1
vote
1answer
31 views

What is the second homotopy group of $R^3 \setminus \{ (0,0,0) \}$

I was told that it was $\mathbb{Z}$, and I can imagine a subgroup isomorphic to $\mathbb{Z}$ of 'wrappings' of the sphere around the point, but I am still convinced there are more homotopy classes. I ...
1
vote
0answers
17 views

Reduced suspension of a CW complex

Is (reduced) suspension an operation on CW-complexes? A naive proof: Reduced suspension is the same as the smash product with $S^1$. $S^1$ is locally compact, so $- \wedge S^1$ preserves colimits, ...
0
votes
1answer
45 views

Mapping $\mathbb R^n - \{0\}$ to $S^{n-1}$

How might one map $\mathbb R^n - \{0\}$ to $S^{n-1}$ ? I found this in a primer on homology where it is proved that the to spaces are homotopy equivalent, as an example of removing a single point ...
2
votes
0answers
33 views

Surjectivity and exactness on higher homotopy groups

If $X$ is a normal algebraic variety over $\mathbb{C}$ and $U$ an open set of $X$ then we have surjectivity on the fundamental groups $ \pi_1(U)\rightarrow \pi_1(X)$. If $f\colon X \rightarrow Y$ is ...
1
vote
1answer
61 views

A map $f: X\rightarrow Y$ is a homotopy equivalence if and only if $h\circ f,f\circ k$ are homotopy equivalences of $X,Y$ respectively.

Show that $f\colon X \rightarrow Y$ is a homotopy equivalence if and only if there exist maps $k,h\colon Y\rightarrow X$ such that $f\circ k$ is a homotopy equivalence of $Y$ to itself, and $h\circ f$ ...
0
votes
1answer
25 views

$[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets

how to show: if $F \to E \to B $ is a fibration then for any space $X$ the sequence $[X,F] \to [X,E] \to [X,B] $ is exact sequence of pointed sets. any hints, thanx.
0
votes
1answer
21 views

$\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$

I need help with this problem Let $\sum X =C^+X \cup_X C^-X$ be the union of two cones on $X$ with a common base. Show that $\tilde{H}^i(\sum X) \cong \tilde{H}^{i-1}(X)$. Dose this give an ...
1
vote
1answer
50 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
1
vote
1answer
31 views

Spaces homotopy equivalent to $A_{\infty}$-spaces

I ask this question after reading Peter May's "Geometry of Iterated Loop Spaces", where the problem is definitely hinted at but I couldn't find a definite answer. Recall a symmetric operad ...
5
votes
0answers
42 views

Homotopy type vs. weak homotopy type, and repercussions for EG

This is another basic question. I'm aware that weak homotopy equivalence is strictly weaker than homotopy equivalence. For one thing, there is a weak homotopy equivalence from $S^1$ to a four-point ...
0
votes
1answer
42 views

A certain homotopy equivalence…

A few friends and I have been stuck on this old qualifying question for quite some time now... Let $D$ be the diagonal subspace of $\Bbb S^2 \times \Bbb S^2$. Show that the projection onto the first ...
0
votes
1answer
39 views

Homotopy equivalences

If $x\in X$, let $C(x)$ the path component of $x$ (the biggest path connected set containing $x$), and similarly if $y\in Y$. Let $C(X)$ and $C(Y)$ the family of all path components of $X$ and $Y$. ...
1
vote
2answers
56 views

Proving the existence of some deformation retract

I am trying to find out if the set $ S ^n \times S^n \setminus \left\lbrace (x,x) \mid x \in S^n \right\rbrace$ deformation retracts onto the subspace $\left\lbrace (x,-x) \mid x \in S^n ...
2
votes
1answer
78 views

Solving an exercise in Milnor-stasheff's “characteristic classes”

I am trying to solve the following exercise (which is an exercise in Milnor-Stasheff's book). It basically says the following: Let $ M =S^n $ be the $n$-sphere and let $TM$ be its tangent ...
1
vote
0answers
21 views

homotopy - maintaining curvature signs

I have the following dilemma! Say, $f_1=\sqrt{1-x^2}$, and $f_2=-\sqrt{1-x^2}$ are two continuous functions on $[-1,1]$ Lets define another function by $F = tf_1 + (1-t)f_2$ where $t=[0,1]$ ...
6
votes
1answer
61 views

Cohomological Whitehead theorem

Let $X$ and $Y$ be CW complexes (resp. Kan complexes) and let $f : X \to Y$ be a continuous map (resp. morphism of simplicial sets). The following seems to be a folklore result: Theorem. The ...
3
votes
1answer
47 views

Does $\pi_0$ preserve infinite products?

