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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6
votes
4answers
387 views

The circle is not contractible

I know that the circle is not contractible because I know that $\pi_1(S^1)\cong \mathbb Z$. But something is going wrong in my head. Choose a basepoint $*$ on the circle and chose an orientation (...
2
votes
0answers
11 views

Does the inclusion of $n$-connected $\infty$-groupoids into $(n-1)$-connected $\infty$-groupoids admit adjoints?

Denote the $(\infty ,1)$-category of $n$-connected $\infty$-groupoids by $\operatorname{Grpd}^{\infty}_{n}$. There is clearly a suitable inclusion $\iota:\operatorname{Grpd}^{\infty}_{n}\...
0
votes
2answers
30 views

Homotopy: equivalence relation (continuity of homotopy in symmetry)

Let $f_0, f_1: X \rightarrow Y$ be continous on topological spaces $X,Y$. Let $F$ be a homotopy between $f_0, f_1$. Which argument shows that $G(x,t):=F(x,1-t)$ is continous? Seems to be hard to ...
0
votes
1answer
35 views

$f: S^n \to S^n$ is the restriction of a continuous mapping $F: \overline{B^n_1} \to S^n$ iff $deg(f) = 0$.

this is a theorem from wikipedia (https://en.wikipedia.org/wiki/Degree_of_a_continuous_mapping#Properties): A self-map $f: S^n \to S^n$ of the $n$-sphere is extendable to a (continuous) map $F: ...
0
votes
0answers
16 views

Isomorphisms between images of the Honda formal group law on Morava $K$-theory

Let $L = MU_*$ be the Lazard ring, which represents formal group laws, and let $W = MU_* MU$ be the ring that represents strict isomorphisms between formal group laws. Let $K(n)_* = \mathbb{F}_p[v_n^\...
0
votes
1answer
32 views

$\pi_0(SO(N))$ and $\pi_0(O(N))$: Inconsistency between Bott periodicity and basic understanding of $\pi_0$

I need to know the homotopy groups of the oriented Grassmannian $\widetilde{Gr}(\infty,\infty) \cong \lim_{N \rightarrow \infty} SO(2N)/(SO(N) \times SO(N))$, and I've run into an inconsistency. It ...
3
votes
1answer
62 views

If two maps are homotopic, are the images homotopy equivalent?

My question is; If two continuous maps $f,g:X\rightarrow Y$ are homotopic, are the images $Im(f),Im(g)$ homotopy equivalent? Clearly, the converse is false. If it is false, is there any condition ...
1
vote
0answers
50 views

Homotopy type of some lattices with top and bottom removed

There was an interesting question on MO which OP removed by some reason. Here is a (more or less) equivalent form. Take a finite cartesian product of finite linear orders, and remove top and bottom. ...
2
votes
1answer
69 views

Counter-example for $\tilde{H} (X/A) \cong H (X, A)$?

Yo! I was not able to find a counter-example to $$\tilde{H} (X/A) \cong H (X, A)$$. It's a well known fact that for cofibrations $A \hookrightarrow X$ (or more generally whenever $A$ is a deformation ...
3
votes
0answers
53 views

Why all differentials are $0$ for Serre Spectral Sequence of trivial fibration?

Consider the fibration $F \hookrightarrow F \times B \to B$. I understand that if I take kunneth's theorem for granted that the group extensions $F_{n-i,i} \to F_{n-i+1,i-1} \to F_{n-i+1,i-1}/F_{n-i,...
1
vote
1answer
29 views

Let $f:S^1 \to X$ a continuous function $X$ a topological space. Then $f$ is homotopic to a constant iff $f$ extends to $D$.

