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

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4
votes
1answer
279 views

Simplicial homology of a tetrahedron

I'm studying the basics of homology on Nakahara, Geometry, Topology and Physics and I'm trying to work out the (simplicial) homology group of the tetrahedron described by the complex $K=\{0,1,2,3,(01),...
1
vote
1answer
77 views

Simple question concerning the properties of the fundamental group

I need to prove that every element of the fundamental group has an inverse. First we define a map $\phi:I\to I$ homotopic to $\operatorname{Id}_I$. If $\phi$ is the constant zero function isn't it ...
1
vote
1answer
126 views

Lifting homeomorphisms covering

Hello I had a question regarding a lemma from the paper: http://www.math.columbia.edu/~jb/bir-hilden-annals.pdf I don't understand the proof of Lemma 5.1. Notation: $T_{0,0}$ is the 2-sphere, $T_{...
1
vote
0answers
39 views

Relative homology of interlevel set

Let us consider a function $f\colon \mathbb{R}^3\to\mathbb{R}$, $f(x,y,z) = x^3+y^3+z^3 - 5yz$. Can anybody drop a hint how to compute relative homology of interlevel sets with coefficients in $\...
0
votes
1answer
151 views

Homeomorphism under subspace topology in Hausdorff space

Let $Y$ be a Hausdorff space, and $U,V \subset Y$ are homeomorphic under subspace topology. Does this imply if $U$ is open(or closed) then $V$ is open(or closed) under original topology? I can't ...
1
vote
0answers
434 views

Find all the covering spaces of the Klein Bottle

I am studying for a qualifying exam and it asks for all the covering spaces of the Klein Bottle. Does anyone have any suggestions? I am not sure how to go about finding ALL of them.
4
votes
2answers
83 views

How to use Mayer-Vietoris to show $\chi(X)=2\chi(M)-\chi(\partial M)$ where $X$ is the double of $M$?

I'm in trouble with the following problem: Let $M$ be a manifold with compact boundary $N$ and let $X$ be the double of $M$, that is, the manifold without boundary one gets by glueing $M$ with ...
3
votes
0answers
53 views

eilenberg-steenrod for pairs in any model category?

The Eilenberg Steenrod axioms are functors on the homotopy category of pairs of "spaces" $(X, A)$. Typically they are introduced when $X$ and $A$ are some sort of topological spaces. My question is ...
2
votes
0answers
102 views

Visualize a projective curve $X^3+Y^3=Z^3$ in $P_2(C)$ as a torus

Let $P_2(C)$ be the 2 dimensional complex projective space, I want to prove that the projective curve defined by $M=\{[X,Y,Z]\in P_2(C)|X^3+Y^3=Z^3\}$ is a torus. I know that there is a theorem saying ...
6
votes
1answer
419 views

To prove a vector bundle $\xi$ is orientable $\iff w_1(\xi)=0$

I went about it this way: $\xi$ an $n$-rank vector bundle over a topological space $M$ is orientable. $\iff$ its top exterior power $\bigwedge^n\xi$ is trivial $\iff$ the 1-frame bundle $V_1(\...
0
votes
1answer
186 views

There is more then one Homology Theory for spaces, which are not Hausdorff

All of us know: if we have a CW-complex, then using an arbitrary homology theory (with the axioms of disjoint union and dimension axiom) we always get the same homology groups up to an isomorphism. ...
1
vote
1answer
93 views

Isomorphism relative homotopy groups

Suppose that we have $Y \subset X$ topological space such that $$ \pi_i(X,Y)=0$$ for all $0 \le i < k$. How can I prove that the homomorphism induced by inclusion $i: Y \hookrightarrow X$, say $i^*:...
3
votes
1answer
62 views

Why not extending to the whole disk implies have a zero

For any complex polynomial $p(z)$ of order $m$, we showed earlier that on a circle $S$ of sufficiently large radius $r$ in the plane, $$\frac{p(z)}{|p(z)|}\quad \text{and}\quad \frac{z^m}{|z^m|}=\...
2
votes
1answer
63 views

fundamental group of a graph is free

Let $X$ be a connected graph, and $T$ its maximal tree. Via covering spaces and deck-transformations, how one can prove that $\pi_1(X)= \pi_1(X/T)$?
1
vote
1answer
226 views

helix and covering space of the unit circle

Does a bounded helix; for instance $\{(\cos 2\pi t, \sin 2\pi t, t); -5\leq t\leq5\}$ in $\mathbb R^3$ with the projection map $(x,y,z)\mapsto (x,y)$ form a covering space for the unit circle $\...
3
votes
0answers
107 views

About Linking Number

I'm looking for some references about Linking Number. I already know these http://mathworld.wolfram.com/LinkingNumber.html http://www.matapp.unimib.it/~ricca/publications/2011/JKTR11.pdf Anything ...
7
votes
0answers
72 views

Automorphism on a kahlerian variety which acts as -id on $H^2(X,\mathbb{Z})$

I've read this proposition: "there is no automorphism (biholomorphic map) of a Kahlerian complex manifold $X$ which acts on $H^2(X,\mathbb{Z})$ as $-$identity" so I tried to prove this statement and ...
0
votes
0answers
44 views

Path fibration over a connected manifold.

