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

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2
votes
0answers
62 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
130 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 ...
0
votes
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
150 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 ...
1
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 $S^2\vee S^2$?
1
vote
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 ...
1
vote
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
vote
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 ...
0
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
vote
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 $n=2$)?
1
vote
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
237 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
74 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
vote
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 ...
1
vote
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 ...
1
vote
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 ...
4
votes
0answers
49 views

Does $\pi_0$ commute with sequential colimits?

I'm looking for an example of a sequence of topological spaces $Y_1 \rightarrow Y_2 \rightarrow \cdots$ such that the induced map $$ \text{colim}\, \pi_0(Y_i) \rightarrow \pi_0 (\text{colim}\, Y_i) $$ ...
1
vote
0answers
61 views

Why is this a group action?

Let $G$ be a group and let $H$ be an infinite cyclic normal subgroup of $G$ of finite index. Let $K$ be the centralizer of $H$ in G, $$K=C_G(H)$$ and suppose that the index of $K$ in $G$ is 2. Let $E$ ...
2
votes
0answers
28 views

example of a regular Y such that F(X,Y) with open-compact topology is not regular? [closed]

this question from (elementary topology by s.willard ) page 288 give an example of a regular Y such that F(X,Y) (space of function not space of continuous function) with open-compact topology is not ...
1
vote
0answers
50 views

morphisms of principal bundles with different structure groups

Let $f \,: X \to Y$ be a continuous map between spaces. Let $G$ and $H$ be topological groups. Consider the diagram: \begin{equation} \label{} \begin{array}{ccccccccccccccccccccccccccccccc} E_G ...
9
votes
1answer
104 views

Is the homotopy type of an aspherical space determined by its fundamental group?

Question: Let $X$ and $Y$ be path-connected spaces that admit a contractible universal cover, with $\pi_1(X) \cong \pi_1(Y)$. Is $X$ homotopy equivalent to $Y$? Comments: $X$ and $Y$ are both ...
2
votes
1answer
61 views

Fundamental group of a connected graph

Is this a legit way to prove that a fundamental group of a connected graph $\Gamma$ is a free group? Without using quotient and homotopy extension property from Hatcher's "Algebraic Topology": Take ...
1
vote
1answer
39 views

Prove $j_{*}:\pi_{1}(X,b) \rightarrow \pi_{1}(Y,b)$ is surjective for certain X,Y.

Let $P=(2,0)$ and $O=(0,0)$. Let $Y=\mathbb{R}^2\backslash\{O\}$ and $X = \mathbb{R}^{2}\backslash \{O,P\}$ and let $j:X \rightarrow Y$ be an inclusion. Prove $j_{*}:\pi_{1}(X,b) \rightarrow ...
2
votes
1answer
56 views

the relation between cohomology and homomorphism

I meet a problem, how can I understand $H^1(M,\mathbb{R})\cong Hom(\pi_1(M),\mathbb{R})$? Where $M$ is a compact manifold. Thanks in advance.
0
votes
1answer
34 views

Lifting property of a covering space

A common example of a covering space is $p:\mathbb R \rightarrow S^1 $, $p(t)= e^{2 \pi it}$, which can be looked at as an infinitely long spiral placed over a circle. Consider the double loop ...
1
vote
1answer
49 views

CW-complex of a genus n surface

I'm having some difficulty understanding how genus n surfaces are built as CW-complexes. I understand how a torus is constructed: -Start with 1 point (a 0-cell) -Add two lines whos start and end ...
5
votes
0answers
39 views

How to kill homotopy groups using framed cobordism

Let $M$ be an orientable manifold (with or without boundary), $N$ a framed submanifold in the interior of $M$ and assume (if necessary) that $\dim N<(\dim M)/2$. If some low-dimensional homotopy ...