Let $X$ a topological space, $D$ a open unitary disc on $\mathbb{R}^2$ and $S^1 = \partial D.$ How to show that $f: S^1 \to X$ continuous is homotopic to a constant map iff there is a continuous ...
1
vote
0answers
24 views

basepoint problem: Is there an action of $\pi_1(B)$ on $\pi_1(F)$ for $F$ path connected

I am doing this to try to figure out The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$ . Let the fibration $F \hookrightarrow E \xrightarrow{p} B$ be ...
2
votes
2answers
45 views

Principal bundle as homotopy fiber universally self-trivializes

In this MO answer, I was told the definition of principal bundle as a homotopy fiber of its classifying map precisely says that it's the universal bundle which trivializes itself. However, I'm having ...
0
votes
0answers
26 views

The action of $\pi_1(BK) \curvearrowright H_*(BG)$ for the fibration $BG \hookrightarrow BH \to BK$

**Note:I was extremely confused when I wrote this post. Please see the linked one. I left this one as it is, because what I understand now is so radically different then what I wrote below ** Let $...
0
votes
1answer
23 views

Let $G$ a simple connected topological group and $H$ a normal discrete subgroup, then $\pi_1(G/H,e) = H.$

I know that $G$ is a covering space for $G/H$ and there is a injection between the fundamental group of $G$ and $G/H.$ How to proceed to show that $\pi_1(G/H,e) = H?$.
1
vote
1answer
43 views

Second homotopy of $S^1\vee S^2 \vee T^2$

How can I prove that the second homotopy group of $S^1\vee S^2 \vee T^2$ is infinitely generated? I know that the second homotopu groups of $S^2$ and $T^2$ are finitely generated, so is kind of ...
2
votes
0answers
29 views

Homotopy of closed curves is also a closed curve?

I'm trying to prove the following statement: Let $\gamma_1$ and $\gamma_2$ be two closed curves from $[a, b]$ to $\mathbb{C}$, and let $h: [a,b]\times[0,1] \to \mathbb{C}$ be a homotopy between ...
3
votes
2answers
47 views

A question on the non trivial rank three bundle on $S^2$

We know that number of rank $3$ vector bundles on $S^2$ is just the number of equivalence classes of maps from $S^1$ to $SO(3)$. This implies that there is one and only one non trivial vector bundle ...
0
votes
0answers
28 views

Homotopy on a cylinder

Given a cylinder $C := \mathbb{R} \times S^1$, the fundamental group is $\pi_1 \cong \mathbb{Z}$. My basic question is: Why? I completely fail to see what the set of non-homotopic loops on the ...
0
votes
1answer
34 views

bottom map of pullback square is cofibration $\Rightarrow$top map is cofibration

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I am trying to show that for a given fibration $E \xrightarrow{p} B$, and a cell structure $\{B_n\}$ on $B$ that , that $p^{-1}B_n \to p^{-1}B_{n+1}$ is ...
0
votes
1answer
32 views

Partial Converse to “Pushout of a cofibration is a cofibration”

$\require{AMScd} \newcommand{\RP}{\mathbb{RP}}$ I want a converse of this fact specialized to the case where I am pushing out a map BY a fibration: I.e., if I am given a diagram $ \begin{CD} E_1 @&...
0
votes
1answer
42 views

Smooth homotopy between $\Bbb R^2-\{0\}$ and $S^1$

In Tu's book "An Introduction to Manifolds" he defines smooth homotopy as follows. $M,N$ smooth manifolds, two $C^\infty$ maps $f,g:M\to N$ are smoothly homotopic if there is a $C^\infty$ map $F:M\...
1
vote
1answer
27 views

Homotopy between homeomorphism and the identity

Let $h\colon [0,1] \to [0,1]$ a homeomorphism, and $I \colon [0,1]\to[0,1]$ the identity. I want show a homotopy $H:h \sim I$. I want show it in order to show that parametrization of a path is ...
0
votes
0answers
30 views

The claim that $A \to X$ a cofibration implies $A \times I \to M_{A \to X}$ is an inclusion.

