma$, so that $\mathbb{H}/\Gamma$ is a Riemannian surface, and therefore $\Gamma$ has to be a ...