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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0
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1answer
47 views

Triangle inside a simply connected open subset of the complex plane

Let $U$ be a connected open subset of the complex plane. Suppose $U$ is simply connected, i.e. its fundamental group is trivial. Let $T$ be a triangle whose boundary is contained in $U$. It is ...
2
votes
1answer
38 views

Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
3
votes
0answers
54 views

Jordan curve theorem for a polygon

Let P be a Jordan polygon on the complex plane. I would like to prove Jordan curve theorem for P using an elementary method. It seems that we can prove it using the winding number with respect to P. ...
4
votes
2answers
65 views

Why is $H_5(K(\Bbb Z_n,4))$ finite?

I want to see that the cohomology $H^i(K(\Bbb Z_n,4); \Bbb Z)$ starts with $\Bbb Z_n$ in degree 5. How do we know that $\operatorname{hom}(H_5(K(\Bbb Z_n,4);\Bbb Z), \Bbb Z)$ is zero? I.e. why is $H_5(...
1
vote
1answer
41 views

Computing the homology of the torus with coefficients in $\Bbb F_p$, using two methods

I have some trouble to compute the homology of the torus with coefficients in $\Bbb F_p$ for $p$ a prime number. In particular I have a problem for $H_1$ : 1) The first way to compute it is to use ...
6
votes
1answer
141 views

Does fundamental group distinguish between any two non homeomorphic topological space?

I am new to fundamental group. I was reading Munkres and found that need of fundamental group was to distinguish between non-homeomorphic topological spaces. So my question is, does fundamental ...
1
vote
0answers
25 views

Covering map, end points of two paths equal criterion

I was reading a proof of the following lemma: $\textbf{Lemma}$: Let $p\colon X \to B$ be a covering map and let $\gamma, \gamma'$ be two paths in $X$ beginning at $x_0$. Let $u=p\gamma$ and $u'=p\...
2
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0answers
33 views

Duality between Thom space and a manifold embedded into a sphere

In a document https://www.math.purdue.edu/~gottlieb/Bibliography/53.pdf (s. 19) it is mentioned that there is a map $S^n \to M^+ \wedge \mathrm{Th}\left(\nu \left(M, S^n\right)\right)$, which gives a ...
3
votes
1answer
43 views

Normal bundle of an embedding of a parallelizable manifold into a parallelizable manifold

This question is motivated by an observation of Milnor. Theorem: Let $M^m$, $N^n$ be parallelizable smooth manifolds, and $i:M\to N$ an embedding. If $n>2m$, the normal bundle is trivial. The ...
3
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2answers
55 views

Is there a retraction of $S^1\vee S^2$ onto $S^1$?

I am currently working through a problem that requires me to prove that $\pi_1(S^1\vee S^2)\simeq\mathbb{Z}$ directly by showing that the homomorphism induced by the inclusion of $S^1$ on $\pi_1$ is a ...
0
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0answers
13 views

Existence criterion of $Spin_{\mathbb{C}}$ structure

In deriving the existence criterion of $Spin_{\mathbb{C}}$ structure Ralph Cohen in his notes on the topology of fiber bundle (pp.169) uses the following diagram $\require{AMScd}$ \begin{CD} BSpin_{\...
3
votes
2answers
138 views

The homology of wedge sum

This is an exercise of Bredon (pg. 190) which I tried to do but got stuck at one part. He asks the following: Let $X$ be a Hausdorff space and let $x_0 \in X$ be a point having a closed ...
0
votes
1answer
19 views

Explicit formulation of hermitian form and corresponding alternating form

I can't understand the following basic thing: Let $X$ be a complex manifold of dimension $n$ and $T_xX$ its tangent space in $x\in X$. Then on $T_xX$ we can define a hermitian form $\sum_{i=1}^ndz_id\...
1
vote
2answers
61 views

injective open map between two euclidean spaces [closed]

Does there exists an injective function from $\mathbb R^2$ to $\mathbb R$ such that image of every open set is open ? Where $R^2$ and $R$ are usual euclidean spaces. Please help. Thank you.
1
vote
1answer
23 views

Properly discontinuous group actions - Hausdorffness

I was told to prove the following: If an action is free and satisfies that each point has a neighborhood $U$ satisfying $U \cap gU=\emptyset$ except for finitely many $g\in G$, and moreover the space ...
1
vote
1answer
62 views

Hatcher's exercise 1.2.22 on the Wirtinger presentation

Here exercise 1.2.22 is recalled, but the asker seems to know how to solve it assuming the "geometry is valid". I however, do not know to use the van Kampen theorem in order to find the relations $...
3
votes
3answers
105 views

Why bother showing $S^{1}$ covers itself?