Akhil Matthew claims in https://amathew.wordpress.com/2010/10/07/cofibrations/ in parenthesis that given a cofibration, $A \xrightarrow{i} X$, the map $A \times I \to M_i$ into the mapping cylinder, ...
3
votes
1answer
38 views

prove that elements in $K_1(A)$ coincide

Let $A$ be a unital $C^*$-algebra $u\in A$ unitary and $s\in A$ isometry. I already proved that $sus^*+(1-ss^*)$ is an unitary. Why is $[u]_1=[sus^*+(1-ss^*)]_1\in K_1(A)$? Basic definitions: ...
1
vote
0answers
46 views

Obstruction to lifting a map from the base space to the total space.

Suppose $\pi :E \to B$ is a fibration with fibre $F$ above a chosen base point. Then I am trying to solve when a map $f$ from a manifold $M$ to $B$ lift to a map $g:M \to E$. The answer given is they ...
2
votes
1answer
77 views

Classifying continuous maps from closed 2-manifolds to various closed manifolds

I believe my question should be simple. The question is more physically oriented and originated from one of Witten's papers, "On Holomorphic Factorization of WZW and Coset Models", where he considered ...
0
votes
1answer
26 views

Is the subgroup of homotopically trivial isometries a closed subgroup of the isometry group?

Let $(M,g)$ be a connected Riemannian manifold. Then according to the Steenrod-Myers-Theorem, the isometry group $\text{Isom}(M,g)$ of $(M,g)$ is a compact lie group with the compact-open topology. ...
2
votes
2answers
52 views

$\mathbb{R}^n - B[0,r]$ is simply connected if $n>2$

Question:$\mathbb{R}^n - B[0,r]$ is simply connected $\iff$ $n>2$. I have to prove or disprove. I know prove that for $n \in \{1,2\}$, $\mathbb{R}^n - B[0,r]$ is not simly connected. So I want ...
1
vote
1answer
13 views

On linear homotopy of operators

Let $F$ be an isomorphism of euclidian space $E$, with orthonormal basis $\{e_1,\ldots e_n\}$. Let $F'$ be orthogonalised $F$. Is any operator $F_t$ from linear homotopy of $F$ and $F'$ an ...
3
votes
1answer
57 views

Topology of non-degenerate $n\times n$ Hermitian matrices

.. where I guess by topology I mean its homology / homotopy groups. Here "degenerate" means having repeated eigenvalues. This is interesting because it defines the space that can be explored via ...
0
votes
1answer
50 views

Triangle inside a simply connected open subset of the complex plane

Let $U$ be a connected open subset of the complex plane. Suppose $U$ is simply connected, i.e. its fundamental group is trivial. Let $T$ be a triangle whose boundary is contained in $U$. It is ...
4
votes
2answers
65 views

Why is $H_5(K(\Bbb Z_n,4))$ finite?

I want to see that the cohomology $H^i(K(\Bbb Z_n,4); \Bbb Z)$ starts with $\Bbb Z_n$ in degree 5. How do we know that $\operatorname{hom}(H_5(K(\Bbb Z_n,4);\Bbb Z), \Bbb Z)$ is zero? I.e. why is $H_5(...
0
votes
0answers
36 views

Finding the boundary of the continuous image of a compact simply-connected Lie group

Statement of the problem Given a continuous map $f:G \rightarrow D^2$ where $G$ is a compact simply connected Lie group and $D^2$ is the unit disk in the plane, I have shown that: There exists a ...
1
vote
0answers
33 views

Prove of homotopy equivalence using differential equation

I have to prove that $R^3 \setminus L_1,...,L_n $, where $L_1,..,L_n$ are non intersecting lines, is homotopy equivalent to a wedge sum of $n$ circles. So once I've managed to show that it is possible ...
0
votes
1answer
35 views

Proving homotopy equivalence of a torus with points removed

Suppose I'd like to show that a Torus with $n$ points removed is homotopy equivalent to a wedge sum of $n+1$ circles. I depict it in a usual way - as a rectangle with $n$ points removed inside. Now it ...
2
votes
0answers
36 views

geometric realization of a simplicial complex

Let $K$ be a simplicial complex and let $|K|$ be the geometric realization of $K$. Suppose that the vertices ($0$-simplices) of $K$ are $v_0,v_1,\cdots,v_k$, $k$ finite. Let $\Delta^n$ be the standard ...
1
vote
1answer
44 views

