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

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4
votes
2answers
284 views

Examples of fundamental groups which we can use the Van-Kampen Theorem

I'm starting to study the Van-Kampen Theorem and I realized there are some questions we can use this theorem directly. I find this theorem difficult for a beginner. Do some of you know some questions ...
1
vote
1answer
315 views

Higher homotopy groups of the Klein bottle

How would you show that $\pi_n, n>1$ of the Klein bottle is the trivial group? I was thinking Seifert-Van Kampen could be applicable?
2
votes
1answer
114 views

Ramified coverings of a manifold

Maybe someone could help me with a bit of alebraic topology. Take $M$ a $n$-manifold with $n \geq 3$ , and $V$ a submanifold of codimension $2$ in $M$. Assume $H_{n-2}(V) = 0$. I've read that under ...
6
votes
1answer
130 views

question about disconnected normal covering map

I have stuck on the problem in Hatcher, Algebraic topology, which claim that if the covering map $q\circ p:X\rightarrow Y \rightarrow Z$ is normal, then the covering $p:X\rightarrow Y$ also is normal. ...
0
votes
1answer
34 views

Let $\Gamma$ be discrete in S. Then for any region $Ω$ in $S$, $Ω \cap \Gamma$ is discrete in Ω

I'm trying to understand this existence of triangulation's proof in this book. I have problems to understand the lemma 8.2.6: Let $\Gamma$ be discrete in S. Then for any region $Ω$ in $S$, $Ω \cap ...
1
vote
1answer
130 views

Homology of a $3$-manifold obtained by rational surgery on an $m$-component link

I was trying to understand homology of a $3$-manifold $M$ obtained by rational surgery on an $m$-component oriented link $L$. I have a few questions regarding the following paragraph in the book ...
4
votes
1answer
251 views

About first Chern class and Poincaré duality in case of an ample divisor

Let $D$ be a very ample divisor in $X$ projective variety. I can't understand why the first chern class $c_1(\mathscr{O}_X(D))$ equals the Poincarè dual of D, $\mathscr{P}(D)$
0
votes
1answer
60 views

differential forms proving degree can't be greater than dimention and 1-forms are linearly independent.

Every differential form w, say of degree k on an open set in R^n, can be written uniquely as $w = \sum_I a_I (x)dx^I$ , where the sum is over all possible lists $I$ of $k$ increasing indices, say $I ...
0
votes
1answer
153 views

Implicit function theorem => continuously differentiable functions

Consider the curve in $R^3$ consisting of the intersection of the paraboloid $z=x^2 + y^2$ and the cylinder $x^2 + y^2 = 1$. Near which points of this curve does the implicit function theorem say we ...
6
votes
0answers
103 views

How difficult is it to impose a differential structure on a fractal?

Assuming we have a 4-dimensional smooth manifold $M$ embedded in $\mathbb{R}^{m}$ that is difficult to understand its differential structure. For convenience's sake we can assume it is compact, simply ...
1
vote
1answer
623 views

The wedge sum is a retract of the torus?

I'm trying to prove or disprove that the wedge sum of two circles is a retract of the torus. Intuitively it seems true, because the torus is defined as $S^1\times S^1$. I tried to disprove also, but ...
4
votes
1answer
128 views

Seek results in group theory obtained by applping algebraic topology tools

After studying the first section, especially the section 1.3 (covering spaces) in Hatcher's book, it is obvious that many classical topics in group theory, especially in infinite group theory, have ...
1
vote
2answers
275 views

$h_*$ is the trivial homomorphism, the $h$ is homotopic to a point

Let $h:S^1\to X$ a continuous map. If $h_*:\pi_1(S^1)\to \pi_1(X)$ is the trivial induced homomorphism, then $h$ is homotopic to a point. I'm starting to study fundamental groups and induced maps by ...
1
vote
1answer
239 views

Prove that $[0,\infty)$ is not a manifold.

Prove that $[0,\infty)$ is not a manifold. Using diffeomorphisms and the implicit function theorem perhaps.
7
votes
1answer
327 views

Intuition for cofibration

The notion of a fibration has a nice geometric intuition of one topological space (a fiber) being parametrized by another topological space (the base) -- this is taken from the Wikipedia entry on ...
1
vote
1answer
582 views

The proof of Brower Fixed Point Theorem for 2-dimensional case

Note the followin theorem Theorem : any continuous map from a unit two dimensional disk $E^2$ into itself has a fixed point. To prove this theorem, Harper and Greenberg's book use the following ...
1
vote
0answers
143 views

How do I prove this surface is homeomorphic to the sphere

Let $S$ a compact and connected surface. If $S=U_1\cup U_2$, where $U_1,U_2$ are of finite character and the boundary of $U_1$ is in $U_2$. How can I prove that S is homeomorphic to the sphere? ...
3
votes
2answers
172 views

Fundamental group of $G/H$

Let $G$ be a connected topological group. And $H$ be a discrete subgroup. Theorem : $\pi_1(g/H)=H$ This is the content in the book Algebraic Topology -Greenberg and Harper I want to know the proof. ...
2
votes
1answer
92 views

How compute $\pi_3(S^2)$ ?

