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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1answer
150 views

Definition of cohomology with compact support

We can define a cohomology on open manifold: Define a simplicial cochain group $$ \Delta^i_c(X;G)$$ consisting of cochains that are compactly supported in the sense that they take nonzero values on ...
7
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2answers
885 views

Contractible spaces has trivial fundamental group.

I have to prove the following: Show that if $X$ is contractible (the def. I have is that $I:X\rightarrow X$ the identity function is homotopic to the constant function $p$ for some $p\in X$), then its ...
2
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1answer
352 views

Is the product of covering maps a covering map?

I have a question about covering maps. If $\phi_1: X_1 \rightarrow Y_1$ is a covering map, and $\phi_2: X_2 \rightarrow Y_2$ is a covering map, then is it true that $\phi_1 \times \phi_2: X_1 \times ...
3
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1answer
57 views

Universal covering space of connected open subset of $\mathbb R^n$

Is the universal covering of an open connected subset $U$ of $\mathbb{R}^n$ homeomorphic to $\mathbb{R}^n$?
4
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2answers
228 views

'homotopy' between morphisms of a 'topological' or 'algebraic' category (Stanley-Reisner ring)

In what follows, a homotopy is a congruence $\simeq$ on a given category. Given such a homotopy, objects $X$ and $Y$ of the given category are homotopy equivalent when there exist morphisms ...
5
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2answers
100 views

Two defintions of simply connected

I'm showing that the first definiton here implies the second (the other implication is obvious). My thoughts: Let $p,q$ be two paths in the space $X$. Then since $X$ is path connected there are two ...
2
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0answers
57 views

Quotients of infinite dimensional sphere

Recall $$S^\infty = \cup S^n,\ {\bf RP}^\infty = \cup {\bf RP}^n,\ {\bf CP}^\infty = \cup {\bf CP}^n$$ Hence $S^\infty / {\bf Z}_2 ={\bf RP}^\infty$. And I think that the following is possible : ...
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0answers
46 views

Geometric interpretation of $R$-orientations

For an $n$-dimensional manifold $M$ and a ring $R$ one can define $R$-orientability, e.g. by choosing local orientations, i.e. generators of $H_n(M, M - \{x\}; R)$ for every point $x$ in the manifold, ...
3
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1answer
108 views

Visualizing the group operation in higher homotopy groups

I'm having trouble picturing the homotopy group operation of concatenation between two pointed spaces. For $n$-spheres, we have for $f,g: S^n \to X$ $$(f * g)(s_1,\ldots, s_n) = \begin{cases} ...
5
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1answer
133 views

Different point-set level definitions of spectra

I've been trying to understand the Adams spectral sequence and one of the more accessible sources is the (unfinished) book on spectral sequences by Hatcher. The usual (only?) construction of the ...
28
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3answers
758 views

How did we know to invent homological algebra?

Update: Qiaochu Yuan points out in the comments that the title of the question is misleading, as homological algebra did not begin with long exact sequences as I'd thought. (Original question ...
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1answer
82 views

What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus?

What would be an example of a nontrivial, i.e. multi-sheeted, covering of the torus? I would greatly appreciate any help that you could give me.
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0answers
168 views

Product of two Kähler manifolds

Let $M$ and $N$ be Kähler manifolds. I know that the cartesian product $M\times N$ is a kahler manifold. I was wondering which form has the Kähler form on $M\times N$. Here's what I thought: Let ...
3
votes
2answers
415 views

covering space of a particular CW complex

I am trying to find all connected covers of the following space $X$ (up to isomorphisms) $X$ has one $0$-cell, two $1$-cells labeled $a$ and $b$, and three $2$-cells attached via $a^2$, $b^2$ and ...
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0answers
48 views

Intersection between cycles and dominant maps

i must say in advance that i'm not very familiar with algebraic cycles and intersection theory, so i hope my question is not too trivial. Let $X$ and $Y$ be two K3 surfaces. Let ...
12
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2answers
676 views

Motivation of stable homotopy theory

A stable homotopy category can be obtained by modifying the category of pointed CW-complexes: objects are pointed CW-complexes, and for two CW-complexes $X$ and $Y$, we take $$\lbrace X,Y \rbrace = ...
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0answers
55 views

Vector field and a solution of an ODE

I have this: And my questions are : 1)what is :"The local theory of differential equations in a Banach space" 2)Why it's implies that each solution of (6) is equal to $\eta$ Please Thank you . ...
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0answers
778 views

When is a fibration a fiber bundle?

