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

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3answers
1k views

Homeomorphism between real projective plane and disc

Let $D = \{\mathbf{x} \in \mathbb{R}^2 \ | \ \|\mathbf{x}\| \le 1\}$ and let $\mathbb{R}\mathbb{P}^2$ be the real projective plane Let $X = D/\sim$ where $\sim$ identifies antipodal points in the ...
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1answer
264 views

Fundamental group of Klein Bottle with 2 points removed.

Does anyone know what this group is. Just want to know what the group is. Thanks.
3
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1answer
113 views

long exact sequence in k theory

I am studying the basics of K-theory and given a CW pair $(X,A)$ I understand how to construct an long exact exact sequence $\cdots \rightarrow K(SX) \rightarrow K(SA) \rightarrow K(X/A) \rightarrow ...
5
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1answer
397 views

composition of certain covering maps

This problem was posted before, but not the proof (because the asker knowed the answer), only a counterexample without the hypothesis of finite fibres. I want to know how to prove this proposition: ...
11
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2answers
300 views

Is the map $S^{2n+1}\rightarrow \mathbb{C}P^n \rightarrow \mathbb{C}P^n/\mathbb{C}P^{n-1}\cong S^{2n}$ essential?

This question is motivated solely by idle curiousity. There is a natural map $p:S^{2n+1}\rightarrow \mathbb{C}P^n$ mapping a point on $S^{2n+1}\subseteq \mathbb{C}^{n+1}$ to the unique complex line ...
30
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10answers
4k views

A really complicated calculus book

I've been studying math as a hobby, just for fun for years, and I had my goal to understand nearly every good undergraduate textbook and I think, I finally reached it. So now I need an another goal. ...
3
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1answer
65 views

condition on existence of quillenization

$X_{Ab}$ is the Quillenization of a path-connected space $X$ if $X_{Ab}$ has abelian fundamental group, and there exists a continuous map $X\rightarrow X_{Ab}$ inducing an isomorphism ...
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1answer
106 views

If a Klein bottle covers a manifold $M$, then $M$ is the Klein bottle

I have to prove that if a Klein bottle covers a manifold $M$, then $M$ is the Klein bottle. Any suggestions? Thanks.
3
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0answers
52 views

does a method exist to distinguish two component link consisting of just two unknots from an unlink?

Clearly, linking number is not enough as there are links like whitehead. There is the enhanced linking number based on conway polynomial that can distinguish whitehead (and infinite family of such ...
1
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1answer
143 views

why the boundary of a contractible, simply connected 2 dimensional simplicial manifold is connected?

Why the boundary of a contractible, simply connected 2 dimensional simplicial manifold is connected? The conclusion is false for simplicial complexes if you consider the two cones $CS^1$ with their ...
10
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1answer
449 views

Prove that a covering map is a homeomorphism

I got stuck in the following exercise: Let $p:\widetilde{X}\rightarrow X$ be a covering map with $\widetilde{X}$ connected and $p^{-1}(x)$ finite, for every $x\in X$. Show that if there exists a ...
7
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0answers
154 views

Poincaré duality and intersection

Let's take $X$ and $Y$ K3 surfaces and $Z\subset X\times Y$ an algebraic cycle of dimension 2. I know that the Poincarè dual of $Z$, namely $[Z]$, is in $H^4(X\times Y,\mathbb{Z})$ and by Kunneth ...
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0answers
67 views

Extending simplicial complex

Let $X$ be a simplex and $Y\subseteq |X|$ a simplicial complex. Can I construct a simplicial complex $X'\supseteq Y$ s.t. $|X'|=|X|$? Can I do it without introducing new vertices apart from ...
1
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2answers
55 views

a proof concerning fundamental group and lifting of paths

Well, I put the theorem $54.3$ only to show that $\phi$ is well defined. I'm not sure why $\phi([f])= e_1 $ only by definition. Because to check that, I have first to lift $f$ and then compute $ ...
8
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2answers
357 views

Is every open, bounded and simply connected subset of $\mathbb{R}^n$ essentially a ball?

Let $\Omega\subset \mathbb{R}^n$ be open, bounded and simply connected. I wonder if the answer to the following question is known: Is there a homeomorphism $\Omega\to \operatorname{B}_1(0)$, where ...
8
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2answers
245 views

Homology of the fiber of a fibration

I was wondering whether the following conjecture is true and, if so, how one would proof this. All spaces are assumed to be pointed spaces but we drop the base point from notation. Conjecture: ...
2
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1answer
91 views

Computing the homology of a subspace of $R^3$

Let $P_1 ,...,P_5$ be mutually distinct planes in $\Bbb{R}^3$ such that: the intersection of any distinct 2 is a line. the intersection of any distinct 3 is a point. the intersection of any distinct ...
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2answers
159 views