My question is described well in the title. As you know, $\pi_3(S^2) = {\bf Z}$
2
votes
2answers
122 views

induced homomorphisms and extended maps

Let $h:S^1\to X$ a continuous map. Show that if $h$ can be extended to a continuous map $H:B^2\to X$, then $h_*$ is a trivial homomorphism. I simply don't know how to use the fact that h can be ...
1
vote
1answer
160 views

If $h:S^1\to X$ is homotopic to a point, then h can be extended to $H:D^2\to X$.

Let $h:S^1\to X$ be a continuous map. Show if $h$ is homotopic to a point, then $h$ can be extended to a continuous map $H:D^2\to X$. Intuitively is a little bit obvious, let $F:S^1\times I\to X$ a ...
1
vote
2answers
193 views

Is there an homeomorphism between $D^2$ and $S^1\times I$

I would to like to know if there is an homeomorphism between the unit disk $D^2$ and $S^1\times I$, where $S^1$ is the unit circle. If I prove this homeomorphism I will be able to solve a question ...
3
votes
1answer
89 views

Definition of singular homology

Let $R = {\bf Z}$ Let $\partial_i : C_i \rightarrow C_{i-1}$ be a boundary map where $C_{-1} = \{ 0 \}$, $C_i$ is the set of all maps $f$ and $f: \Delta_i \rightarrow M$. Let $Z_n = $Ker of ...
8
votes
1answer
160 views

Morse homology of $P^2$

I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is ...
4
votes
2answers
207 views

The empty set in homotopy theoretic terms (as a simplicial set/top. space)

I am currently confused about the empty set in terms of its path components and how this fits into the Quillen adjunction between topological spaces and simplicial sets. Probably, one of my ...
10
votes
4answers
2k views

Examples of fundamental groups

I'm starting to study fundamental groups and I didn't find in the books of Algebraic Topology many examples of them. Can you list the examples you know and the demonstrations? I think it would be ...
8
votes
1answer
264 views

Who first discovered that the torus supports a flat structure?

Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?
2
votes
2answers
717 views

Hatcher Problem 2.1.16 (b)

I am trying to do the stated problem in Hatcher: Show $H_1(X,A) = 0$ iff $H_1(A) \to H_1(X)$ is surjective and each path component of $X$ contains at most one path component of $A$. Now ...
3
votes
1answer
336 views

Uncountable limit point of uncountable Set (Munkres Topology)

Let $X$ have a countable basis and $A\subset X$ is uncountable. Would you help me to prove that uncountably many points of A are limit points of A.
2
votes
2answers
393 views

Degree of map from $S^n$ to $S^n$

This is an excercise in the book Algebraic Topology - Greenberg and Harper Excercise : Let $f$ and $g$ be a map from $S^n$ to $S^n$ such that $f(x)\neq g(x)$ for all $x$ Then $f $ is homotopic to ...
4
votes
2answers
70 views

Homotopy group realization

I am looking for information related to the following question: For which $n \in\mathbb{N}$ can every group $G$ be realized as $\pi_n(X)$ for some space $X$? I have seen in Hatcher that $n=1$ is ...
1
vote
3answers
1k views

Abelian Covering Space of Surface of genus $g$

I am doing Problem 1.3.19 of Hatcher and I come to this part of the problem: For $n = 3$ and $g\geq 3$, describe a normal covering space $\tilde{X}$ of $X=M_g$, the surface of genus $g$ explicitly ...
3
votes
0answers
139 views

Killing successive homotopy groups via fibrations

Let $X$ be some sort of sufficiently nice space, e.g. a (connected) cell complex. Then $X$ has a universal cover $\tilde{X}$. This is simply connected by definition and it is easy to show that ...
6
votes
0answers
118 views

Unicoherence of non-euclidean spaces

My question concerns the notion of unicoherence, which is a property that a topological space may or may not have. The definition (from Wikipedia) is: "A topological space $X$ is said to be ...
3
votes
0answers
83 views

