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

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0
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1answer
33 views

To prove that the projection map $S^2 \to \mathbb RP^2 $ is a covering map via group action

I am reading Algebraic Topology and I got some problem in covering map. Please help me. Thnx in advance. I want to show that the projection map $S^2 \to \mathbb RP^2 (\text{ real projective plane })$ ...
1
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1answer
42 views

Is it (not) possible for two vector fields on the Klein bottle to be a basis?

Lefschetz fixed point theorem (for example) implies that a compact manifold with a nowhere vanishing vector field must have Euler characteristic zero. Is there a way to draw stronger algebraic ...
4
votes
1answer
40 views

Hirzebruch's $L$-polynomial and $\mathbb{C}P^n$

Hirzebruch's $L$-polynomial is the formal power series \begin{equation} L(x) = \frac{x}{\tanh x} = 1 + \frac{x^2}{3} + \cdots \end{equation} This defines a multiplicative sequence and a genus $L(M)$ ...
2
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0answers
32 views

$\Delta$-Complexes Are Hausdorff

I am using the definition of a $\Delta$-complex as given in Hatcher's book here on pg 103. Now on pg. 104, just before the section on Simplicial Homology, Hatcher remarks that if $X$ has a ...
3
votes
2answers
47 views

CW-decomposition of quotient space

Let $X$ be the space that results form $D^3$ by identifying points on the boundary $S^2$ that are mapped to one another by a $180°$-rotation about some fixed axis. I want to calculate the cellular ...
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0answers
29 views

Trouble With the Definition of $\Delta$-Complex in Hatcher's Book

On Pg. 103 of Hatcher's Algebraic Topology the author has defined what a $\Delta$-complex is: This is what I have gathered from what the author writes: A $\Delta$-complex is a collection of ...
6
votes
3answers
185 views

What is the sheafification of the presheaf of the one point compactification?

Okay, so I had this idea for a presheaf that is quite peculiar. Instead of being based on algebraic category (i.e. abelian groups), it is based on a topological one, the category of compact ...
2
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1answer
31 views

Irregular (branched) cover

I need to know the definition of an irregular (branched) cover. I heard this somewhere but I am not able to find any definition on the internet.
1
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2answers
32 views

Isomorphism of chain complexes

In my notes it says $C^{sing}_n(\sqcup_{i\in I} X_i;R) \cong {\bigoplus}_{i \in I} C^{sing}_n(X_i;R)$, where $C^{sing}_n$ denotes the n-th singular chain complex and $R$ is a ring, $S_n(X)$ is the set ...
2
votes
1answer
62 views

Does trivial fundamental group imply contractible?

Let $X$ be a path-connected topological space with a trivial fundamental group: $$\pi_1(X,x_0)=\{e\}.$$ Does $X$ have to be homotopic to a point? I know that the converse is true: a ...
1
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0answers
41 views

Closed unit ball is a retract of $R^2$

I was asked whether a closed unit ball is a retract of the euclidean space $R^2$. I think the answer is yes and the retraction might be defined as follows: for all the points in $R^2$ join them with ...
3
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2answers
52 views

Help finding the fundamental group of $S^2 \cup \{xyz=0\}$

let $X=S^2 \cup \{xyz=0\}\subset\mathbb{R}^3$ be the union of the unit sphere with the 3 coordinate planes. I'd like to find the fundamental group of $X$. These are my ideas: I think the first thing ...
7
votes
2answers
109 views

The homology of $\Omega T^n$

As part of a bigger plan for conquering Europe, I have to compute the integral homology of the loop space of the $n$-torus $T^n = S^1\times \cdots \times S^1$. The plan is: compute $H_*(\Omega ...
2
votes
1answer
32 views

“Winding number”, Chern character and relative signatures of the metric

Anyone answer with good explanation is appreciated. In differential geometry, we discuss about topological quantities like characteristic classes. For example, the first Chern character of some ...
4
votes
1answer
42 views

Relative Homology is not trivial

Let $(H_*, \partial_*)$ be a homology theory satisfying the dimension axiom. Let $A \subset S^n$ be a proper subset. Show that $H_n(S^n, A)$ is not trivial. I tried applying the long exact sequence ...
3
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0answers
34 views

Is the signature of inverse images of diffeomorphic submanifolds (along a homotopy equivalence) the same?

Suppose it is given an orientation preserving homotopy equivalence $h:N\to M$ between closed oriented connected manifolds. Let $X$ and $Y$ be diffeomorphic submanifolds of $M$, and assume $h$ to be ...
4
votes
1answer
53 views

Pontryagin class of a wedge product of vector bundles.

Let $E\to M$ be a real vector bundle over a differentiable manifold $M$ and let $p_{1}(E)$ denote its first Pontryagin class. I would like to know if there is any formula allowing to write ...
4
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1answer
69 views

Casson handles neighborhoods are representable by $D^2$-bundles over $S^2$.

On 250 page of Scorpan's book Wild world of 4-manifolds. there is a construction of an exotic $\mathbb{R}^4$. It starts from taking manifold $M = \mathbb{C}P^2 \# 9 \overline{\mathbb{C}P}^2$ and ...
0
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0answers
29 views

What is the statement of simplicial approximation theorem for homotopies?