In this question I am using Wiki's definitions for fibration and fiber bundle. I want to be general in asking my question, but I am mostly interested in smooth compact manifolds and smooth fibrations ...
5
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1answer
129 views

Explanation of example 3F.7 in Hatcher

The section I am refering to is the following example on page 314 of Hatcher's Algebraic Topology: I'm a bit confused by his statement about relations and can't quite see what he is trying to say. ...
5
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3answers
394 views

Homomorphism/map in both direction implies isomorphism/homeomorphism or not?

I was working on a homework, and my first attempt get me to a deadend, but I was eventually able to solve it using a different method. But the fail attempt make me curious, and I wonder if it could ...
0
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1answer
93 views

Properly discontinuous action on $\mathbb{R}^2$

Given a group $H$ with two generators $a$ and $b$, and one relation $abab^{-1}=e,$ I want to show that $H$ acts properly discontinuously on $\mathbb{R}^2$. I think that I need to show that for every ...
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2answers
528 views

Cup product on torus

I want to calculate the cup product on torus (cf. Hatcher's book) If $\pi_1(T^2) = ([a]) + ([b])$, then by universal coefficient theorem we have a cocycle $\alpha$ (resp. $\beta$) which have a ...
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0answers
63 views

Homomorphism and boundary operator [closed]

I have this : I dont know how to verify 1) and 2)? Please help me Thank you.
5
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1answer
201 views

Sheafification of singular cochains

Let $S^k$ be the presheaf on a space $X$ that assigns to every open set $U$ the abelian group $S^k(U)$ of singular k- cochains on $U$. This is clearly not a sheaf. Consider the sheafification $F^k$ of ...
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1answer
61 views

Isomorphism of Covers

On page 26 of Peter May's A Concise Course on Algebraic Topology, it is claimed that given any two covers of a space $X$, $(E, p)$ and $(E', p')$ are isomorphic iff for any points $e \in E, e' \in E'$ ...
12
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1answer
177 views

The fundamental group functor is not full. Counterexample? Subcategories with full restriction?

Anyone aware of a nice counterexample to "The fundamental group functor is full?" (Which is...false, right?) and are there a nontrivial subcategories on which its restriction is full? I.e. Can you ...
2
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2answers
427 views

CW-pairs are good pairs

Hatcher uses in a proof that every subcomplex of a CW-complex is a deformation retract of some neighborhood. In what way can I see this in the infinite dimensional case?
8
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2answers
921 views

Two spaces homotopy equivalent to eachother, attaching maps, Algebraic Topology.

I have a question regarding algebraic topology with which I was hoping someone could help me with. I've managed to show the following: If $f,g:S^{n-1} \to X$ are homotopic maps, then $X\sqcup_fD^n$ ...
0
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1answer
80 views

Prove $SO(3)$, the group of rotations of $\mathbb{R}^3$, is not homotopically equivalent to $S^1\times S^2$

Prove $SO(3)$, the group of rotations of $\mathbb{R}^3$, is not homotopically equivalent to $S^1\times S^2$. I know that $\pi_1(SO(n))\cong \mathbb{Z}_2$ and I think that $P:\mathbb{R}\times S^2 \to ...
0
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0answers
84 views

Question about an isomorphism result in homotopy theory

I have another question regarding homotopy theory and winding numbers (or degrees). In Manton and Sutcliffe they state the following theorem: $\pi_2(G/H)=\pi_1(H)$ provided $G$ is a compact, ...
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0answers
109 views

Why is the pullback of a connected cover not necessarily connected?

In particular, I read somewhere that the fiber product of the maps $S^1\rightarrow S^1$ sending $z\mapsto z^m$ and $S^1\rightarrow S^1$ sending $z\mapsto z^n$ is disconnected with $\gcd(n,m)$ ...
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0answers
39 views

Question on critical groups

I have this theoreme with it's proof But i don't understand who is $f_0$ ? Can someone help me please ? Thank you .
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1answer
74 views

Finite group acting freely on Haussdorf space- Topology problem

How to prove the following problem: It is given Hausdorff space $X$ and finite group $G$ (with neutral $e$) that is acting freely on $X$. For $g\in G$, $\overline{g}:X\rightarrow X$ is ...
1
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2answers
104 views

horn of a simplex

I'm reading one book of simplicial homotopy. It's just amazing. But I am stuck at the very beginning of the book. He let a simplicial set be a contravariant functor between the category $ \Delta $ of ...
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2answers
65 views

Question about “THE MORSE INEQUALITIES”in Milnor's book

in this paragraph what is $H_{*}$ ? Please help me Thank you .
2
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1answer
120 views

A Borsuk theorem

Let $M$ and $L$ be two subspaces of Banach space $X$ such that $\dim L<\dim M<\infty$. Let $S=\{m\in M : \|m\|=1\}$ and let $g$ be a continuous function from $S$ to $L$ such that $g(-m)=-g(m)$ ...
11
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4answers
921 views

What are two continuous maps from $S^1$ to $S^1$ which are not homotopic?