I've just been introduced to covering spaces, and one of the examples I've been shown is that $p: S^{1} \to S^{1}$, $p(z)=z^{n}$ is a covering map for every $n$. My question is: why would you care? ...
2
votes
0answers
51 views

Topology of CW-complex and attaching map

I think I must have a fundamental misconception in place right now in my mind. When defining a CW-complex, we use inductively continuous maps from $f_{\partial \sigma} :S^n \to K^{(n)}$. We then ...
0
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1answer
45 views

I want to prove that $B=S^2 \cap \{(x_1, x_2, x_3) : x_i \geq 0 \}$ is homeomorphic to the disk $B^2= \{ x \in \mathbb{R^2} : ||x|| \leq 1 \}$

I claim that the map $f:B \to B^2$ defined by $f(x_1, x_2, x_3)=(x_1, x_2, 0)$ is a continious, bijection with $g: B^2 \to B$, $g(x_1, x_2)=(x_1, x_2, \sqrt{1-(x_1^2+x_2^2)})$ it's inverse. So i ...
3
votes
2answers
51 views

Does topological degree generalize to maps that aren't between closed connected orientable manifolds?

From what I gather, the degree of a map originally arose in the context of studying maps $f:S^n\rightarrow S^n$. Since $H_n(S^n)\cong \mathbb{Z}$, the induced map $f_\star$ has the form $x\mapsto kx$ ...
-2
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0answers
19 views

Pictures and illustrations of attaching a 2-cell to a wedge of circles?

I am having a lot of trouble visualizing how to attach a disk "along a word". Where can I find some pictures and illustrations of this procedure?
2
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1answer
515 views

Lifting correspondence in Algebraic topology

Recently I have been studying algebraic topology and came across the notion of lifting correspondence. Here, lifting correspondence definition is the same that Munkres uses in his book. However, I ...
0
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0answers
65 views

Covering map of classifying space

We know that for any $m\in\mathbb{N}$ the map $p_m:S^1\to S^1$ is an $m$-sheeted covering of $S^1$. Suppose that $BG$ is the classifying space of an arbitrary group $G$. Does there exist such a map $...
2
votes
1answer
49 views

how to represent a relative cohomology class

Let $X$ be a topological space, and $A \subseteq X$ a subspace. How to think about an element $u \in H^n(X, A)$? Is the following correct? $u$ can be represented by a function $U$ taking an $n$-cell $\...
6
votes
0answers
131 views

Products of homology groups

In the topology course I attend we work with the following, rather unusual definition of homology groups: For topological spaces $X,Y$ define the presheaf $\mathcal{F}_Y$ on $X$ as follows: Let $\...
2
votes
1answer
26 views

What motivated trying to express the signature of a manifold as a linear combination of pontrjagin numbers?

I have tried reading a proof of the signature theorem but it is way beyond me, is there a way to motivate, in english, why anyone even started searching for such a formula? Why would one assume that ...
1
vote
0answers
58 views

Looking for a right book for Algebraic Topology - is Dieck's textbook a good choice?

I self-study Algebraic Topology. I use Hatcher's textbook Algebraic Topology and soon I'm going to end reading Chapter 3. I know that there is one more chapter about homotopy theory but I'd like to ...
1
vote
0answers
25 views

Covering space, second-countable

Let $p:Y\to X$ be a covering space and $p^{-1}(x)$ countable for every $x\in X$ Show, that if $X$ is second-countable, so is $Y$ Hello, I am a little bit stuck with this question, because I do ...
0
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0answers
39 views

What does the operator “B” applied to an Eilenberg-MacLane space mean?

Let $K(m, \mathbb{Z})$ be the Eilenberg-MacLane space. I've read that $BK(m, \mathbb{Z}) = K(m+1, \mathbb{Z})$, but I'm trying to understand this. I'm familiar with the bar construction $BG$ for a ...
-1
votes
1answer
78 views

Does any technical definition of embedding accept a “non-injective” function as opposed to only “injective”?

Embedding is defined to be a one-to-one structure preserving mapping. My question is if the one-to-one condition is really critical. Like if linear mappings from high-dimensional space to low-...
3
votes
1answer
61 views

Automorphisms action on $\mathbb C$

If we know that the $$Aut \ \mathbb C=\{z\mapsto az+b:a\in \mathbb C^*,b\in \mathbb C\}$$ What are the subgroups of $Aut \ \mathbb C$ that act withiut fixed points and properly discontinuously on $\...
1
vote
0answers
32 views

Prove of homotopy equivalence using differential equation

I have to prove that $R^3 \setminus L_1,...,L_n $, where $L_1,..,L_n$ are non intersecting lines, is homotopy equivalent to a wedge sum of $n$ circles. So once I've managed to show that it is possible ...
2
votes
1answer
69 views