In the wikipedia page simplicial approximation theorem and a former answer related to this theorem, it was mentioned that on simplicial complexes, homotopy between continuous mappings can be ...
1
vote
2answers
78 views

Fundamental group of $X$?

Let $X=X_1\cup X_2\cup X_3$, where $X_1=\{ (x,y,z): x^2 +(y-1)^2+z^2=1\}$ , $X_2=\{ (x,y,z): x^2 +(y+1)^2+z^2=1\}$ and $X_3=\{ (0,y,1): -1\leq y \leq 1 \}$. Find the fundametal group of X. My guess ...
0
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1answer
33 views

Hypothesis in homotopy equivalence inducing isomorphism in the fundamental groups

Let $X$ and $Y$ be topological spaces. If $f\colon X\to Y$ is a homeomorphism, then it induces an isomorphism $f_\sharp\colon\pi_1(X,x_0)\to \pi_1(Y,f(x_0))$. All good. As far as I know, the result ...
3
votes
1answer
47 views

Möbius band as line bundle over $S^1$, starting from the cocycles

The professor asked us to construct a non-trivial line bundle over $S^1$ by giving an open cover of $S^1$ and the cocycles. My idea was to take as open cover $U_1:=S^1\setminus\{0\}$ and ...
4
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0answers
57 views

Showing that a space is not homeomorphic to $\mathbb{R}^4$.

I have a space $X$ which is defined to be the quotient of $[0,1)\times T^3$ ($T^3$ is the 3-torus) by the relation $(0,x)\sim (0,y)$ for all $x,y \in T^3$. It is a kind of cone over $T^3$, except that ...
2
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0answers
39 views

Topological Boundary Map

In May, Concise Algebraic Topology, p. 108-109, for a cofibration $A \rightarrow X$ a "topological boundary map" is defined as the composite: $X/A \xrightarrow{\psi^{-1}} Ci \xrightarrow{\pi} \Sigma ...
3
votes
1answer
48 views

examples for fibration not fibre bundle

We can use path space to make a map into a fibration. Generally, is this construction of fibration a fiber bundle? Or can someone give me some examples of fibration not fiber bundle?
6
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1answer
85 views

Fundamental group of a quotient space

I want to calculate the fundamental group of the space $(S^1 \times [0,1])/$~ where $(z,0)$ ~ $(e^{2\pi i/n}z, 0)$ and $(z,1)$ ~ $(e^{2\pi i/m}z, 1)$. My idea is to find a pushout and then use the van ...
1
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1answer
29 views

Trouble Understanding the Statement of a Theorem in Hatcher's Book

Let $X$ and $Y$ be topological spaces and $A$ and $B$ be subspaces of $X$ and $Y$ respectively. We write $f:(X,A)\to (Y, B)$ as a shorthand for writing $f:X\to Y$ and $f(A)\subseteq B$. Now ...
0
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0answers
44 views

What concept would be independent of path and how do I state it?

Below are lists of theorems I have studied: Theorem1. Let $x,z_0\in S^1\times \mathbb{C}$ and $f,g:S^1\rightarrow \mathbb{C}\setminus\{z_0\}$ be coninuous functions and $\alpha$ is a loop in $S^1$ ...
1
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1answer
55 views

If a line bundle admits a non-vanishing section then it is trivial

Suppose $\pi:E\to B$ is a line bundle. Let $s:B\to E$ a non-vanishing section, i.e. for every $b\in B$ $s(b)\ne 0$ and $\pi\circ s=Id_B$. I have to prove that the line bundle above is trivial. Idea: ...
1
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0answers
25 views

Leray sheaf being constant - what does it mean in terms of singular cohomology?

Let me first admit that I know next to nothing about sheaf cohomology, but I might have encountered a good reason to learn it. Suppose that I have a fibration $F \to E \to B$ and I know that its ...
2
votes
1answer
58 views

A fiber product is a fiber bundle

Let $F,B$ be topological spaces. A fiber bundle $E$ over the basis $B$ with fiber $F$ is a topological space $E$ endowed with a continuous surjection $\pi:E\to B$ such that there exists an open cover ...
0
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0answers
12 views

Explicit presentation of (the fundamental group of) N-coverings of a Riemann surface

I am interested in studying multiple covers of Riemann surfaces. If I understand correctly one possible (algebraic) approach is to take the $\pi_1(S)$ and look for subgroups of a given index. I even ...
3
votes
1answer
71 views

Integral homology of real Grassmannian $G(2,4)$

I would like to compute $\pi_1$ and the integral homology groups of the real Grassmannian $G(2,4)$. (This is a question on an old qualifying exam.) The hint for the computation of $\pi_1$ is to put a ...
6
votes
2answers
93 views

What, and how can, topological invariants can be computed from a space's algebra of functions?

The Gelfrand duality says that the category of locally compact Hausdorff spaces (with proper continuous functions) is equivalent to the category of commutative $C^*$ algebras (with proper ...
2
votes
0answers
82 views

Which sequential colimits commute with pullbacks in the category of topological spaces?