Showing $\Omega^n(X,x_0)\approx M(S^n,*; X,x_0)$

I am trying to prove $$\Omega^n(X,x_0)\approx M(S^n,*; X,x_0)$$ where $\Omega^n(X,x_0)$ is the $n$-loop group and $M(S^n,*; X,x_0)$ is the set of pointed continuous maps from $(S^n,*)$ to $(X,x_0)$ ...
3
votes
0answers
100 views

Wilson lines, boundary condisions, surface defects of TQFTs

I have been studying (extended) topological quantum field theories (in short TQFTs) from the mathematical point of view and I have no background of the physics point of view. Sometimes I encountered ...
8
votes
1answer
183 views

Almost complex structure on $\mathbb CP^2 \mathbin\# \mathbb CP^2$

Why doesn't $\mathbb CP^2 \mathbin\# \mathbb CP^2$ admit an almost complex structure?
1
vote
1answer
789 views

Cell Structure on Sphere with Two points identified

My question is related to this one here but is different in that I am wondering about the CW structure on such a space. I am trying to put a CW structure on $S^2/S^0$ and I think that we have $1$ 0 ...
0
votes
1answer
98 views

Eigenvalues for Hessian evaluated at nondegenerate critical points

Let $f \colon M \to \mathbb{R}$ be a smooth function on a manifold $M$. If $f$ is Morse then all the critical points of $f$ are non-degenerate; that is, if $p$ is a critical point of $f$, then $\det ...
2
votes
1answer
67 views

What happens to homology without cycles?

Let X be a topological space and $C_n(X)$ be the singular chain complex. The homology is defined to be $H_n(X)$ = $ ker \partial_n / im \partial_{n+1}$. What happens if we take $ K_n(X) = C_n(X) / im ...
0
votes
1answer
93 views

Does Euler Characteristic > 2 imply not connected

For any given topological space $X$ does $\chi(X)>2 \Rightarrow $ more than 1 connected component? If not, when does it. And if true, can someone point me towards a proof. Thanks
7
votes
2answers
538 views

Is the center of the fundamental group of the double torus trivial?

I know that the fundamental group of the double torus is $\pi_1(M)=\langle a,b,c,d;a^{-1}b^{-1}abc^{-1}d^{-1}cd\rangle$. How can I calculate its center subgroup $C$? Is $C$ trivial? Let $p$ be the ...
0
votes
1answer
348 views

$S-\{p\}$ admits a bouquet of circles as deformation retract.

Let $S$ be a closed compact surface, $p\in S$ and $X=S-\{p\}$. Show that X admits a bouquet of circles as deformation retract. How many circles? I'm starting to study algebraic topology and I can't ...
3
votes
1answer
278 views

About covering maps!

Can someone post a proof of the statement that if $X$ is compact then the covering map $q:E\rightarrow X$ is finitely sheeted given that $E$ is compact as well.
2
votes
0answers
65 views

Is the boundary map induced by a good pair $(D^2, S^1)$ equal to multiplication by $2$?

Consider the disk $D^2$ and its boundary $ \partial D^2 = S^1$. Then $(D^2, S^1 )$ is a good pair. This means there is a boundary map induced $$ \tilde{H}_2(D^2 / S^1 ) \xrightarrow{\ \partial \ } ...
1
vote
0answers
77 views

Special case for Alexander's Duality

Let $C_1$ and $C_2$ be homeomorphic closed sets in $S^n$.Prove that: $$H^{r}_{dR}(S^n\setminus C_1 )\cong H^{r}_{dR}(S^n\setminus C_2 )$$ Here's what I tried: Let $ p \in S^n\setminus C_1 $ , $ ...
2
votes
1answer
198 views

Deligne tensor product

I would like to know something about a tensor product of categories and it seems Deligne tensor product is what I am looking for. But the paper "Categories tannakiennes" by Deligne is not available ...
2
votes
1answer
116 views

How do I prove this function is not continuous?

Let $\alpha$ a path from $[0,1]$ to a topological space X. Let $\alpha(0)=\alpha(1)=c$, where $c\in X$. The standard function to prove that $\alpha\cdot\bar \alpha$ is homotopic to the constant map ...
2
votes
1answer
57 views

Simple example that $H_{1}(\mathcal{X})=0$ but $\mathcal{X}$ is still not orientable.

Can anyone show me a simple example of a manifold such that $H_{1}(\mathcal{X})=0$ but $\mathcal{X}$ is not orientable? We know the contention hold for $\pi_{1}$, I am not sure if it holds for ...
14
votes
1answer
3k views

Difference between simplicial and singular homology?

I am having some difficulties understanding the difference between simplicial and singular homology. I am aware of the fact that they are isomorphic, i.e. the homology groups are in fact the same (and ...