The functor $\pi_0\colon \operatorname{sSets}\to \operatorname{Sets}$ has value on a simplicial set $X$ the coequalizer of the two boundary morphisms $d_0,d_1\colon X_1\to X_0$. It is easy to see ...
1
vote
0answers
35 views

existence of a cofiber sequence

can anyone help me with this problem. thanx. Show that there are cofiber sequence $S^{n+3} \to S^{n+2} \to \sum^{n}\mathbb{C}P^2$ for each $n \in \mathbb{Z}^+$. Conclude that a space of the form ...
7
votes
1answer
123 views

Explanation for a line from a MathOverflow answer

Sometimes I see questions answered on MathOverflow in such a way that I don't really understand the answers. Sometimes I work out what they mean, and other times I can't. I'd like to ask for more ...
2
votes
2answers
81 views

homotopy between two functions

Let us discuss this problem: Let $A=\{a_{1},a_{2},\ldots,a_{n}\}$, $B=\{b_{1},b_{2},\ldots,b_{n}\}$ and $C=\{c_{1},c_{2},\ldots,c_{n-1}\}$ be discrete finite sets embedded in a unit sphere ...
3
votes
2answers
164 views

Why is $\pi_1(X,x_0)$ a group?

I want to show that $\pi_1(X,x_0)$ is a group. I am told that $e(t) := x_0$ is the identity element. Now, I am struggling to show that it is an identity element, and also that the inverse of an ...
3
votes
0answers
41 views

Path-homotopic definition.

Given two paths $f,g: [0,1] \mapsto X $ Then their product is defined by $f\cdot g := \begin{cases} f(2t) , \space 0\leq t \leq \frac{1}{2}\\ g(2t-1) ,\space \frac{1}{2} \leq t \leq 1\\ \end{cases} ...
2
votes
1answer
44 views

Example for a space that is contractible to precisely one of its points

Give an example for a space that is contractible to one of its points and is not contractible to another of its points. I am really curious about that space, I have thought about tree with $n$ ...
0
votes
1answer
38 views

Why is a spectrum $X$ with only $\pi_0X\neq 0$ equivalent to an Eilenberg-MacLane spectrum?

For an abelian group $G$, the Eilenberg-MacLane spaces $K(G,k)$ assemble to a spectrum $HG$ with $HG_k=K(G,k)$. This spectrum has the property that $\pi_0 HG=G$ and $\pi_{n}HG=0$ for $n\neq 0$ where ...
1
vote
2answers
95 views

What is the difference between multiplication and direct sum on homotopy groups of spheres?

In Allen Hatcher's book Algebraic topology he states that factoring out 2-torsion $\pi_{i}(S^{2n})\cong\pi_{i-1}(S^{2n-1})\times\pi_{i}(S^{4n-1})\:\forall n$ but in his book Spectral Sequences in ...
0
votes
2answers
56 views

Quasi circle is not contractible

I'm trying to show that the quasi circle (picture below) doesn't have the homotopy type of a CW complex. I proved that all homotopy groups are zero. Now I need to show that it is not contractible to ...
3
votes
3answers
134 views

how should I show that it is wedge of infinite circles?

I know that the shape that we see it below is homotopy equivalent of wedge of infinite circles,so the fundamental group of it is $\prod _{1}(\vee _{\alpha \in A}S^{1})=\ast _{\alpha \in A ...
1
vote
2answers
75 views

Spaces with different homotopy type

I want to show that the spaces $ S^1 \vee S^1 \vee S^2$ and $S^1 \times S^1 $ do not have the same homotopy type. I calculated their homologies and cohomologies and they turn out to be equal. So I ...
3
votes
1answer
50 views

Question on the uniqueness of a homotopy colimit up to unique isomorphism

Let me first give an abstract definition of the homotopy colimit. Let $C$ be a cofibrantly generated model category and let $D$ be a small category. There is an adjunction $$ ...
0
votes
0answers
30 views

Book of Pullbacks and Pushouts

what books can I consult for properties of pullback and pushouts in algebraic topology? I need to understand the theory of homotopy in algebraic topology and I started to study pullbacks and push ...
3
votes
1answer
87 views

Fibrations induced by deformation retractions

Given a map $f: A \to B$ and a homotopy equivalence $g: C \to A$, I wish to show that $E_f \to B$ and $E_{fg} \to B$ are fiber homotopy equivalent, where, given a function $f: A \to B$, $E_f$ is the ...
2
votes
0answers
57 views

Integral Homology of $BU$

We know that the integral cohomology of $BU$, $H^*(BU) = Z[c_1,c_2,...]$ where $|c_i| = 2i$ is the $i$th Chern class, with coproduct $\Delta c_i = \Sigma c_j\otimes c_{i-j}$. And at almost ...
2
votes
0answers
24 views

map to product of eilenberg-maclane spaces

Given a space $X$, and an Eilenberg-MacLane space $K(G,n)$ (hereafter referred to as $K$), and two maps $f: X \to K$ and $g:X \to K$, let $f \times g:X \to K \times K$ map $x \in X$ to $(f(x),g(x))$. ...
2
votes
1answer
34 views

Multiplication on a K(G,n)

Suppose that, given an abelian group $G$, there is a multiplication map $\mu:K(G,n)\times K(G,n) \to K(G,n)$ defined such that the induced map on the homotopy group $\mu_*:\pi_n(K(G,n) \times K(G,n)) ...
2
votes
0answers
30 views

