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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28 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 $...
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0answers
28 views

Requirement for Algebraic topology.

I am graduate student. I wil learn algebraic topology next semester which is my second semester. Book learned is written by Hatcher. In vacation , I want to study requirements for algebraic topology....
3
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1answer
26 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(...
2
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0answers
19 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 ...
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0answers
8 views

Local isometry between non-positively curved cube complexes

Let $X$, $Y$ be non-positively curved cube complexes and suppose there is a local isometry $X \to Y$. If $Y$ is special, then so is $X$. This is Exercise 4.32 in M. Sageev's notes "CAT(0) Cube ...
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1answer
37 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
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2answers
43 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
18 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
512 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 ...
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0answers
51 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 $...
3
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1answer
47 views

Under what type of transformations are characteristic classes and characteristic numbers of a manfold invariant?

When I say "characteristic class of a manifold" I mean the characteristic class of the tangent bundle. I assume that all Chern/Pontrjagin classes/numbers are invariant under diffeomorphism, if they ...
2
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1answer
45 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 $\...
5
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0answers
126 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
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1answer
22 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 ...
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0answers
45 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 ...
4
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1answer
68 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
1
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0answers
23 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 ...
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0answers
35 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 ...
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1answer
76 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
57 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 $\...
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0answers
30 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
58 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
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1answer
62 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
29 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. ...
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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
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1answer
15 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
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1answer
12 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
624 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 $x$. ...
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1answer
65 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? ...
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1answer
38 views
8
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2answers
991 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
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1answer
21 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 ...
0
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1answer
38 views

Construction of the classifying space of a group $G$.

Assume that $G$ is a given (topological) group, then we know that there is an induced $G$-bundle of the form $ G \hookrightarrow EG \rightarrow BG$. I do know some properties and I do understand into ...
3
votes
3answers
83 views
+50

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
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1answer
33 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
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1answer
33 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 ...
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0answers
32 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
31 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
271 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)_\...
3
votes
1answer
48 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
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0answers
41 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 ...
4
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1answer
748 views

When is Quotient Map a Covering Map

Group $G$ acts on topological space $X$. Also, $x,x'\in X$ not in the same orbit of $G$ have open $U$, $U'$ such that $g(U)\cap U'=\varnothing$ for all $g\in G$. I have shown that $X/G$ is Hausdorff....
3
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1answer
53 views

Difference between product projections and split epis in $\mathbf{Top}$

I don't understand the excerpt below. The trivial bundles were defined as split epimorphisms, and since the ground category was additive with kernels (in fact abelian), they are the same as ...
1
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1answer
42 views

A deformation retract that is not a strong deformation retract

In Lee's Introduction to Topological Manifolds, problem 7-12 asks to show that $\{(1,0)\}$ is a deformation retract, but not a strong deformation retract of the subspace of the plane $$ X = \bigcup_{...
1
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1answer
18 views

Is a $\Omega$-Spectrum a connective one?

I can't find this result anywhere, but it seems pretty straightforward. I want to avoid silly mistakes, but I can't see any fault. I'd love to receive some feedback Let $X$ be a $\Omega$-spectrum (of ...
3
votes
2answers
153 views

projective space, constant function, homotopy

Show, that every continious function $f:\mathbb{R}P^2\to S^1$ is a homotopy to a constant function. (Hint: Has $f$ a lift $\tilde{f}:\mathbb{R}P^2\to\mathbb{R}$) Hello, I want to solve this ...
1
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1answer
23 views

Show that Borsuk lemma need not hold if $f$ is not injective

The following lemma is called Borsuk lemma which can be found in Munkres' topology (Lemma 62.2). (Borsuk lemma) Let $a$ and $b$ be points of $S^2$. Let $A$ be a compact space, and let $f:A\to S^2\...
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1answer
48 views

Simplicial Complexes/homology Reference request

I am taking an introductory Algebraic Topology course, and we have just finished talking about the fundamental group/ covering spaces in Munkres' Topology. However, his treatment of simplicial ...
3
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2answers
36 views

If $C$ is a simple closed curve lying in a simply connected open set $U$, then its interior also lies in $U$

Let $U$ be a simply connected open set in $\mathbb{R}^2$. If $C$ is a simple closed curve (a space homeomorphic to unit circle $S^1$) lying in $U$, then each bounded component of $\mathbb{R}^2\...
1
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1answer
20 views

A question about inherited orientation of simplices

I have the 2-manifold $[v_2,v_0,v_3]$. My books says that on removing $v_3$, the orientation of the face that we end up with is $[v_0,v_2]$. I don't understand how this happens. What is the ...