Two functions are homotopic, if one of them can by continuously deformed to another. This gives rise to an equivalence relation. A group called homotopy group can be obtained from the equivalence classes. The simplest homotopy group is fundamental group. Homotopy groups are important invariants in ...

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1answer
82 views

Showing $S^1$ is not a retract of $D^2$ using homotopy

I'd like to know if my argument below, in which I try to show that the 1-sphere is not a retract of the 2-disk using homotopy, is valid. Suppose there is a retract $r:D^2 \to S^1$. Then we can ...
1
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0answers
46 views

Stack on commutative ring spectra?

One approach to stacks to call a stack a "sheaf of groupoids" which means a functor $$ \mathcal{C}^{\text{op}} \rightarrow \mathcal{G} $$ from a category $\mathcal{C}$ with a Grothendieck topology to ...
3
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0answers
42 views

Generalisation of Adams spectral sequence to triangulated categories

We have the Adams SS with $$ E_2^{p,q} = Ext^{p,q} _{E^*(E)}([S,E],[S,E]) $$ where $E$ is the Eilenberg-Maclane Spectrum yielding $\mathbb{Z}/p$ coefficients. I was wondering if there is a SS for ...
2
votes
1answer
63 views

Use Hurewicz Theorem to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$

Want to calculate $\pi_3(\mathbb{R}P^4 \vee S^3)$, using Hurewicz theorem. This is one of the questions on the previous topology qualifying exams. Any help will be appreciated! I am thinking in stead ...
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0answers
53 views

Why the Objects of Homotopy Category not Homotopy Classes of Spaces?

A homotopy category is a category whose objects are topological spaces and whose morphisms are homotopy classes of continuous functions. I wonder why the objects are spaces, instead of homotopy ...
1
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0answers
94 views

extending maps from spaces to their whitehead towers

Let $f \,: X \to Y$ be a map between connected spaces. Let: $$ X^{(k)} \to \ldots \to X^{(0)} \approx X $$ and $$ Y^{(k)} \to \ldots \to Y^{(0)} \approx Y $$ be whitehead towers for $X$ and $Y$. What ...
2
votes
2answers
44 views

Prove sequence has limit in $\gamma (S^1)$

This is a seemingly interesting exercise from my topology notes, but I can't solve it for the life of me. It's like this: take a closed curve $\gamma : S^1\to \mathbb R^2$, and a sequence ...
0
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1answer
157 views

Example of non-homotopic functions

I start to study homotopy theory and I've trouble with following "proof": "If $f_0$, $f_1$ are continous functions $X \rightarrow Y$, then one can consider the continous function $F=(1-t)\cdot f_0+ ...
1
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1answer
51 views

Does ''homology vanishes eventually'' imply ''homotopy vanishes eventually''?

Let $X$ be a connected CW complex. Assume there is an integer $N\geq 0$ such that the singular homology $H_n(X)=0$ vanishes for all $n\geq N$. Is there an integer $M\geq 0$ such that $\pi_m(X)=0$ ...
2
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0answers
34 views

How do I prove that $U(r) \to S(r,n) \to G(r,n)$ is a fibration?

$U(r)$ here is unitary group of $\mathbb{C}^n$, $S(r,n)$ is the Stiefel manifold of $r$-frames in $\mathbb{C}^n$ and $G(r,n)$ is the Grassmannian manifold of $r$-planes in $\mathbb{C}^n$. I've tried a ...
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2answers
103 views

Homotopy equivalence between $X/A$ and $X$?

Consider the following definition: Definition: Let $(X, A)$ be a topological pair. We say $A$ has the homotopy extension property with respect to a space $Y$ if given any continuous map ...
0
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0answers
36 views

existance of loop with finitely many point of intersection

for every loop on compact orientable surface exists freely homotopic loop with finitely many points of intersection. I see that it have to be true, but I can't prove it. I know Thom's theorem, Sard's ...
1
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0answers
197 views

Homeomorphic or Homotopic

Q1: Are the Fig (a) and (b), the equivalence "=" is Homeomorphic or Homotopic? ps. for details of the figures see Ref here. I learned that "The characterization of a homeomorphism often ...
5
votes
1answer
101 views

Why are the total spaces of two Serre fibrations equivalent when the bases and the fibers are equivalent?

