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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Homology of $X/\{x\sim f(x)\}$ where $f\colon X\to X$

Let $X$ be a space and $f\colon X\to X$ a continuous map. What tools do we have to compute the homology $H_n(X/\sim)$ where $\sim$ is defined by $x\sim f(x)$? Relative homology was my first though, ...
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31 views

Two definitions of the first Chern class

there are two definitions of the first Chern class that I don't know how to relate - hints and references are both welcome. So, first approach: say that I have a complex vector bundle $E\to M$; I can ...
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1answer
20 views

Excercise 1(b) on Zero-dimensional Homology in Munkres

If $\phi:C_0(K)\to \mathbb{Z}$ is an epimorphism such that $\phi\circ \partial_1=0$ then show that $$H_0(K)\cong \frac{ker\phi}{im\partial_1}\oplus\mathbb{Z}.$$ My working is since $C_0(K)$ is ...
6
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1answer
453 views

Homology groups of orientable surfaces.

Edit: I have a proof here but when I spoke last with my professor, she told me something was close, but not quite. Can someone help me patch this proof? I've been trying to get this down for quite a ...
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2answers
36 views

Intuition of product spaces

So I have a product space of the form: $X=X_1 \times \ldots \times X_n$ and I take two elements of it, say $x=\{x_1,\ldots,x_n\}$ and $x'=\{x_1',\ldots,x_n'\}$. Now suppose I take the following ...
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23 views

Does the induced actions of a continuous map on the covering space determine it?

If I have continuous maps $f,g : X \rightarrow Y$ between topological spaces which induce the same actions of the fundamental group of X on the covering space of Y, are they necessarily homotopic?
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2answers
52 views

Show that $\Bbb R^{2n+1}$ is not a division algebra over $\Bbb R$ for $n>0$

This is an exercise from Hatcher's Algebraic Topology (exercise 2.B.8). Here is the problem statement: Show that, for $n>0$, $\Bbb R^{2n+1}$ is not a division algebra over $\Bbb R$ by showing ...
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31 views

Reason why “Cut and Paste” of Fundamental Polygon is allowed

Background: I am studying this question by Hatcher: What familiar space is the quotient $\Delta$-complex of a 2 simplex $[v_0,v_1,v_2]$ obtained by identifying the edges $[v_0,v_1]$ and $[v_1,v_2]$, ...
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1answer
30 views

Isomorphism on top cohomology implies isomorphism on homology

Let $F$ be a finite field (for example I could take $\mathbb{Z}_2$) and $f:X\longrightarrow Y$ a continuous map between compact, orientable and connected manifolds of dimension $n$. Suppose I have an ...
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1answer
30 views

Constructing a map $H^{k}(M,\mathbb{Z})\to H^{k}(M,\mathbb{C})$

I read that on a compact oriented manifold, there is a map $$H^{k}(M,\mathbb{Z})\to H^{k}(M,\mathbb{C}).$$ I want to be sure that I have the right map in mind. We don't have an inclusion, since ...
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25 views

Homology and triangulation of open surfaces

For example I have an open disk, or an open annulus. How do I triangulate open surfaces to find their (simplicial) Homology? Well, I know that open disk and closed disk are both homotopic to a ...
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1answer
28 views

Some basic question on pasting map from a square to a Klein bottle and homology

Consider a square $S$ which edges identified as follows Let $K$ be a Klein bottle and $p:S\to K$ be pasting map. Let $X$ be the image of the interior of $S$ under $p$ and let $Y$ be the image of a ...
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0answers
34 views

Relative Homology (Question about Example 2 in Munkres)

I have no problems for $p=0$ case and for $p\geq 2$ it is quite obvious since $C_p(K,v)=C_p(K),\forall p\geq 2$. Now the tricky is for $p=1$. Since the elements of the kernel now not anly map to $0$ ...
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1answer
33 views

Bott&Tu Definition: "Types of Forms:

In Bott&Tu's well-known book "Differential forms in Algebraic topology", they note -(p34): every form on $\mathbb{R}^n \times \mathbb{R}$ can be decomposed uniquely as a linear combination of two ...
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2answers
46 views

Isomorphism in fundamental group implies isomorphism is homology

Let $X$ be a connected space and $f:X\longrightarrow X$ a map. Suppose $\pi_1(X)$ is an abelian group and that $\pi_1(g):\pi_1(X)\longrightarrow\pi_1(X)$ is an isomorphism. I know we can deduce that ...
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0answers
35 views

Are complex subvarieties cycles in the sense of singular homology?

