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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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 ...
4
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0answers
106 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 ...
3
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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 $\...
2
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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 ...
2
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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 $\...
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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 ...
3
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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. ...
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33 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 ...
3
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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 $\sigma,\tau\in H^1(X,\mathbb{C})$?...
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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 ...
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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-...
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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? ...
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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 ...
0
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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, ...
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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 ...
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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
35 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 ...
2
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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 ...
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1answer
44 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_{...
4
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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-...
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1answer
27 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
24 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 ...
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18 views

A question about orienting a simplicial complex.

A complex may be oriented by assigning, in a completely arbitrary fashion, an orientation to each of its simplices I've always been confused about this point. Say we take a tetrahedron (a $3$-...
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0answers
39 views

What are some examples of cohomology theories without a corresponding classifying space?

The general nonsense of cohomology theories is that each one "should" be presented by a classifying space, so that maps into this space give the cohomology (before passing to connected components). ...
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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 $...
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1answer
28 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 ...
9
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2answers
146 views

Is there a subject in mathematics like topological Algebra?

I would consider myself an algebraic topologist and there is a lot of influence from algebra into topology and without this input from the algebraic site I would say that a lot of topological theorems ...
2
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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 ...
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0answers
33 views

Why is topological K-theory equivalent to nonabelian cohomology with respect to the stable unitary group?

I was reading on the $n$Lab page for topological K-theory that taking cohomology of a smooth space with respect to the smooth $\infty$-stack $\mathbf{Vect}$ is equivalent to taking its cohomology with ...
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1answer
23 views

Isomorphic homotopy groups of universal cover?

In Ralph Cohen's notes on the topology of fiber bundle he says (1) on pp.167, $BSO(n) \to BO(n)$ is a universal cover thus $\pi_i(BSO(n)) \to \pi_i(BO(n))$ is an isomorphism for $i \geq 2$ (2) on pp....
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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 ...
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 ...
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1answer
41 views

tom Diecks's proof of $H_1(X)\cong \pi_1(X,x_0)^{ab}$

My question is about tom Dieck's proof of Theorem 9.2.1 on page 227, which states that if $X$ is path connected, then the induced map $$h:\pi_1(X,x_0)^{ab}\to H_1(X)$$ is an isomorphism. Specifically,...
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2answers
48 views

How to construct a homotopy equivalence between a mobius band and a circle?

A mobius band is homotopic equivalent to a circle because the mobius band can deformation retract onto a circle. I am wondering how could we understand this fact from the definition of being ...
1
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1answer
19 views

homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
1
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1answer
59 views

A proposition about free product

My question is about a proposition that I found on Munkres book: Topology (page 419). My definitions are based on this book. I give you the definition of free product that he uses. $\textbf {...
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0answers
44 views

$H_{k+1}(X \cup_f D^{k+1},X) = ?$

I am stuck with the calculation of the following homology group: $H_{k+1}(X \cup_f D^{k+1},X) = ?$ where $X$ is a simply-connected CW complex and $f: S^k \to X$ is a continuous map (attaching map of ...
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1answer
45 views

CW complex= cell complex?; compact metrizable spaces

I have a question about finite cell complexes and compact metrizable spaces. In a paper I read the statement: Let $X$ be a compact metrizable space. Then $X$ is a countable inverse limit $\varprojlim\...
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1answer
24 views

Seifert matrix, linking numbers, generators

I have been asked to compute the seifert form of a knot, the twist knot. I know how to compute the seifert surface, and then the seifert matrix seems to be defined accordingly (according to all the ...
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1answer
57 views

Which group homomorphisms induce the action of the fundamental group on the fiber?

Given topological spaces $X$ and $Y$ and a covering map $p: X \rightarrow Y$, we know that the group $\pi_1(Y,y_0)$, where $y_0\in Y$, acts on the fiber $F=p^{-1}(y_0)$. Also, we know that the set ...
5
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2answers
113 views

$[X,Y]$ is finite where $X$ is finite connected CW-complex, and $Y$ has finite homotopy groups

I have read this question in Allen Hatcher's book Algebraic Topology, (exercise 20, page 359): Show that $[X,Y]$ is finite if $X$ is a finite connected CW complex and $ \pi_i(Y) $ is finite for $ ...
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1answer
45 views

Homology group of an open set on $S^1$

Let $U$ be an open set which is constructed as intersection of $S^1$ and open ball in $\mathbb{R}^2$. And $x$ is just a point contained in $U$. My opinion: By long exact sequence, $H_n(U, U-x)$ is ...
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1answer
54 views

A relation between the index of a fundamental group and the covering map

My professor just enunciated this statement: $|\pi_1(X,x_0):\pi_1(\tilde X,\tilde x_0)|=\#P^{-1}(x_0)$ where $P$ is the covering $P:\tilde X\rightarrow X$, such that $P(\tilde x_0)=x_0$. I've ...
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0answers
27 views

Differences among the Group Cohomology with coefficients over a commutative ring and coefficients over an arbitrary $G$-module

Assume that $G$ is a finite group and $k$ is an arbitrary commutative ring. From the general theory we know that the group cohomology $H^{*}(G ; k)$ becomes a graded ring (with the cup product). ...
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1answer
33 views

Deck transformation, covering space

Let be $X=(S^1-1)\cup (S^1+1)\subset\mathbb{C}$ (shaped like the "eight") and $u(t)=e^{2\pi it}-1, v(t)=1-e^{2\pi it}$. Give every deck transformation $\Delta(p)$ and $p_{\ast}(\pi_1(Y, y_0))\...
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0answers
53 views

Is the cohomology ring of a CW complex computable?

There is a well-developed technology for computing the cohomology groups of a CW complex, cellular cohomology. It reduces the problem of computing cohomology to the two simpler problems of (1) ...
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1answer
28 views

Is there a common way, to find all deck transformations $\Delta(p)$

Suppose $p:E\longrightarrow B$ is a covering projection. I have a general question, on how to find the group of all deck transformations $\Delta(p)$. Is there a common way to do this, or what could be ...
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0answers
15 views

Covering spaces of surface sphere glued to the mobius strip at one point on its boundary.

I have determined the universal covering space, but I am having trouble finding two-sheeted and three sheeted covering spaces. Any help would be greatly appreciated on how to approach this!
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34 views

definition of mod p k-theory

The (topological) complex K-theory is a cohomology theory, i.e can be represented by a spectrum $K$ whose $2n$-th space is $BU \times \mathbb{Z}$ and whose $2n+1$-th space is its loop space (and is ...
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41 views

Projective space, fundamental group

Let $g:\mathbb{R}P^2\to\mathbb{R}P^2$ continuous. Let $q:S^2\to\mathbb{R}P^2$ die quotient map. Show, that: If $g_{\ast}(\pi_1(\mathbb{R}P^2, x))$ is not trivial (therefore contains more ...