This is an exam question I encountered while studying for my exam for our topology course: Give two continuous maps from $S^1$ to $S^1$ which are not homotopic. (Of course, provide a proof as ...
20
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2answers
2k views

Curious remark of D. Ravenel

In his beautiful (but difficult) book "Complex cobordism and stable homotopy groups of spheres", concerned mostly with methods of computing homotopy groups of spheres, D. Ravenel describes a general ...
6
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1answer
153 views

Perverse sheaves (or D-modules) on vector spaces, constructible with respect to a hyperplane arrangement

Let $V$ be a finite dimensional complex vector space and let $\mathcal{A}$ be a finite collection of hyperplanes in $V$. Stratify $V$ by the intersections of elements of $\mathcal{A}$, and consider ...
2
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2answers
166 views

Complex projective line homeomorphic to $2$-sphere

Define an equivalence relation $\sim$ on $X={\bf C}^2\setminus \{(0,0)\}$ by $(x_1,y_1)\sim(x_2,y_2)$ if and only if there exists $t \in C\setminus\{0\}$ such that $(x_1,y_1)=(tx_2,ty_2)$ show that ...
7
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1answer
575 views

Wedge product $S^1 \vee S^2$

I am trying to compute $\pi_1(S^1 \vee S^2$) by Van Kampen. I know Hatcher has a solution but I need to verify if my approach is correct and rigorous. I have seen a previous post on this topic, but I ...
0
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2answers
133 views

How to find the induced map $f_{*} : \pi_1 (S^1 , (1,0)) \to \pi_1 (S^1 , (1,0) ) \ \ ? $

I came across this old exam question while studying for my own exam for our topology course. Let $f : S^1 \to S^1 $ be the map $z \mapsto z^n$. What is the induced map $$f_{*} : \pi_1 (S^1 , (1,0)) ...
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0answers
48 views

Algebra homomorphism in Steenrod algebra.

Let $\psi$ be the map of generators $\psi : \mathfrak{a}(2) \rightarrow \mathfrak{a}(2) \otimes \mathfrak{a}(2)$. $$\psi(Sq^k) = \sum_{i=0}^k Sq^i \otimes Sq^{k-i}$$ Let $\underline{\mathfrak{a}}$ be ...
0
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1answer
78 views

function lifting on $S^1 \times S^1$

Let $f:S^1 \times S^1 \to S^1 \times S^1$ a continuous function and $p:\mathbb{R}^2 \to S^1 \times S^1: (t,s) \mapsto (e^{2\pi i t},e^{2\pi i s})$ a covering map. if $F: \mathbb R ^2 \to \mathbb R ^2 ...
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0answers
86 views

Principal $G$-bundles as pull back bundles. [duplicate]

Let $G$ be a compact Lie group and consider a $G$-universal bundle $\pi: EG \to BG $ where $BG$ is the classifying space for the group $G$ and the bundle $\pi: EG \to BG $ is defined as the principal ...
3
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1answer
164 views

Dropping the orientable condition from the Thom isomorphism theorem.

My first question is: what are some examples of un-oriented n-plane bundles over $ \mathbb{Z}$ where it is easy to see that there is no Thom class? I would like to know some examples because the real ...
7
votes
1answer
406 views

Retract and Homotopy extension property

See picture below the following picture. According to Hatcher, homotopy extension property implies that for a pair $(X,A)$ where $A$ is a subspace of $X$, $X\times I$ should retract to ...
2
votes
1answer
80 views

a counterexample of covering projection

Let $X=S^1\times S^1\times\cdots$ be a countable product of 1-spheres and for $n\geq 1$ let $\tilde{X_n}=R^n\times S^1\times S^1\times\cdots$.Define $p_n:\tilde{X_n}\rightarrow X$ by ...
5
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3answers
1k views

Why doesn't the circle retract to a point?

OK, this appears to me like perhaps a dumb question. I am reading Allen Hatcher's Algebraic Topology. I've seen bits and pieces of further material here and there before, now I'm restarting from the ...
0
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1answer
131 views

the nonexistence of a division algebra on $\mathbb R^{2n+1}$

Prove that there is no division algebra structure on $\mathbb R^{2n+1}$. The hint says Suppose that there is such a structure on $\mathbb R^{2n+1}$. Take a nonzero $a \in \mathbb R^{2n+1}$. ...