Let $M$ be a differentiable manifold. We can consider $P(M):=\{\gamma:[0,1] \to M\}$, so we have a natural projection on $M$ $$ P(M) \to M $$ $$ \gamma \mapsto \gamma(1) ,$$ in the fibre of this ...
5
votes
1answer
210 views

If $i\colon A\to X$ is a cofibration then $1\times i\colon B\times A\to B\times X$ is a cofibration for any space $B$. Is that true?

In Algebraic Topology (Hatcher, pg 14) I find: A pair $\left(X,A\right)$ has the homotopy extension property if and only if $X\times\left\{ 0\right\} \cup A\times\mathbb{I}$ is a retract of $X\times\...
2
votes
1answer
145 views

Classifying space for finite-dimensional torus

Note that $K({\bf Z},1)=S^1$ but $BS^1 = {\bf CP}^\infty$. For finite groups $H$, $G$, $$K(G\times H,1) = K(G,1)\times K(H,1)$$ Does it works for classifying spaces of continuous groups ? As far ...
8
votes
3answers
1k views

Local homeomorphisms which are not covering map?

I am trying to find examples of maps between topological space which are local homeomorphism but not covering maps. Especially, how twisted has to be such a counterexample : can it be a local ...
1
vote
1answer
54 views

For which $n$ can one infer from these assumptions that there is a $y \in S^n$ such that $f(y) = y$? [closed]

In this problem $S^n$ means $\{x \in \mathbb{R}^{n + 1} \ | \ \|x\| = 1\}$ and $f\colon S^n \rightarrow S^n$ is a continuous map that satisfies $f(x) = f(-x)$ for every $x \in S^n$. For which $n$ can ...
0
votes
1answer
244 views

Regarding 3-fold connected coverings of the $S^1 \vee \mathbb{R} P^2$

As in the question, I need to determine all of the 3-fold connected coverings of the wedge of the unit circle and the real projective plane. Here's what I think: I know that the fundamental group ...
0
votes
0answers
85 views

Clarification needed - the fundamental group of the circle

I am reading the proof of $\mathbb{Z}\approx\pi_1(S^1)$ from Hatcher and didn't understand the last paragraph in the picture (the homomorphism part): Isn't $\tau_m\tilde{\omega}_n:I\to \Bbb{R}$ ...
2
votes
0answers
64 views

Toric Varieties as quotient of Lie groups

let $X_\Sigma$ be the toric variety of a smooth and complete fan $\Sigma$. Then $X_\Sigma$ has no toric factors and can nicely be represented as the quotient $$X_\Sigma \simeq \left( \mathbb{C}^r - Z\...
2
votes
0answers
81 views

Hurewicz isomorphism in equivariant stable homotopy

Let $G$ be a finite group and let $X$ be a $G$-CW-complex. Denote by $\pi_{\ast}^G(X)$ the $G$-equivariant stable homotopy groups of $G$ and by $\mathrm{H}_{\ast}^G(X,A(-))$ the Bredon homology of $G$ ...
2
votes
1answer
95 views

Path space vs loop space

If we consider the path fibration over a topological space $X$ we have $$ \Omega(X;p,p) \hookrightarrow P(X) \to X .$$ Where I denote with $\Omega(X;p,p)$ the set of paths $\omega:[0,1] \to X$ such ...
0
votes
1answer
212 views

Can't understand proof of excision property

I was reading the proof here. I understood almost all the parts after Exercise 4, such as defining $bs_n ^X$ inductively, a chain homotopy $(R^X )$, and that we get a small chain $(bs_n ^X)^k $ by ...
2
votes
0answers
132 views

Torsion of fundamental group is abelian

On Riemannian manifold with Ricci curvature bounded below (For example, flat torus. It has ${\bf Z}^2$ as fundamental group, which is nilpotent (=almost abelian). In fact Ricci curvature bouned ...
10
votes
0answers
633 views

Homology of a compact manifold is finitely generated [duplicate]

Perhaps this is obvious and I am overlooking something, but why are the homology groups of compact manifolds finitely generated?
1
vote
0answers
54 views

Definition of HEP correct?

In Algebraic Topology (Allen Hatcher, pg 14) I read a definition of HEP: ...Suppose one is given a map $f_{0}:X\rightarrow Y$, and on a subspace $A\subset X$ one is also given a homotopy $f_{t}:A\...
0
votes
1answer
140 views

Prove that there exists a long exact sequence…

Let $f, g : X \to Y$ be two continuous maps. Consider the space $Z$ which is obtained from the disjoint union of $Y$ with $(X \times [0, 1])$ by identifying $(x, 0) \sim f (x), (x, 1) \sim g(x),$ for ...
3
votes
0answers
151 views

Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel: Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $ \...
2
votes
1answer
67 views

Pointed space mapping clarification and isomorphisms between different ordered Homotopy groups.