Given a $p$-codimensional complex subvariety $Z\subset M$ of a non singular complex projective variety $M$ of dimension $n$ we can define an element $$\int_\hat{Z}i^*\in ...
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1answer
90 views

Non-injective monomorphisms

I am reading Borceux, vol. 1, I found this example at page 27: we consider the category whose object are the pairs $\langle X,x\rangle$ where $X$ is a topological space and $x$ a point of $X$ (base ...
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1answer
35 views

Does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form?

Question: On a $C^\infty$ manifold, does every $C^1$ closed differential form differ from some $C^\infty$ closed form by an exact form? Motivation: This result holds for $C^1$ closed 1-forms on a ...
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1answer
24 views

Understanding proof of Universal coefficient theorem for cohomology

I am working through Cohomology chapter on Hatcher's book and I am having trouble with the proof of Universal Coefficient theorem for Cohomology. To be concrete I don't understand the last part of the ...
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0answers
12 views

Continuous maps to $S^n$ without antipodal pairs are homotopic [duplicate]

Let $S^n$ denote the unit sphere in the Euclidean space $\Bbb R^{n+1}$, $X$ a topological space, $f,g:X\to S^n$ are both continuous and there doesn't exist $x\in X$ such that $f(x)=-g(x)$, show ...
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1answer
64 views

Odd map implies odd degree with homology and cohomology

Suppose $n$ is odd. Let $f:S^n\longrightarrow S^n$ be an odd function. Then it induces a map $g:P^n\longrightarrow P^n$ such that the diagram $$ \begin{array}[c]{ccc} ...
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0answers
25 views

Homotopy sets for a pushout of spaces (Seifert-van-Kampen?)

my problem is the following: I have two bordisms $M : \Sigma_0 \to \Sigma_1$ and $M' : \Sigma_1 \to \Sigma_2$, so I can glue them along $\Sigma_1$ to get $M'\circ M$. The manifold $M'\circ M$ is the ...
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1answer
41 views

Classification of homotopy classes of lifts

Consider the following diagram: $$ \begin{array} & & & F & \to & * \\ & & \downarrow & & \downarrow\\ X & \overset f\to & E & \overset g\to ...
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2answers
79 views

de Rham cohomology on finitely smooth manifolds

In all of the places I've looked, de Rham cohomology is defined on $\mathcal{C}^\infty$ manifolds with $\mathcal{C}^\infty$ differential forms. What about de Rham cohomology on $\mathcal{C}^r$ ...
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0answers
30 views

Is Engelking and Sieklucki's “Topology: A Geometric Approach” a Good Introduction to Algebraic Topology?

I only found this book incidentally while looking at Engelking's more well-known "General Topology". I posted a link here. ...
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1answer
482 views

Definition of the algebraic intersection number of oriented closed curves.

Can anyone point me to a reference (book/paper) where I can read up on the the algebraic intersection number of closed curves on an orientable surface? In this paper by John Franks it is used to ...
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0answers
35 views

Given a CW-structure on a space, when can a subspace be realized as the n-skeleton?

So I've been working with CW-complexes in my algebraic topology classes for about a year now, and while I certainly feel that I understand them pretty well at this point, there have been some ...
2
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1answer
23 views

Free groups on simply connected space : application of Van-kampen theorem

This comes from the application of Van-Kampen theorem. Note Van-kampen theorem, states for $U$, $V$ and $U \cap V$, are open and path connected space we have \begin{align} \pi_1 (U \cup V) =\pi_1 ...
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2answers
152 views

Question about Relative Homology

I am reading through Hatcher and I came across the following statement and am having trouble making sense of it. I am not sure why elements may be written this way. Any help will be appreciated. By ...
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1answer
35 views

Coverings of a three-manifold

He guys, I have two questions regarding the following: Consider the three-manifold $\mathbf{T}^3 = S^1 \times S^1 \times S^1$ and let $S_n$ be the permutation group acting on $n$ letters. Let ...
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2answers
385 views

Can a finite group act freely (as homeomorphisms) on $\mathbb R^n$

I am asking if whether or not a finite group acts freely (as homeomorphisms) on $\mathbb R^n$. To answer in the negative, it suffices to show: for any homeomorphism $f$ such that ...
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0answers
18 views

Every vector bundle can be induced from a principal bundle? its frame bundle?

