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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2
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1answer
37 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
vote
2answers
53 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 ...
4
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1answer
118 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 ...
1
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1answer
50 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 ...
0
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0answers
23 views

What if a line integral is independent of the function?

This concerns the proof that the $n$ times punctured plane has $\mathbb Z ^n $ as his second homotopy group. We choose for each puncture $a_i$ a loop $\sigma_i$ which circles it once counterclockwise, ...
2
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1answer
68 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) ...
-1
votes
1answer
28 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 ...
2
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0answers
46 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 ...
2
votes
1answer
85 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
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2answers
123 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
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1answer
49 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
1answer
39 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
54 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
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0answers
37 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
75 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 $h\circ f$ ...
1
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0answers
35 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
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1answer
28 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
35 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
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0answers
16 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 ...
0
votes
3answers
86 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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votes
1answer
43 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
63 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
votes
1answer
32 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 ...
1
vote
1answer
42 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
51 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 ...
1
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0answers
32 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
60 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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votes
2answers
44 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
48 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 ...
3
votes
1answer
73 views

Why do we ask that condition in Kan complex?

Let $\{X_n\}_{n=0}^\infty$ be simplicial set with faces $d_i:X_n\to X_{n-1} $, a simplicial set is called a Kan Complex if for any $x_0,...,x_{k-1},x_{k+1},...,x_{n+1}$ if $d_i(x_j)=d_{j-1}(x_i)$ for ...
0
votes
1answer
47 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 $Y$. ...
0
votes
0answers
11 views

Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals?

Proof: The dual of the Homology $(H_{n-k})^{*}$= Homology $H_{n-k}$ over the reals ? So by dual, I mean the linear maps on $H_{k}$. I need this to understand the Poincare duality i.e. $H_{k}\cong ...
1
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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
115 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 ...
3
votes
1answer
60 views

Relating Ext groups of abelian groups and group cohomology

One can define $\mathrm{Ext}$-groups in the category of abelian groups (not $\mathbb{Z}[G]$-modules) and group cohomology in very similar ways. The second, group cohomology, can be computed in the ...
1
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1answer
30 views

Elementry collapses implies same homotopy type.

Let $\Delta$ be a simplicial complex, and suppose that $\sigma \in \Delta$ is a proper face of exactly one maximal simplex $\tau \in \Delta$. A simplicial collapse of $\Delta$ is the removal of all ...
2
votes
0answers
23 views

simplicial commutive rings as algebra objects in a symmetric monoidal category

Do simplicial commutative rings arise as the commutative algebra objects in some symmetric monoidal infinity category? If I understand correctly, I think $E_\infty$ rings are the commutative algebra ...
1
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0answers
22 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
95 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
57 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 ...
0
votes
1answer
38 views

“Homotopy theory” on finite topological spaces?

My question concerns finite sets carrying a not-necessarily-discrete topology. I'm wondering if there's an analogue for homotopy theory where the role of $S^n$ is played by some other, finite set. (My ...
0
votes
1answer
33 views

Homotopy between two homomorphisms and homology

If I have two chain complexes $C$ and $D$ and I suppose that there is a homotopy between $\phi, \psi:C \rightarrow D$ (i.e there is a sequence of homomorphisms $(K_n: C_n\rightarrow D_{n+1})$ such ...
1
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0answers
42 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
176 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 ...
1
vote
1answer
45 views

Eilenberg-Mac Lane and classifying spaces

What can we say about An Eilenberg-Mac Lane space $K(G,n)$ is a classifying space $BG$. When it could be true? For what kind of $G$? For what values of $n$? References are welcomed.
2
votes
2answers
90 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
170 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
44 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} ...
0
votes
1answer
35 views

Is there any standard terminology for the quotient of a topological group by the connected component of the identity?

If $G$ is any topological group, then the connected component of its identity is a closed normal subgroup $H$. It follows that $G/H$ is a totally disconnected topological group. Often, $G$ will be ...
2
votes
1answer
58 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 $n$ ...