Group structure on pointed homotopy classes [X,S^1]

Let $[X,S^1]$ denote the set of pointed homotopy classes of maps $f:X\to S^1$. I need to show that, when $S^1$ is viewed as a subset of $\mathbb{C}$, complex multiplication induces a group structure ...
4
votes
0answers
37 views

what can you say about the degree of $f:\mathbb{C}P^n \to \mathbb{C}P^n$

Any thoughts on this problem: If $M$ and $N$ are simply-connected, $n$-dimensional manifolds, then $H^n(M;\mathbb{Z}) \cong \mathbb{Z} \cong H^n(N;\mathbb{Z})$. A map $f:M \to N$ induces a map ...
0
votes
0answers
46 views

additive isomorphism from $H^*(S^2 \times S^4 ; \mathbb{Z})$ to $H^*(\mathbb{C}P^3 ; \mathbb(Z))$

I need help in solving this problem: show that there is an additive isomorphism from $H^*(S^2 \times S^4 ; \mathbb{Z})$ to $H^*(\mathbb{C}P^3 ; \mathbb(Z))$. Then determine whether or not $S^2 \times ...
2
votes
0answers
24 views

isomorphic cohomology rings for the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times *)$ and $(S^3 \times \mathbb{C}P^{\infty})$

I need to show that the spaces $(S^1 \times \mathbb{C}P^{\infty})/(S^1 \times *)$ and $(S^3 \times \mathbb{C}P^{\infty})$ have isomorphic cohomology rings. any ideas ..... thanx!
5
votes
2answers
78 views

Is there an analogue of the universal cover for higher homotopy groups?

The universal cover $U$ of a topological space $X$ is a simply-connected covering space of $X$. As the 'universal' moniker implies, this space is universal in the category of covering spaces of $X$ ...
0
votes
1answer
81 views

Showing that two spaces are homotopy equivalent

Let $x_0 \in S^1 \times S^1$. I want to show that $(S^1 \times S^1) - \{x_0\}$ and $S^1 \vee S^1$ are homotopy equivalent. We have to show that $\exists$ maps $f: X \rightarrow Y$ and $g: Y ...
0
votes
1answer
59 views

How to prove “Homotopy is an Equivalence Relation”

Reading Allen Hatchers book (available online via this link) on Algebraic Topology, it states on page 3 that homotopy type defines an equivalence relation. The symmetry and reflexiveness are ...
1
vote
1answer
44 views

Covering spaces and homotopical equivalence

I have this simple question: if $X$ and $Y$ are two topological spaces homotopically equivalents, have they the "same" covering spaces? (and if yes, in which sense?) This question derive from an ...
1
vote
1answer
58 views

Prove fundamental group is the direct product

Suppose that $A$ is a retract of $X$ with retraction $r : X \rightarrow A$. Also suppose that $i_*(\pi(A,a))$ is a normal subgroup of $\pi(X,a)$. Prove that $\pi(X,a)$ is the direct product of the ...
0
votes
1answer
25 views

Regarding an arbitary fibration as an inclusion

I'm reading through Allen Hatcher's Algebraic Topology, and he mentions that, given a Postnikov tower, the fibration $X_n \rightarrow X_{n-1}$, where $X_n$ and $X_{n-1}$ are CW complexes, can be ...
3
votes
1answer
43 views

Trivial Cohomology Group->Lower-Dimensional Homotopy?

Calculating the (de-Rham) cohomology of a tee connector (Picture), I got $H^0=R,H^1=R^2,H^2=0$. Furthermore, just from looking at it, I assume the tee connector is homotopic to a circle with an arc ...
1
vote
1answer
49 views

Inverse limit of sequence of fibrations

I'm reading through the proof of Proposition 4.67 in Hatcher's Algebraic Topology, and I've come to something that I'm having trouble understanding. For an arbitrary sequence of fibrations $... ...
5
votes
1answer
61 views

Are there spectral sequences for calculating homology or cohomology of homotopy (co)limits?

Suppose my nice topological space $X$ is the homotopy colimit $$\operatorname{hocolim}D\cong X$$ of a diagram $D\colon I\to \mathbf{Top}$ and the homotopy limit $$\operatorname{holim}E\cong X$$ of a ...
4
votes
1answer
77 views

Model structure on sSet

Which is the model structure on $ \text{sSet} $ (category of simplicial sets) that makes $\text{sSet}$ Quillen equivalent to the category $ \text{Cat} $ (of small categories) by the adjunction ...
1
vote
0answers
51 views

Is it possible to consider this property of “being nice” as a homotopy property?

Consider a Model (Quillen) Category $ M $ (possibly with all oobjects being fibrant or cofibrant (or both)). I'm wondering if the following property is a homotopy property: "Let $ p $ be a morphism ...
5
votes
1answer
70 views

Fat geometric realization weakly equivalent to the usual one

Let $X$ be a simplicial set. Recall that we can associate to it a certain topological space, the geometric realization, given by $ | X | = \int ^{n \in \Delta} X_{n} \times \Delta _{n} \simeq ...