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

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3
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1answer
78 views

Monodromy representation of Airy equation

Let $K=\Bbb{C}(z)$ with the usual derivation and consider the Airy dierential equation $y^{(2)}-zy$=0. How to determine the monodromy representration? Airy equation is not Fuchsian diferential ...
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0answers
69 views

differential forms and orientation in Bott and Tu

I am reading Bott and Tu book on my own. If you have the text you can answer my question perhaps, if not it would be too long to explain it. I have gotten to page 29. I am confused about proposition ...
1
vote
2answers
57 views

boundary of $M \times I$ where $M$ is the Möbius band

Let $M$ be the Möbius band and $I$ be the closed interval $[0,1]$. What is the boundary of $M \times I$? Is it orientable? What can I do when I want to know the boundary of such space? Please give an ...
0
votes
2answers
52 views

How to compute Euler characteristic from polygonal presentation?

How can I compute the Euler characteristic of a compact surface from its polygonal presentation $\langle S | W_1 , \ldots , W_k \rangle$? I guess that the number of edges is the number of different ...
2
votes
0answers
26 views

A correspondence between generators of $H_n(\mathbb{R}^n,\mathbb{R}^n-\{0\})$ and eq. classes of orthonormal frames

The problem is about (topological) orientation of $\mathbb{R}^n$: Define an equivalence relation on orthonormal frames in $\mathbb{R}^n$ by declaring two frames equivalent if the matrix expressing ...
4
votes
1answer
35 views

an arc is not a retraction of the Klein bottle

I want to use homology to solve the following problem: Prove that the circle represented by the blue arc in the picture is not a retract of the Klein bottle. (See the attached picture of the Klein ...
2
votes
1answer
53 views

The term “Homotopy” was given by whom?

I want to know the names who define the term 'Homotopy' in algebraic topology in 1907. Are they Dehn and Paul Heegaard? What is the full name of Dehn? Thank you in advance.
2
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0answers
39 views

Question about proof of van Kampens theorem

I found this proof of van Kampen's therem on page 2. link I don't understand the part where text 'since this is simply connected' occurs. Why does it follow that if $H_{i,0} $ is in $U_2$ that it is ...
3
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0answers
36 views

Minimum regularity Of Stoke's theorem to hold in smooth manifold.

Stokes’ Theorem on Manifolds is often express as follows: Given a differential m-form $\omega$ whose support is the $C^{\infty}$ $m$-dimensional compact manifold ${\cal{M}}$ with boundary ...
3
votes
1answer
42 views

Local topological properties

Could we define the terms locally connected/compact/contractible/simply-connected/whatever to mean that there is a basis (for the topology on our space) of ...
3
votes
3answers
88 views

covering map $S^n \rightarrow P^n$ is not null homotopic

Here is the problem: Prove that the covering projection $S^n \rightarrow P^n$ is not null-homotopic. This problem is from Algebraic Topology by Harper and Greenberg. There is a suggestion: The lifting ...
2
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0answers
63 views

pullback is injective on picard groups?

Let $E \rightarrow X$ be a rank two holomorphic vector bundle over a complex manifold $X$. I was recently asked on exam to prove an assertion that I believe boils down to showing that the pullback map ...
2
votes
1answer
134 views

mapping torus eqivalent definition

Let $X$ be a topological space and $f:X\to X$ a homeomorphism. I need to find a continuous, properly discontinuous $\mathbb{Z}$-action on $X\times\mathbb{R}$, such that the quotient ...
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0answers
40 views

Cohomology of Hawaiian earring?

Do the infinite wedge of circles and the Hawaiian earring have the same cohomology? I am happy that they have different homologies (the first is countably generated, the second uncountably).
0
votes
1answer
30 views

CW complex with no cells in dimension $n$

Hi need some help with the following problem: if $X$ is a CW complex with no cells of dimension $n$ then $\tilde{H}^n(X,G)=0$. where $G$ is any group. thanx.
6
votes
0answers
63 views

Origins of the name “Q” and “R” for cofibrant and fibrant replacement functors.

In a model category $\mathscr M$ (in the modern sense, i.e. closed and with functorial factorizations), there is a notion of fibrant and cofibrant replacement functors. Specifically, for any object ...
1
vote
1answer
40 views

Is $S^2/\sim$ a $CW$-complex?

Consider the equivalence relation on $S^2$ define by $x\sim -x$ if $x\in S^{1}$ (we are supposed to see $S^1\hookrightarrow S^2$ as the equator) and $x\sim x$ otherwise. I have some questions ...
1
vote
1answer
64 views

Decomposition of cohomology group on $S^{n}$

If we have decomposition of cohomology group on $S^{n}$ it looks like $H^{n}(S^{n})=H^{n}(S^{n})_{+}\oplus H^{n}(S^{n})_{-}$, where $H^{n}(S^{n})_{\pm}$ cohomology of invariant or anti-invariant $n$ ...
2
votes
1answer
40 views

Good resource for learning braid theory?