Given diagrams of topological spaces $$X_0\rightarrow X_1\rightarrow\ldots$$ $$Y_0\rightarrow Y_1\rightarrow\ldots$$ $$Z_0\rightarrow Z_1\rightarrow\ldots$$ and maps $X_i\rightarrow Z_i$, ...
0
votes
0answers
23 views

Path lifting property proof

I'm reading these notes on algebraic topology. At the bottom of page 6 (page 8 by the pdf numbering), there is a statement of the homotopy lifting property. It assumes that $K$ is a complete metric ...
7
votes
1answer
98 views

Composition of homotopy classes with self-maps of spheres

Are there some general rules/formulas on the relation between the homotopy class $[f]\in \pi_i(S^n)$ and the homotopy class of the composition $S^i\stackrel{a}{\to} ...
2
votes
0answers
38 views

Union of a sphere and a diameter.

Im doing a question from the textbook Algebraic Topology by Hatcher, and its starts off by saying 'consider a space $X \subset R^3$ that is the union of a sphere and a diameter'. I'm confused by what ...
5
votes
0answers
137 views

Is $\mathbb{S}^1 \wedge E$ a cofinal subspectra in $\Sigma E$?

I'm following the proof of Switzer's "Algebraic Topology and Homotopy" of the known result Theorem. Let $E$ be a (CW-pre)spectra. There is a natural (up to homotopy) homotopy equivalence $E \wedge ...
0
votes
1answer
46 views

Need help to understand Uniqueness of Lifts theorem's proof.

Theorem: Let $p:E \to B$ be a covering map. Fix $b_0 \in B$ and $e_0 \in p^{-1}(b_0)$. Let $f: X \to B$ be a continuous map with $f(x_0)=b_0$ and $X$ is connected. Suppose $g_1,g_2: X \to E$ are ...
2
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0answers
40 views

cohomology ring of cross-section space of fibre-bundles

Given an $m$-dimensional manifold $M$, let $TM$ be the tangent bundle of $M$ and $SM$ be the $m$-sphere bundle over $M$ obtained by fibre-wise one point compactification of $TM$. Let $\Gamma(SM)$ be ...
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1answer
40 views

Cohomology of Grassmannian of 2-planes in $\mathbb C^4$

The cohomology of the Grassmannian of 2-planes in $\mathbb C^4$ can be deduced from computations in section 4.D of Hatcher's book, using the Leray-Hirsch theorem for fiber bundles. However, I was ...
0
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0answers
16 views

Calculation of G-CW(V) structure of equivariant disc

Let $G = Z/p,$ where $p$ denotes a odd prime.Let $\{ 1, \xi , \xi^2, \cdots , \xi^{p-1}\}$ be the set of irreducible representations of $Z/p.$ Question : What is the $Z/p$-CW($\xi$) complex structure ...
10
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2answers
163 views

What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration?

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? Many thanks in advance.
4
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0answers
30 views

What is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? [duplicate]

As the question suggests, what is an intuitive explanation of the Hopf fibration and the twisted Hopf fibration? I not a topologist by trade and I find the concept kind of hard to understand... Thanks ...
7
votes
1answer
140 views

Universal coefficient theorem and multiplication on cohomology

Let $X$ be a topological space and $R$ is a commutative ring. For $H^*(X)$ we have $$0\to H^n(X,\mathbb Z)\otimes R\to H^n(X,R)\to \mathrm{Tor}(H^{n+1}(X,\mathbb Z), R)\to0.$$ Is it true that we ...
0
votes
1answer
54 views

When is there a 1-1 correspondence between relative discs and those in a cover?

Let $X$ be a topological space with covering $p : \tilde{X} \to X$ and $A \subset X$. Consider the set of maps $M := \{u: (D^n, \partial D^n) \to (X,A)\}$ taking $n$-dimensional discs to $X$ with ...
1
vote
2answers
86 views

Cohomology ring of $\mathbb RP^n$ with integral coefficient.

I know cup product structure on $H^*(\mathbb{R}P^n;\mathbb{Z}_2)= \mathbb{Z}_2[\alpha]/(\alpha^{n+1})$. How to get $H^*(\mathbb{R}P^n;\mathbb{Z})$ from this? I have two cochain complexes for two ...
5
votes
1answer
70 views

Proving that the tangent vector of a simple closed curve rotates by $ 2 \pi$

I am trying to prove that if $\gamma(t)=(x(t),y(t))$ ,a function from the closed interval $[0,1]$ to $\mathbb{R^2}$ is a simple closed unit speed curve such that $\gamma '(0)=\gamma '(1)$. Then the ...
1
vote
1answer
53 views

Why does the inclusion $X\vee Y\rightarrow X\times Y$ induce an isomorphism on homology?

Let $X$ and $Y$ be nice spaces (connected, path-connected, locally contractible, etc). We do not assume they are CW complexes. There is a natural inclusion map $$X\vee Y \rightarrow X\times Y.$$ ...