If it is a theorem could somebody tell me the name? If it is wrong could somebody give a counterexample to illustrate what the obstruction is? I am wondering this because in Clifford Taubes' book on ...
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0answers
28 views

Cohomology exact sequence induced from exact sequence of coefficient

This might be a trivial question, but I was wondering if the following was true or not. Suppose that we have an exact sequence of groups $$0\rightarrow G\rightarrow H\rightarrow K\rightarrow 0$$ Does ...
2
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1answer
141 views

Motivation for the proof of the associativity of multiplication of equivalence classes of paths

After having defined the equivalence classes of paths in a topological space in chapter two of the book A Basic Course in Algebraic Topology, William S. Massey proves the lemma The multiplication ...
4
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1answer
107 views

Group Action and Smooth Manifolds

I was wondering if it is sufficient for a compact (i.e. Hausdorff) smooth manifold $M$ to have a free group action of a finite group $G$ in order to conclude that $M/G$ is a compact smooth manifold? ...
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36 views

Questions about complexes and homology

I just learn about the simplicial and delta complexes and computing homology group. But I have a few questions: Is there any topological space which cannot be given a delta compplex structure? Is ...
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1answer
27 views

Two different kinds of actions on fibers of a universal cover

This concerns Hatcher's exercise 1.3.26. It says almost this: Given a universal cover there are two actions of $\pi_1$ on the fibers, one given by lifting loops and one given by restricting deck ...
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2answers
79 views

Expressing homotopy groups of spaces of (unpointed) maps $S^1\to M$ in terms of homotopy groups of spaces of pointed maps.

I came across the following problem while studying for a topology exam: Let $M$ be a topological space, let $\Lambda(M)=M^{S^1}$, the space of continuous maps $S^1\to M$ with the compact-open ...
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1answer
47 views

Homotopic type of $GL^+(n)$, $SL(n)$ and $SO(n)$

Question: Consider $GL^+(n) \supset SL(n) \supset SO(n)$ the groups of matrices $n \times n$ with positive determinant, determinant $1$ and orthogonal with positive determinant, respectively. Show ...
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1answer
20 views

Covering Torus by torus

This question concerns example 1.41 in Hatcher's Algebraic topology. There he constructs a covering for the genus 3 surface by the genus 11 surface by shaping it like a star with 5 arms with two holes ...
2
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2answers
48 views

Existence of a (n-1)-connected map beween CW-spaces

I have two finite CW-spaces $K$ and $L$ (K is n-dimensional and L is (n-1)-dimensional), a topological space $X$ and two maps $\phi:K\to X$ and $\psi: L\to X$, while $\phi$ is n-connected and $\psi$ ...
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1answer
31 views

Connected Linear Graph not Path-Connected

Given a set of vertices $\{x_\alpha\}$ whose cardinality exceeds $\aleph_1$, (assume the axiom of choice) connect each vertex with its successor by an edge, forming a linear graph. Choose two vertices ...
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Constructing normal covering spaces

This is an exercise from Hatcher 1.3.12. It says given a,b the generators of $\pi_1(S^1\vee S^1)$, draw a picture of the covering space that corresponds to the normal subgroup generated by ...
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156 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. It is true that if ...
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0answers
35 views

Universal Cover of a non-Hausdorff space

It is widely known that if $X$ is a path-connected Hausdorff space, and the universal cover $\widetilde{X}$ is compact, then $\pi_1(X)$ is finite. What happens if we don't assume that $X$ is ...
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2answers
64 views

Homology of the $n$-torus using the Künneth Formula

I'm trying to apply the Künneth Formula $$H_{n}(X \times Y) \simeq \displaystyle \bigoplus_{r+s=n} H_{r}(X) \otimes H_{s}(Y)$$ to compute the homology groups of the $n$-torus. For the double torus, ...
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4answers
859 views

Homology Whitehead theorem for non simply connected spaces

(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true? Suppose ...
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2answers
36 views

injective open map between two euclidean spaces

Does there exists an injective function from $R^2\ to\ $R such that image of an open set is open ? Where $R^2$ and $R$ are usual euclidean spaces. Please help.Thank you.
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Gluing Balls to get $CP^k$?

As we all know, the $2$-sphere can be obtained by gluing together two discs along their boundary. One way to generalize $S^2 \simeq CP^1$ is a $CP^k$. Does there exists a generalised construction of ...
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1answer
62 views

Normal bundle associated to $\mathbb{R}P^n\hookrightarrow \mathbb{R}P^{n+1}$ is the tautological line bundle

I'm interested in proving the fact of the title and I was following the reasoning in page 8 here (at the end). There is a step which are totally unclear to me, namely the identification of the normal ...