I recently heard about braid theory and read the Wikipedia article on it, and it seems really beautiful. What is a good resource for learning more about it? I have a background in mathematics at the ...
9
votes
1answer
185 views

Relative de Rham Cohomology is Homotopy Invariant

Suppose $ f:N\rightarrow M$ is a smooth map between two manifolds. Relative de Rham cohomology is defined through the complex $ \Omega^{q}(f)=\Omega^{q}(M)\oplus\Omega^{q-1}(N)$ with ...
8
votes
0answers
151 views

Am I reading Bott - Tu right?

Summary: I'm finding Bott - Tu to be too brief and terse. I constantly have to look elsewhere to fill in details. This is not time-efficient. Am I missing something? If not - what other books do ...
2
votes
0answers
58 views

Reduction of structure group of real vector bundles

I'm trying to show that the structure group of real vector bundles can be reduced to the orthogonal group. This is an exercise in Differential Forms in Algebraic Topology by Bott and Tu. The book ...
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vote
0answers
22 views

existence of a closed curve and sequence that…

a) Prove that there is a curve $\alpha$ and sequence $\{x_n\}_{n \geq 0}$ in $\Bbb R^2$ such that $wind_{xn}(\alpha)=n$ for all $n \geq 0$. ($wind_{x_n}(\alpha)$ refers to the winding number of ...
0
votes
2answers
64 views

Is it true that $X\simeq S^2\vee S^2$?

Let $X$ be the quotient space of $S^2$ under the identifications $x\sim -x$ for every $x$ in the equator $S^1$. Is it true that $X\simeq S^2\vee S^2$, that is, $X$ is homeomorphic to $S^2\vee S^2$?
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0answers
10 views

Fibrewise normal, but not functionally normal space

In general topology, if a space is normal, then exist a continuous function which separates two closed sets. This is because on a normal space, you can "put" an open set and it's closure between an ...
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0answers
45 views

A Question About Notation (Homology with Local Coefficients)

I am currently reading A J Berrick’s An Approach to Algebraic K-Theory, and I am stuck at one of the propositions there because he does not define homology with local coefficients. Proposition: ...
1
vote
1answer
44 views

Determining if certian properties of a topological space pass to its image under a quotient map.

A property $P$ of topological spaces is said to "pass to quotients" if whenever $p : X \rightarrow Y$ is a quotient map and $X$ has property $P$ then $Y$ has property $P$. For the following ...
3
votes
1answer
61 views

Question about Hatcher's book CW complex

I am currently reading in Hatcher's book at page 522 about the construction of open sets in a CW complex. They start with an arbitrary set $A \subset X$ and want to construct an open neighborhood ...
5
votes
1answer
104 views

Poincare dual of unit circle

I'm trying to self-study Differential Forms in Algebraic Topology by Bott and Tu. I've come across this exercise: Show that the closed Poincare dual of the unit circle in $ R^2-\{0 \} $ is zero, ...
0
votes
1answer
48 views
1
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1answer
27 views

Proving the left lifting property for a map

I want to prove the left lifting property for the inclusion map of the sphere into the disk for any fibration $q:X \rightarrow Y$, where $q$ is a weak equivalence. I don't know how to draw a square ...
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votes
0answers
16 views

Box complex of joint of graphs

What is relation between $coind_{\mathbb Z_2}(G\ast H)$ and $coind_{\mathbb Z_2}(G)+coind_{\mathbb Z_2}(H)$, where $G$ and $H$ be two arbitrary graph and $\ast$ is the joint operation of two graphs.
3
votes
0answers
68 views

Winding number and homotopy

Given two maps $f,g : S^1 \rightarrow S^1$, I want to show that if they have the same winding number, then there is a homotopy between them. Well, we know that we can write them as $f(\exp(2 \pi i ...
1
vote
1answer
35 views

Winding number from complex analysis and differential geometry

I showed that for a differentiable function $f:S^1 \rightarrow S^1$, the winding number is given by $\frac{1}{2 \pi i } \int_{S^1} \frac{f'(z)}{f(z)} dz$. Now I want to show that the winding number ...
0
votes
0answers
40 views

Turning a torus inside-out

Smale's paradox is now famous, and great videos can be found illustrating it. Similarly, there is a video showing how to turn a torus inside-out. The solution seems to be simpler, but is the proof ...
1
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1answer
24 views

In the definition of $n$-equivalence, what is the motivation for only requiring surjectivity on the $n$th dimension.