A deformation retract that is not a strong deformation retract

In Lee's Introduction to Topological Manifolds, problem 7-12 asks to show that $\{(1,0)\}$ is a deformation retract, but not a strong deformation retract of the subspace of the plane $$ X = \bigcup_{...
1
vote
1answer
44 views

Can't see why particular homotopy is continuous

I'm checking the group laws for the fundamental group of $(X,x_{0})$: in particular I'm trying to show that $\gamma \simeq \gamma \cdot e$ , where $\gamma$ is a loop based at $x_{0}$, $e$ is the ...
2
votes
1answer
39 views

Hopf fibration and exact sequence in homotopy

I have encountered the Hopf fibration $S^1\hookrightarrow S^3\twoheadrightarrow S^2$ when studying smooth principal $G$-bundles. Whenever I google the Hopf fibration, I encounter a remark which boils ...
1
vote
1answer
41 views

tom Diecks's proof of $H_1(X)\cong \pi_1(X,x_0)^{ab}$

My question is about tom Dieck's proof of Theorem 9.2.1 on page 227, which states that if $X$ is path connected, then the induced map $$h:\pi_1(X,x_0)^{ab}\to H_1(X)$$ is an isomorphism. Specifically,...
1
vote
2answers
48 views

How to construct a homotopy equivalence between a mobius band and a circle?

A mobius band is homotopic equivalent to a circle because the mobius band can deformation retract onto a circle. I am wondering how could we understand this fact from the definition of being ...
1
vote
0answers
82 views

Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
0
votes
1answer
59 views

How to prove that the Lefschetz number is invariante under homotopy?

How to prove that the Lefschetz number is invariante under homotopy? We define the Lefschetz number as the number of $f : M \to M$ as the number of intersection of the map $g(x) = (x,f(x))$ with the ...
3
votes
0answers
46 views

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then:

Let $\gamma_1, \gamma_2 : I \to M$ two homotopic curves, $M$ a $C^{\infty}$ manifold. If $\omega$ is a closed $1-$form then: $$\int_{\gamma_1(I)} \omega = \int_{\gamma_2(I)}\omega.$$ Then, conclude ...
1
vote
1answer
18 views

How do I see if the induced homomorphism from the inclusion map $S^{1'}\to S^1\times S^3$ is injective

Let $S^1\times S^3$ be parametrised as $\{(\alpha,\beta, \gamma)\in \mathbb{C}^3||\alpha|^2+|\beta|^2=1, |\gamma|=1\}$ and let $S^{1'}=\{e^{i\theta}(1,0,1)|\theta\in [0,2\pi]\}$. I would like to see ...
0
votes
1answer
22 views

Proving weak homotopy equivalence of a map

This is a modification of the question I previously asked here. Consider the category of pointed topological spaces $C$. Suppose objects $a,b,c,d,e,f,g,h \in C$. Suppose we also have the commuting ...
0
votes
1answer
43 views

Which theorem of homotopy theory states that if two objects have different genus then they are not homotopy equivalent?

I'm quite new inexperienced in the field but from what I see two objects with different genus are not homotopy equivalent. Question: Which theorem of homotopy theory states that if two objects have ...
0
votes
0answers
18 views

Isomorphism of Homotopy groups across a filtration

Let $X$ and $Y$ be CW complexes. Let $sk_{\bullet}(X)$ and $sk_{\bullet}(Y)$ denote the canonical skeleta filtrations of $X$ and $Y$, respectively. Suppose that we have isomorphisms on homotopy groups ...
2
votes
2answers
33 views

Compositions of homotopic maps are homotopic

I'm reading some lecture notes on homotopy, and the author has just proved the theorem: If $f_{0} \simeq f_{1}$ and $g_{0} \simeq g_{1}$, then $g_{0} \circ f_{0} \simeq g_{1} \circ f_{1}$ ...