Suppose $B$ is a pointed space and suppose $f\colon E\to B$ and $f\colon E'\to B$ are two Serre fibrations. Let moreover a map $g\colon E\to E'$ be given such that $f=f'\circ g$ which is a weak ...
8
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1answer
517 views

Easier proof about suspension of a manifold

For what manifolds $M$ is the suspension $\Sigma M$ also a manifold? By the suspension of a topological space $X$ (not necessarily a manifold), I mean the space $$\Sigma X = (X \times [0,1])/{\sim}$$ ...
2
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0answers
38 views

Contractible as an Unbased Space but Not Contractible as a Based Space

An unbased space $X$ is contractible if $id_X$ is homotopic to a constant function, that is, any function which carries all of $X$ to single point. Is there an unbased contractible space $X$ such ...
1
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1answer
70 views

Contractible vs. Contractible in a space

I am reading Introduction to Homotopy Theory by Arkowitz Martin and on page 9 it reads: More generally, if $A$ is a subset of $X$ with inclusion map $i : A \to X$; then $A$ is contractible in $X$ ...
4
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1answer
208 views

Is a Simply Connected Space Homotopically equivalent to a point

If X is a simply connected Space is it homotopically equivalent to a point in the space, I know this holds in $ \mathbb{R}^n $ but only because of its algebraic properties does it hold for general ...
4
votes
1answer
84 views

Topology of space of symmetric matrices with fixed number of positive and negative eigenvalues

Let $M$ be real non-singular symmetric $n \times n$ matrix with $p$ positive and $n-p$ negative eigenvalues. What is the topology of the space of such matrices? For a trivial case $n=1$ the matrix is ...
1
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1answer
42 views

homotopical equivalence of projective real space less a line

let $r$ a projective line of the projective real space. How can i prove that $\mathbb{P} ^3(\mathbb{R}) - r$ is homotopical equivalent to $S^1$?
2
votes
2answers
77 views

Foundamental group of $n+1$ spheres in $\mathbb{R}^{n+1}$ that touch two by two

How can i calculate the foundamental group of three $S^2$ in $\mathbb{R}^3$ that touches two by two in one point (if you take any two spheres, they touch only in one point) ? I know that is ...
2
votes
1answer
50 views

Elementary proofs of $\pi_k(S^n)=0$ for $1\leq k<n$.

Is there an elementary proof of the triviality of the first homotopy groups of spheres (i.e. the statement that for $1\leq k<n,\;\pi_k(S^n)=0$)? By elementary I mean without using the tool of ...
1
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2answers
79 views

The center of the fundamental group of closed surface [duplicate]

$S^g$ is a closed surface with genus $g$, we know that the fundamental group $\pi_1(S^g)=\{a_1,a_2,\dots ,a_g,b_1,\dots,b_g|a_1b_1a_1^{-1}b_1^{-1}\dots a_gb_ga_g^{-1}b_g^{-1}=1\}$, how to calculate ...
5
votes
1answer
198 views

Homotopy quantum field theories as functors

A homotopy quantum field theory is a symmetric monoidal functor $\tau:\mathrm{HCobord}(n,X)\to\mathrm{Vect}_{\mathbb{K}}$, with $X$ a path connected space with basepoint $\ast$. There is the following ...
2
votes
1answer
126 views

Higher homology group of Eilenberg-Maclane space is trivial

I'm trying to solve the following exercise from Algebraic Topology by Hatcher (self-study): Show that $ H_{n+1}(K(G,n);\mathbb{Z}) = 0 $ if $ n > 1 $. $ K(G,n) $ is the Eilenberg-Maclane ...
2
votes
1answer
93 views

Homotopy group of Lens space minus point

I'm trying to solve the following exercise from Algebraic Topology by Hatcher, self-study. Let $ X $ be obtained from a lens space of dimension $ 2n+1 $ by deleting a point. Compute $ \pi_{2n}(X) ...
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votes
1answer
68 views

Homotopic maps between spheres

I have read somewhere that two maps $f,g:S^n\rightarrow S^n$ satisfying $$ |f(x)-g(x)|<2 \qquad \forall \ x\in S^n $$ are homotopic. How can one show this (or does someone have a reference)? I ...
3
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0answers
60 views

An equality of some equalizers of simplicial sets.

