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 ...

learn more… | top users | synonyms

0
votes
0answers
25 views

Homotopy sets for a pushout of spaces (Seifert-van-Kampen?)

my problem is the following: I have two bordisms $M : \Sigma_0 \to \Sigma_1$ and $M' : \Sigma_1 \to \Sigma_2$, so I can glue them along $\Sigma_1$ to get $M'\circ M$. The manifold $M'\circ M$ is the ...
0
votes
1answer
50 views

Homotopic type of $GL^+(n)$, $SL(n)$ and $SO(n)$

Question: Consider $GL^+(n) \supset SL(n) \supset SO(n)$ the groups of matrices $n \times n$ with positive determinant, determinant $1$ and orthogonal with positive determinant, respectively. Show ...
5
votes
0answers
194 views

Homotopy of space of immersions, Smale-Hirsch theorem

If $M$ and $N$ are simply connected manifolds with $\dim M< \dim N$, we denote by $Imm\left(M,N\right)$ the space of immersions of $M$ in $N$. Let $M$ and $M'$ manifolds of dimensions $m>0$. ...
0
votes
1answer
49 views

Do homotopy equivalences operate over discrete spaces?

My understanding is limited and I'm trying to learn more about how homotopy forms the notion of equivalence. I can grasp its definition as "continuous", but my understanding of homotopy falls away in ...
0
votes
0answers
15 views

Generalize equivalence of simply connected stated for subspaces of the complex plane

In complex theory we have the following proposition: Let $A \subset \mathbb{C}$ . Then A is simply connected (in the topological sense, i.e., that it's path connected and fundamental group is trivial)...
3
votes
2answers
53 views

Existence of a (n-1)-connected map beween CW-spaces

I have two finite CW-spaces $K$ and $L$ (K is n-dimensional and L is (n-1)-dimensional), a topological space $X$ and two maps $\phi:K\to X$ and $\psi: L\to X$, while $\phi$ is n-connected and $\psi$ ...
2
votes
0answers
20 views

Homotopy continuations for solving systems of equations over a finite field

A way of solving systems of polynomial equations over $\mathbb{R}$ or $\mathbb{C}$ is using homotopy continuation. Roughly speaking this method uses a homotopy that starts from some system of ...
1
vote
2answers
61 views

Does $f^{\ast}$ homotopic to $g^{\ast}$ imply $\int f^{\ast} w = \int g^{\ast} w$?

Let $f,g: M^{k} \to N$ ($M$ and $N$ with out boundary ) such that they are homotopic then for $\omega$ a $k$-form on $N$ do we have that $$ \int_M f^{\ast} \omega = \int_M g^{\ast} \omega$$ as ...
1
vote
1answer
65 views

Why does this have to be $f(0)=g(0)$?

For the problem, I am not given any solution so no idea Prove that any two continuous maps $f,g; I \to X$ such that $$f(0)=g(0) \in X$$ are homotopic where $I=[0,1]$ is the unit line. ...
4
votes
1answer
49 views

What do paths have anything to do with homotopy equivalence?

I don't understand how to solve this problem, it seems disconnected from the definition of homootpy equivalence Let $X,Y$ be spaces with the underyling set $\{a,b\}$ for both but $\tau_=\{\phi,\{a,...
3
votes
1answer
25 views

Contradictory; Homotopy equivalence and deformation retract problem

The question Show that $S^1$ is a deformation retract og $D^2\setminus\{(0,0)\}$ the unit punctured disc. The solution the inclusion map $i:S^2 \to D^2\setminus\{(0,0)\}$ and $$j:D^2\...
0
votes
0answers
34 views

I don't understand what a Fundamental group is

I have been staring at the definition for days, drew diagrams but I don't understand as to what its elements are The fundamental group $\pi_1(X,x)$ at a base point $x$ is a set of rel $\{0,1\}$ ...
1
vote
2answers
14 views

Continuous deformation of loop to point.

Suppose I have a homotopy from a loop around the origin to a constant loop which is not the origin. Prove that the origin is in the image of the homotopy. Basically prove that if I deform a loop to ...
0
votes
1answer
35 views

“Reduction to finite case” arguments in algebraic topology

Hello I was studying the corollary to the excision property in Homotopy theory (Hatcher 4K.2) and the thing I can't understand is why the injectivity argument works when moving from an infinite ...
1
vote
0answers
27 views

Lifting paths in a fibration in families

Given a fibration (or a fiber bundle or whatever might make my question true) $f\colon E\rightarrow B$ and basepoints $e$ in $E$ and $b$ in $B$, such that $f$ preserves them. By the homotopy lifting ...
0
votes
1answer
24 views

Mind numbing homotopies; why is this a homotopy rel?

Proposition. every map $\alpha: [s_0,s_1] \to \mathbb{R}^n$ is homotopic rel $\{s_0,s_1\}$ to the linear map $$\beta=\frac{(s_1-s_0)\alpha(s_0)+(s-s_0)\alpha(s_1)}{s_1-s_0}:[s_0,s_1] \to \mathbb{...
2
votes
0answers
31 views

Smoothing a continuous section in a fibre bundle.