There is no antipode-preserving map $f:S^{n+1}\rightarrow S^n$

I am trying to prove that there is no antipode-preserving map $f:S^{n+1}\rightarrow S^n$. Here is what I have: Suppose there is an antipode-preserving map $f:S^{n+1}\rightarrow S^n$. If we restrict ...
3
votes
1answer
69 views

First cohomology group on a Riemann surface with all wedge products equal to zero

Sorry for the strong edit, but I realized my question had a easier formulation: Can there be a Riemann surface $X$ with the property $\sigma\wedge \tau=0$ for every $\sigma,\tau\in H^1(X,\mathbb{C})$?...
3
votes
1answer
34 views

Ratio of holomorphic forms on a Riemann surface

Let $R$ be a Riemann surface of genus $g>1$ and $\omega$, $\sigma$ two holomorphic 1-forms on $R$. This means that locally we can write $\omega=fdz$ and $\sigma=gdz$ with $f$ and $g$ holomorphic. ...
0
votes
0answers
7 views

reference for simplicial complexes and triangulation

I am trying to understand simplicial complexes and triangulations, am following Basic Algebraic topology by Prof. Anant R Shastri and not very comfortable with the way it has been explained. Suggest ...
0
votes
1answer
16 views

Edge identification implies vertex identification (but not vice versa)?

This may be an obvious question, but just to ask it to be sure. Q1) When we identify edges, is it automatically assumed that we identify the vertices as well? E.g: We have a 2-simplex with vertices $...
0
votes
1answer
13 views

n-cubes vs simplices when defining chains

When defining chains, the standard definition is formal linear combinations of n-simplices. However in Calculus on Manifolds by Spivak, he defines chains as formal linear combinations of n-cubes. I ...
6
votes
1answer
626 views

Local homology group: a homeomorphism takes the boundary to the boundary

Let $X \subset \mathbb{R}^{n}$ be the subspace $\{ x_{1},...,x_{n} \mid x_{n}\geq 0 \}$, and let $Y$ is the subspace with $x_{n}=0$. Let $x\in Y$, calculate the local homology of $X$ at $x$. ...
-2
votes
1answer
38 views
8
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2answers
1k views

What does “splitting naturally” mean in the Universal Coefficients Theorem

The Universal Coefficients Theorem states that $0\rightarrow H_n(X)\otimes G\rightarrow H_n(X;G)\rightarrow\operatorname{Tor}(H_{n-1}(X),G)\rightarrow 0$ splits, but not naturally. In all the ...
0
votes
1answer
24 views

Map of degree two from $S^2$ to the torus $T^2$. [duplicate]

Prove that there is no map of degree two from $S^2$ to the torus $T^2$. I'm struggling with this problem. I've tried lifting the map to the covering space but I'm not sure what to do from there. I ...
3
votes
3answers
108 views

category-theory, right group action

Let $G$ be a group. We observe the category $(Set)_G$ of right group actions. a) Let $X\times G\to X$ and $Y\times G\to Y$ be two transitive right group actions with $x\in X$ and $y\in Y$. Find a ...
0
votes
1answer
41 views

Existence of a universal cover of a manifold.

Suppose $M$ is a manifold, topological or smooth etc. As a topological space $M$ is required to be primarily locally homeomorphic to $\Bbb R^n$, with some added things that don't come along with this, ...
0
votes
1answer
35 views

Proving homotopy equivalence of a torus with points removed

Suppose I'd like to show that a Torus with $n$ points removed is homotopy equivalent to a wedge sum of $n+1$ circles. I depict it in a usual way - as a rectangle with $n$ points removed inside. Now it ...
0
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0answers
37 views

Hatcher Covering Spaces Ex. 11 & 31 and Surjectivity of the Covering Map

I am confused by the statements of a couple of the exercises in Section 1.3 of Hatcher. I think they need additional hypotheses that are not reflected in Hatcher's errata. Exercise 11: Construct ...
2
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0answers
34 views

geometric realization of a simplicial complex

Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard ...
5
votes
2answers
274 views

Excision via simplicial sets

Let $X$ be a topological space, covered by a collection of open sets $\{U_\alpha\}$ (or more generally a cover by subspaces whose interiors cover $X$). Consider the singular simplicial set $S(X)_\...
4
votes
1answer
61 views

Natural isomorphism $\tilde H_i(X) \xrightarrow{\cong} \tilde H_{i+1}(\Sigma X)$ where $\Sigma X$ is the suspension of $X$.

Define $\Sigma X$ to be the quotient space of $[-1,1]\times X$ obtained by identifying ${0}\times X$ and ${1}\times X$ to two points respectively. For any homology theory (satisfying Eilenberg-...
2
votes
0answers
45 views

singular homology of a simplicial complex [duplicate]

On Page 108 of the book Algebraic Topology, Allen Hatcher, the singular homology of a topological space $X$ is defined to be the homology of the chain complex by setting the $n$-chains $C_n(X)$ as the ...