$\newcommand{\cosimp}[1]{{#1}^\bullet} \newcommand{\sSet}{\mathsf{sSet}} \newcommand{\Set}{\mathsf{Set}}$ [I realize that the post is imposing.However, all the content of the problem is in the gray ...
3
votes
1answer
155 views

The fundamental group of $U(n)/O(n)$

Since $O(n)$ is a subgroup of $U(n)$, we can consider the quotient space $L_n=U(n)/O(n)$. The quotient space is homotopic to the ($n$-th) Lagrangian Grassmannian, and it is known that its fundamental ...
3
votes
2answers
136 views

the mapping class group of the disk is trivial proof

Proof : Identify $D^2$ with the closed unit disk in $\mathbb{R}^2$. Let $\phi : D^2 \rightarrow D^2$ be a homeomorphism with $\phi_{\partial D^2}$ equal to the identity. We define, $F(x,t) = ...
1
vote
1answer
68 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
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1answer
46 views

Basic localizers contain adjoint functors

I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences. Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category ...
1
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1answer
64 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
44 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 ...
2
votes
1answer
85 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 $h\circ f$ ...
2
votes
0answers
61 views

Combinatorial definition of the homotopy groups of a quasi category?

The homotopy groups of a Kan complex are defined combinatorially, and they coincide with the homotopy groups of the geometric realization. For a general complex $X$, homotopy isn't an equivalence ...
0
votes
1answer
38 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.
1
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1answer
45 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 ...
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0answers
21 views

Action of Homeomorphisms on Proper Arc system.

Let $S_{g,n}$ be a surface of genus $g$ and with $n$ punctures. By an essential arc we mean an embeded arc (end points are in punctures) which is: Homotopically non-trivial i.e. not homotopic to a ...
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votes
3answers
300 views

Homotopy on the unit circle

I am trying understand why the identity function on the unit circle $X=\{(x,y): x^2+y^2=1\}$ is not homotopic to $f: X \to X$ where $f(z)=(1,0)$ for all $z\in X$.
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1answer
69 views

problems with proving that f and g are homotopic.

i need to give an example of 2 continuous functions $f,g: X \rightarrow Y$ which are not homotopic, with: $X = [0,1] \times [0,1]$ and $Y = [0,1] \cup [2,3]$ and i need to show how many homotopical ...
1
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1answer
73 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 ...
0
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1answer
63 views

Showing two things are homotopic to each other

I want to show that $\mathbb{C} - \{0\} \simeq S^1$ and the unit square is $\simeq S^1$ where $\simeq$ is homotopic in this case. In other words I want to find an equation for each sort of speak that ...
2
votes
1answer
79 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
102 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 ...
3
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0answers
97 views

Geometric realization of function complexes of simplicial sets

Let $\operatorname{sSet}$ be the category of simplicial sets and $\operatorname{Top}$ the category of (say, compactly generated) topological spaces. There is a pair of adjoint functors called ...
0
votes
1answer
65 views

Fundamental Group equaling 0

Let $X$ be a space for which $\pi(X,x)=0$. If $f,g$ are two paths in $X$ with $f(0)=g(0)=x$ and $f(1)=g(1)$, why is $f$ equivalent to $g$?
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2answers
55 views

Proving homotopy of paths

Let $f$ be a path in $X$ and $h:[0,1] \mapsto [0,1]$ a continuous mapping with $h(0)=0$ and $h(1)=1$. How can I prove that $f$ and $fh$ are homotopic relative to the endpoints?
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1answer
45 views

Can one drop the retracts in the definition of anodyne extensions?

Definition: Anodyne extensions(i.e. acyclic cofibration) of simplicial sets are the closure of horn inclusions under transfinite composition, pushouts, and retracts. The composition and pushouts can ...
0
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1answer
67 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 ...