Let $\pi: X \rightarrow M$ be a smooth fibre bundle and let $p^{1}_{0} : X^{(1)} \rightarrow X$ be its 1-jet bundle. Suppose there is a $\mathcal{C}^{1}$ section $h: M \rightarrow X$ such that it is ...
0
votes
2answers
35 views

$f$ is null homotopy if and only if $f_*$ is the trivial induced homomorphism

I want to show that $f:S^1\to S^1$ is homotopic to the constant map if and only if $f_*:\pi_1(S^1,x_0)\to \pi_1(S^1,x)$ , $f_*([\gamma])=0$ This seems like it should be an obvious fact but I am having ...
0
votes
0answers
14 views

Abelianization in relative Hurewicz theorem for a non-simply connected subspace

I have found two (probably equivalent) versions of relative Hurewicz theorem. The first one is from Hatcher, and the second one is from a lecture note of MIT (page 9 of this link). $(\pi_n)_{ab}$ ...
0
votes
1answer
49 views

Showing $\mathbb{R}^3$ minus $n$ parallel lines is homotopic to $\mathbb{R}^2\setminus\{p_1,\dots,p_n\}$

I want to show that if I remove $n$ parallel lines from $\mathbb{R}^3$ then I get $\mathbb{R}^2\setminus \{p_1,\dots,p_n\}.$ There is also some underlying structure I wish to also understand. That is,...
1
vote
1answer
61 views

Reference/Definition of Homotopy in an Abstract Category

Let $\mathscr{C}$ be a complete and cocomplete category, and let $W$ be the collection of weak equivalences relative to some model on $\mathscr{C}$. We can form the homotopy category by localizing at ...
5
votes
0answers
37 views

Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
0
votes
1answer
74 views

Unit sphere without a point is contractible

Let $a$ be a point on the unit sphere $S=\{(x,y,z)|x^2+y^2+z^2=1\}$. How do I show that $S\backslash\{a\}$ is contractible? How do I show that a non-surjective loop $\phi\in P(S,s)$ with ...
0
votes
0answers
5 views

Proof of decomposition of homotopy into elementary decompositions?

In my Complex Analysis notes, the following lemma is stated without proof: If $G$ is an open connected domain, and $C$ and $C'$ are homotopic in $G$, then the homotopy can be decomposed into a finite ...
4
votes
0answers
30 views

$\mathbb{CP}^2$ realizable as boundary of compact smooth $5$-manifold? [duplicate]

As the title says, can $\mathbb{CP}^2$ be realized as the boundary of a compact smooth $5$-manifold?
1
vote
1answer
16 views

Enriching categories of simplicial objects

Let $C$ be a cocomplete category and $Simp(C)=C^{\triangle^{op}}$ the category of simplicial objects in $C$. I want to show that $Simp(C)$ is simplicially enriched but I don't understand how the ...
2
votes
1answer
39 views

Simplicial homotopy's exponential law

The following was cited from Simplicial Homotopy Theory by John F. Jardine and Paul Gregory Goerss. We have an adjunction $$\hom_{\mathbf{S}}(X\times Y,Z)\simeq \hom_{\mathbf{S}}(Y,\mathbf{Hom}(X,Z))$$...
0
votes
0answers
24 views

Is this game explained with knot theory or with homotopy theory? Or both?

The question is stated here. Obviously there exists an homotopy from the twist to the 'normal' circle because we are in $\mathbb{R}^3$, but I don't think there is always a solution to the game because ...
3
votes
0answers
46 views

Obstruction for extending a map along a CW-inclusion: sufficient conditions?

I'm looking for a clarification of the highlighted comment taken from "A User's Guide to Algebraic Topology" by Dodson & Parker. In order to make the setting clear, I uploaded definition $8.1.6$ ...
1
vote
1answer
76 views

Hatcher's proof of the van Kampen Theorem (injectivity of $\Phi$ – unique factorizations of $[f]$)

I am trying to understand the details of Allen Hatcher's proof of the Seifert–van Kampen theorem (page 44-6 of Algebraic Topology). My question is regarding the same part of the proof mentioned in ...
0
votes
0answers
58 views

Homotopy between $x\mapsto x$ and $x\mapsto x/|x|$

1 Let $a,b:\mathbb{C}^\times\rightarrow\mathbb{C}^\times$ with $$a(x)=x,$$ $$b(x)=x/|x|.$$ 2 Let $c,d:\mathbb{C}^\times\rightarrow\mathbb{C}^\times$ with $$c(x)=\bar{x},$$ $$d(x)=1/x.$$ 1 How do I ...
0
votes
0answers
34 views

Two self maps $f,g:S^n\to S^n$ are homotopic if there is no $x\in S^n$ with $f(x)=-g(x)$

What I want to prove: Let $f,g:S^n\to S^n$ be continuous. If there is no point $x\in S^n$ with $f(x)=-g(x)$ then $f\sim g$, then . I am using this as a lemma to prove a slightly bigger result: ...
4
votes
1answer
38 views

Which surface is homotopy equivalent to $\Bbb{R}^4$ minus the planes $x=y=0$, $z=w=0$?