An $n$ equivalence $f\colon X \to Y$ such that the induced map on the homotopy group $f_* \colon\pi_m(X) \to \pi_m(Y)$ is an isomorphism for $m<n$ and an epimorphism for $m=n$. What's the ...
2
votes
1answer
31 views

If Y dominates X and Y is a CW complex, then X has the homotopy type of a CW complex

Let $f\colon X \to Y$ and $g \colon Y \to X$ be maps such that $g \circ f \simeq \mathrm{id}_X$, and suppose $Y$ ix a CW complex. Then show that $X$ has the homotopy type of a CW complex This is ...
0
votes
1answer
43 views

CW complex topology

I am looking at the real projective plane and I am supposed to show that is possesses the structure $\mathbb{R}P^n = e_0 \cup\cdots\cup e_n$. Well, I know that $\mathbb{R}P^n = S^n/(x \sim -x)$ I ...
2
votes
2answers
87 views

Definition of Normal Bundle

I'm reading Differential Forms in Algebraic Topology by Bott and Tu. I reached the point where the book defines the normal bundle of a submanifold and uses the tubular neighborhood theorem. I can't ...
4
votes
1answer
106 views

Compute $\pi^n(S^1\times S^{n+1})$.

What is the space of homotopy classes of maps $S^1\times S^{n+1}\to S^n$? Is there a simple way to compute it, if we know $[S^{n+1}, S^n]\simeq\mathbb{Z}^2$ (resp. $\mathbb{Z}$ for $n=2$)?
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0answers
43 views

Numbers of $m$-simplices in the barycentric subdivision of an $n$-simplex ($m \leq n$).

Can someone indicate me how to count the numbers of $m$-simplices in the barycentric subdivision of an $n$-simplex (m $\leq n$) in an efficient way? For $m = n$, I have come up with the following ...
10
votes
1answer
239 views

What is the Atiyah-Singer index theorem about?

I was just a little bit curious about the general statement of this theorem. Honestly, I am not at all interested in fully understanding this, so it is not that I am too lazy to read plenty of books ...
2
votes
1answer
28 views

CW complex adjunction map

In topology we defined a quotient topology for glueing in the following way: Let $(X,O)$ and $(Y,O)$ be topological spaces and $f:A \subseteq X \rightarrow Y$ a continuous map, then we have that $X ...
4
votes
1answer
75 views

Homology and Fundamental group of $\mathbb{R}^4\setminus S^1$

I was attempting to help someone with this problem and realized I could not solve it myself. Let $S^1=\{(x,y,0,0):x^2+y^2=1\}$ be the unit circle in $\mathbb{R}^4$ and consider $M=\mathbb{R}^4 ...
1
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1answer
65 views

Number of connected components of this complement

Let $X$ be a locally finite simplicial complex and let $K$ be a finite subcomplex of $X$. Why is the number of connected components of the complement $X-K$ finite?
3
votes
2answers
61 views

Is $\Omega \tilde X \simeq \Omega_0 X$?

Let $\tilde X\to X$ be a universal covering of a based space $X$, with a chosen basepoint. Is $\Omega \tilde X \simeq \Omega_0 X$? Here $\Omega$ denotes the loop space, $\Omega_0$ denotes the ...
2
votes
1answer
62 views

Can you compute relative homology using simplicial chain complex?

So I had my Algebraic Topology exam yesterday and one of the questions asked to compute the homology groups $H_*(M)$, $H_*(\partial M)$ and $H_*(M, \partial M)$ where $M$ is the Möbius strip and ...
0
votes
1answer
35 views

Isomorphism between the cohomology of projective spaces and spheres

An invariant form on $S^{n}$ is a form $\omega$ such that $i^{*} \omega=\omega$,where $i$ is antipodal map. The vector space of invariant forms on $S^{n}$, denoted $\Omega^{*}(S^{n})^{I}$, is a ...
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0answers
58 views

Cofiber Sequences in Reduced Homology Theory

While going through axiomatic treatments of homology theories I got a bit stuck on this problem. Consider given a reduced homology theory, i.e. functors $(\tilde{E}_q:Top_* \to Ab)_{q \in ...
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0answers
53 views

Is there a “standard” way to compute the fundamental group of the $CP^n$?

I know that $\pi_1(CP^n)=0$ here is a possible proof: Notation: for a CW complex, denote by $X^k$ the $k$-skeleton of $X$. I will show that $\pi_1(CP^n)$ is contained in $\pi_1(S^2)=0$. Let $f:S^1 ...