Proof that a continuous map to $S^n$ whose image is a proper subset of $S^n$ is null-homotopic

I am attempting to prove the following: If $g:X \to S^n$, $n \ge 1$, is a continuous map whose image $g(X)$ is a proper subset of $S^n$, then $g$ is null-homotopic. Just before this I proved that if ...
13
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2answers
712 views

Suspension of a product - tricky homotopy equivalence

Let $(X,x_0), (Y,y_0)$ be well-pointed spaces (inclusion of the basepoints is a cofibration). Show the following homotopy equivalence $$ \Sigma (X\times Y) \simeq \Sigma X \lor \Sigma Y \lor \Sigma ...
2
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0answers
39 views

Approximating triangles by squares

I have a simplicial complex $X$ embedded in $\mathbb{R}^n$ s.t. $|X|$ is an $n$-manifold. I would like to algorithmically construct a finite set $\Omega$ of $n$-cubes ("cube"=product of intervals) ...
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3answers
2k views

Topology Prerequisites for Algebraic Topology

Note: There is another question of the same title, but it is different and asks for group theory prerequisites in algebraic topology, while i want the topology prerequisites. I am a physics ...
5
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1answer
118 views

Sullivan model of the odd sphere

I want to determine the Sullivan model of an odd sphere $S^{2n+1}$. Let $(A,d)$ be a cdga such that $H^*(S^{2n+1};\mathbb Q)\cong H^*((A,d_A))$ as graded algebras. Hence $$H^*((A,d))\cong ...
0
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1answer
73 views

Understanding the proof that if two loops in $S^1$ are equivalent then their degrees are equal

I am trying to understand the proof of the following: Theorem: For loops $\alpha$, $\beta$ in $S^1$ with base point $1=(1,0)$, $[\alpha]=[\beta]$ if and only if ...
3
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1answer
148 views

Is there any deformation retraction between a point and a circle?

A point is a retraction of every topological space but how it isn't a Deformation Retraction of s1(circle)?
1
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2answers
249 views

How is the general linear group a topological group?

How to see if the general linear group GL($n$), of non-singular $n$-square matrices over the real (or complex) numbers under matrix multiplication, is a topological group? How to show that matrix ...
2
votes
1answer
121 views

cohomology groups and the pontryagin construction

I'm just beginning to learn about cohomology groups, and have been told that at this point, it definitely behooves me to step away from the crutch of geometric intuition, but I'm going to try to lean ...
1
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1answer
158 views

Homotopy cofibre of a homotopy equivalence contractible?

Given a homotopy equivalence $f:X\to Y$ and its homotopy cofibre $C(f)$ (that is cofibre of the cofibration $i_1: X\to Z(f)$, where $Z(f)$ is $X\times I$ glued by $f:X\times \{0\}\to Y$ to $Y$) prove ...
3
votes
2answers
95 views

Are the maps 'in between' two homotopic paths paths themselves?

We say that paths $\alpha, \beta: I \to X$ with common initial point $\alpha(0)=\beta(0)$ and common terminal point $\alpha(1)=\beta(1)$ are homotopic provided that there is a continuous function $H:I ...
1
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0answers
136 views

Homomorphisms induced by inclusions

Let $X$ be a topological space and $U, V ⊂ X$ open sets such that $X = U ∪ V$; $U ∩ V$ is path-connected; $V$ is simply connected. Let $x_0 ∈ U ∩ V$. Is the homomorphism $i_* : π_1 (U, x_0)\to π_1 ...
2
votes
2answers
319 views

Fundamental group of Klein Bottle generated by two elements

The problem asks me to show that the fundamental group of the Klein bottle is generated by "latitudinal" loops $a$ and "longitudinal" loops $b$ where $a$ and $b$ obey the relation $aba^{-1} = b^{-1}$. ...
0
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0answers
38 views

Showing that equivalence of loops is preserved by the product of loops

I have tried to prove that the equivalence of loops is preserved by the $\ast$ product, i.e. if $\alpha \sim_{x_0} \alpha'$ and $\beta \sim_{x_0} \beta'$, then $\alpha \ast \beta \sim_{x_0} \alpha' ...
1
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0answers
112 views

Subcomplexes of the infinitely-dimensional sphere

I'm trying to understand better what Hatcher does in the beginning chapter on cell complexes and so in this sense I would really like someone to elucidate for me what are the subcomplexes of ...
1
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0answers
114 views

2-sphere union with unit 2-cell is homotopy equivalent to one point union of two 2-spheres.