If I am given a pointed pair of spaces $(X,A,x_{0})$ and define $P(X;x_{0},A) \subset X^{I}$ as the subspace given by the paths $\alpha$ in $X$ such that $\alpha(0) = x_{0} \text{ and } \alpha(1) \in ...
4
votes
2answers
122 views

Derivation and meaning of a long exact sequence of a Homotopy Groups for pairs of spaces

I read that there are long exact sequences of homotopy groups for each pair of pointed spaces $(X,A,x_{0})$. Now I know that for an exact sequence that, as the example below denotes $f \text{ and } g$...
0
votes
2answers
78 views

Surgery on $S^m$

On page 4 of the book "ALGEBRAIC AND GEOMETRIC SURGERY" by Andrew Ranicki, after the definition of surgery has written: Example View the $m$-sphere $S^m$ as $$S^m=\partial (D^{n+1} \times D^{m-n})=S^...
3
votes
1answer
196 views

Topics of Group Theory Required to Understand Betti Numbers

I am studying Group Theory. I made sure I have a problem at hand, as part of the motivation for my study. I have chosen the problem as being able to understand as well as compute Betti Numbers where ...
3
votes
1answer
83 views

Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$

Let $F_n$ denote the free group on $n$ elements. Prove that any subgroup of $F_5$ of index 3 is isomorphic to $F_{13}$. I noted that the wedge product of 13 copies of $S^1$ is a 3 fold covering of ...
3
votes
1answer
78 views

singular (co)homology over various fields of same characteristic

Is the following true: if $K$ and $F$ are fields with the same characteristic and $X$ is a topological space, then for any $n$ there holds $$\dim_K H_n(X;K) = \dim_F H_n(X;F)\text{ and }\dim_K H^n(X;K)...
4
votes
1answer
751 views

Fundamental group of the complement of the solid torus

Let $T$ be a solid torus, how to calculate the fundamental group $\pi_1(\mathbb R^3- T)$? intuitively, I think it's a free group with one generator. So if it is so, how to obtain it, and what the ...
1
vote
1answer
163 views

The fundamental group of the unit disc with one point removed from its boundary

If $y\in \partial (\mathbb D^2)$, then how to find $\pi_1(\mathbb D^2-\{y\})$? I know that if $y$ was an interior point then the answer will be $\mathbb Z$. But why both cases would be similar ?
4
votes
1answer
111 views

Yoneda's lemma and $K$-theory.

The K-theoretic form of Bott periodicity is that $\tilde{K}(X)\cong \tilde{K}(\Sigma^2 X)$, i.e., $\langle X, BU\times \mathbb{Z} \rangle \cong \langle \Sigma^2 X, BU\times \mathbb{Z} \rangle$. The (...
6
votes
1answer
139 views

$K$-theoretical interpretation of Bott periodicity.

We consider complex Bott periodicity $\pi_{i-1}(U) \simeq \pi_{i+1}(U)$. Why we can say that the $K$-theoretical interpretation of this assertion is $\tilde{K}(X) \cong \tilde{K}(\Sigma^2(X))$? Where ...
3
votes
3answers
101 views

Definition of the fundamental group

Why are the elements of the fundamental group of a space equivalence classes? Why isn't the group defined to be the set of all possible loops at a base point with the product operation of paths? What ...
4
votes
1answer
689 views

Loop space suspension/adjunction

Let be $X,Y$ Hausdorff spaces. I denote with $\langle \cdot, \cdot \rangle$ the basepoint preserving homotopy classes of maps, with $\Sigma$ the reduced suspesion and with $\Omega$ the loop space. How ...
0
votes
1answer
92 views

Why is this homotopy an isotopy?

I am trying to follow the proof of Willem's quantitative deformation lemma and I get everything except the justification for the (iv) property which states: $\eta_t(u)$ is a homeomorphism for ...
6
votes
1answer
100 views

A step in computing the cohomology ring of $\mathbb{C}P^n$

On page 250 of Hatcher's Algebraic Topology, he uses a certain corollary to compute the cohomology ring of $\mathbb{C}P^n$. The relevant section is below for convenience: I understand the proof ...
7
votes
1answer
805 views

The loop space of the classifying space is the group: $\Omega(BG) \simeq G$

Why does delooping the classifying space of a topological group G return a space homotopy equivalent to G. In symbols, why $\Omega(BG) \cong G$, where $G$ is a topological group and $BG$ its ...
2
votes
1answer
65 views

Graphic interpretation of path fibration.

Let $S^2$ the unit sphere. We can consider the associated path fibration $$ \Omega(S^2) \rightarrow P(S^2) \rightarrow S^2 .$$ I have to explain path fibration so I think that it is useful to make a ...
3
votes
3answers
108 views

Is $\langle abab^{-1}\rangle$ a normal subgroup of $\langle a,b|\varnothing\rangle$?

Let $G=\langle a,b | \varnothing\rangle$ and let $H\leq G$ s.t $H=\langle abab^{-1}\rangle$. Is $H\triangleleft G$? I'm asking this question in order to understand the fundamental group of the Klein ...