In completing an exercise I have shown that $\Bbb{R}^3$ minus the axes $x=0$, $y=0$, and $z=0$ is homotopic equivalent to the cube graph $Q_3$. To visualize this, $\Bbb{R}^3-0$ is homotopy equivalent ...
0
votes
1answer
34 views

Continuous functions between topological spaces and their homotopy equivalence relations

Let $A,B,C$ be topological spaces and $\alpha,\alpha':A\rightarrow B$ continuous and $\beta,\beta':B\rightarrow C$ be continuous. Let $\sim$ be the homotopy relation (which I know/can use to be an ...
1
vote
1answer
48 views

Every “star-shaped” set is simply connected

This is based off of a question in Serge Lang's Complex Analysis book, though much harder than the version of the question in the text. Call a set $S\subseteq\Bbb{C}$ star-shaped if there exists a ...
1
vote
0answers
34 views

Which of these graphs are homotopic but not homeomorphic?

I am struggling with the 'homotopic' part of the question Which of these are homotopic but not homeomorphic? The number of vertices of degree $\neq 2$ is a topological invariant, thus is true ...
1
vote
0answers
25 views

Do $G$-spaces with equivalent orbit categories also have equivalent fundamental categories?

I have heard it mentioned before that $G$-spaces which have equivalent orbit categories must then have equivalent fundamental categories (sometimes called the equivariant fundamental groupoid). This ...
0
votes
1answer
32 views

Homology of contractible space

I understand that if $f,g: X \to Y$ are maps and $f$ is homotopic to $g$, then the induced maps on the homology groups $f_*$ and $g_*$ are equal. Why does this imply that if $X$ is contractible then $...
1
vote
0answers
11 views

Homeomorhism of pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})$

I'm having trouble following an argument in Bredon's Topology and Geometry. He says that the pairs $(D^n \times I,\mathbb{S}^{n-1} \times I \cup D^n \times \{0\})$ and $(D^n \times I,D^n \times \{0\})...
0
votes
0answers
15 views

Properties of free suspensions and free cones

In the category of based topological spaces, suspension $- \wedge S^1$ is adjoint to loop spaces $\text{Hom}(S^1,-)$ and the based cone $- \wedge I$ (where $I$ has zero as basepoint) is adjoint to the ...
2
votes
1answer
28 views

The significance of CW-complexes in homotopy theory

I try to understand the significance of CW-complexes in homotopy theory, in particular with respect to the classical models structure on $\mathbf{Top}$. Why do we chose Serre cofibrations for the ...
0
votes
0answers
59 views

Is this octogon topologically equivalent to the Klein Bottle?

Note: this is an extension of a previous problem (identify the topological type obtained by gluing sides of the hexagon ) where a hexagon was considered. Is the space below also a Klein bottle ($...
4
votes
1answer
46 views

identify the topological type obtained by gluing sides of the hexagon

Identify the topological type obtained by gluing sides of the hexagon as shown in the picture below Clearly the boundary is encoded by the word $abcb^{-1}a^{-1}c$ I do not understand how the ...
0
votes
1answer
35 views

configuration spaces in mathematics and in physics

On the Wekipedia website Configuration space , there are two configuration spaces defined. One is Configuration spaces in physics, the other is Configuration spaces in mathematics. Question. Do ...
2
votes
1answer
112 views

Badly explained solution

My algebraic topology class is very bad at teaching, it just doesn't explain what's needed. Let me be specific, I am looking at this question, Q. Find the degree of $f_0 :S^1 \to S^1$ the constant ...
2
votes
1answer
36 views

Infinite wedge sum of circles and a space homotopy equivalent

Given a space $G = \bigcup_{n=1}^{\infty} A_n$ where $ A_n $ is a circle $ C[ (n,0),n] \in R^{2}$ I'd like to show that its fundamental group is an infinately generated free group. So let's say $a_i $ ...
2
votes
0answers
37 views

stable homotopy groups of the projective plane

The projective plane $\mathbb{R}P^2$ is obtained by attaching a $2$-cell to $S^1$ via a degree $2$ map $$ S^1\overset{[2]}{\longrightarrow}S^1\longrightarrow\mathbb{R}P^2. $$ Question 1: the ...
2
votes
2answers
59 views

Why are bordism groups of a point nontrivial

My definitions of Bordism are from Tom Dieck's Algebraic topology book. Briefly, a singular manifold, $M \xrightarrow{f} pt $, for a closed smooth oriented manifold $(M, \omega)$ without boundary is ...
5
votes
0answers
30 views

The dimension of $H_i(X_n, \mathbb{Q})$ is linear in $n$ with a bounded nonnegative error, where the error is periodic?

Let $X$ be a finite simplicial complex with a simplicial map to $S^1$. Take covers of $X$ associated to the subgroups $n \mathbb{Z}$ of $\mathbb{Z} = \pi_1(S^1)$. This defines a sequence of spaces $...
0
votes
0answers
7 views

pointed simplicial set as coequalizer

Im studying simplcial sets and homotopy theory. I found this statement that seems quite immediate but for me it is not. Let $X$ be a pointed simplicial set, then $X$ can be realized as the ...