This is problem 2 in Bredon I.14, on homotopy. I need to prove that $X$ = union of the 2-sphere with the unit 2-cell going through the origin is homotopy equivalent to $Y$ = one-point union of two ...
3
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0answers
304 views

suggestion for video lectures on algebraic topology

can anyone suggest me any good video lecture series forr algebraic topology other than N.J.wildberger video.If it is equivalent to munkres topology(algebraic topology section) it should be great. ...
2
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1answer
416 views

Isotopy and Homotopy

What is the difference between homotopy and isotopy at the intuitive level.Some diagrammatic explanation will be helpful for me.
2
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2answers
94 views

Question on part of the proof that $\pi_1(X,x_0)$ is isomorphic to $\pi_1(X,x_1)$

I am hung up on part of the proof of the following theorem: Theorem: Let $X$ be a path connected space and $x_0$, $x_1$ points of $X$. Then the fundamental groups $\pi_1(X,x_0)$ and $\pi_1(X,x_1)$ ...
2
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1answer
105 views

Vector bundles on algebraic 2-sphere

Let $R=k[x,y,z]/(x^2+y^2+z^2-1)$ be an algebraic sphere over some field $k$. Is it true that any projective module of rank 1 is isomorphic to $R$? More generally, what is the structure of ...
2
votes
1answer
119 views

Confusion about commutative differential algebra

Here we read the following Let $(A,d)$ be a commutative differential graded algebra such that $H^0(A,d)=\mathbb Q$, $H^1(A,d)=0$ and $\dim H^p(A,d)<\infty$ for each $p$. There exists then a ...
12
votes
1answer
559 views

A simply-connected closed surface is a sphere

From the Classification Theorem for closed (i.e. compact and boundaryless) surfaces, it follows that $S^2$ is the only closed surface with trivial $\pi _1$. That's easy because the fundamental group ...
1
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1answer
87 views

Connection between spaces of cosets and homotopy cofibers

Suppose we have an inclusion of a closed sub-Lie group $H\to G$ and take the space of left cosets of $H$, $G/H$. How is this related to the homotopy cofiber of the inclusion of topological spaces ...
27
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3answers
1k views

Roadmap to study Atiyah-Singer index theorem

I am a physics undergrad and want to pursue a PhD in Math (geometry or topology). I study it almost completely by myself, as the program in my country offers very less flexibility to take non ...
16
votes
1answer
392 views

Twisted Cech cohomology

Let $X$ be a CW-complex with contractible universal cover $\tilde{X}$ and fundamental group $\pi = \pi_1X$. Twisted (co)homology is found by lifting the cell structure on $X$ to a $\pi$-invariant ...
4
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2answers
394 views

Hatcher Ch.0 (P18) #5 Inclusion Map is Nullhomotopic

Question: Show that if a space $X$ deformation retracts to a point $x ∈ X$, then for each neighborhood $U$ of $x$ in $X$ $\exists$ a neighborhood $V ⊂ U$ of $x$ such that the inclusion map $V ...
6
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1answer
213 views

Connectedness of the orthogonal subgroup $O^+_+(k,l)$

Let $O(k,l)$ be the orthogonal group associated to the quadratic form $q$ on $\mathbb{R}^{k+l}$ with signature $(k,l)$. Let $O^+_+(k,l)$ be the connected component of the identity, i.e. the connected ...
3
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1answer
101 views

From (algebraic) topology to geometry

I am thinking about a "correct" didactic way of linking topology (algebraic topology) to geometry. Usually, we are taught introducing geometry first, then topology, almost as an abstraction of ...
6
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1answer
339 views

Exercise 4-A “Characteristic Classes” by Milnor and Stasheff

Exercise 4-A of Milnor and Stasheff's book "Characteristic Classes" reads: Show that the Stiefel-Whitney classes of a Cartesian product are given by $w_k(\xi\times\eta) = \sum^k_{i=0} ...
2
votes
1answer
182 views

Cohomology $SO(3)$

We have that De Rham cohomology of $SO(3) \simeq \mathbb{R}P^{3}$ is $\mathbb{R}$ in degree $0$ and $3$ and $0$ in degree $1$ and $2$. But I saw that $H^{*}(SO(3)) \simeq \mathbb{Z}_{2} $ in degree 2. ...
0
votes
1answer
95 views

Cellular cohomology complex projective spaces

I have to calculate the cohomology of complex projective spaces $\mathbb{C}P^{n}$ using cellular cohomology (I know that we have a CW decomposition of $\mathbb{C}P^{n}$ in $n+1$ cells of even ...
0
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1answer
81 views

Cohomology and exterior algebra

Why I can look at $H^{*}(S^{3})$ as the exterior algebra $\Lambda(x_{3})$? Where $x_{3} \in H^{3}(S^{3})$ is a cohomology class (I suppose...).
1
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1answer
192 views

Homology group of mapping cylinder for map of degree m

Hatcher claims on p. 148 that $H_{n}(M_{f},S^{n})=\mathbb{Z}/m\mathbb{Z}$, where $f$ is a degree $m>1$ map of $S^{n}$, and $M_{f}$ is the associated mapping cylinder. Why is this?