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

4
votes
1answer
49 views

Showing that $\mathbb S^1$ is a deformation retract of the Mobius strip, rigorously.

Intuitively, I can see why this is. I've found a few threads about this, but they only provide, for example, a deformation retraction of $I \times I$ to its diagonal $D = \{ (x,x) \in I \times I \}$, ...
2
votes
0answers
33 views

Convergence of continuation scheme for fixed-point via homotopy

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be a non-expansive map, i.e. $$\|f(x) - f(y)\| \leq \|x - y\|$$ for all $x,y\in\mathbb{R}^n$. Further, assume $f$ has at least one fixed-point $x^\star$. ...
5
votes
1answer
77 views

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$

Prove that the Torus is not homotopy equivalent to $S^1\vee S^1\vee S^2$. I need to show that a homotopy equivalence between them doesn't exist, but it seems like the homology groups of the spaces ...
4
votes
1answer
43 views

isotopy equivalence between manifolds

The definition below is from Encyclopaedia of Mathematics: Volume 6. Question: For any $n\geq 1$, is the $n$-dimensional closed cube $$[0,1]^n=[0,1]\times [0,1]\times\cdots \times[0,1]$$ isotopy ...
4
votes
0answers
48 views

Composition, fibrations?

Let $p: D \to B$ and $q: E \to B$ be fibrations and let $f: D \to E$ be a map such that $q \circ f = p$. Suppose that $f$ is a homotopy equivalence. My question is, does it follow that $f$ is a fiber ...
1
vote
1answer
36 views

Deleting a contractible subspace is the same as deleting a point

Let $X$ be a topological space and $A$ and $B$ are subspaces of $X$. Suppose that $A$ is contractible. I know that taking the quotient does not affect the homotopy type, that is $X/A$ is homotopy ...
0
votes
0answers
33 views

equivalent characterizations of manifolds such that configuration spaces are homotopy equivalent

Let $M$ and $N$ be manifolds. If $M$ is homeomorphic to $N$, then the $k$-th configuration spaces $F(M,k)$ is homeomorphic to $F(N,k)$. If $M$ is homotopy equivalent to $N$, then the $k$-th ...
2
votes
0answers
87 views

Deformation retract of a triangle

Let $X \subset \mathbb{R}^2$ be a triangle equipped with the topology induced by the euclidean topology on $\mathbb{R}^2$ and let $Y \subset X$ be the subset made of two sides of the triangle. I need ...
0
votes
0answers
47 views

$\Bbb R^5$ without two circles and line

How to prove that $\Bbb R^5$ without two $S^1$ and one $\Bbb R^1$ is homotopiclly equivalent to wedge of $(\Bbb R^5\backslash \Bbb R^1)$ and two copies of $(\Bbb R^5\backslash S^1)$? The only idea ...
3
votes
0answers
44 views

Number of path components of a function space

Let $X,Y$ be compact topological spaces. $Map(X,Y)$ is the set of continuous functions from $X$ to $Y$ with the compact-open topology (but any reasonable topology should do, am I wrong?). What ...
1
vote
2answers
43 views

Pullbacks and homotopy equivalences

Say I have a map between pullback squares $(Y \rightarrow Z \leftarrow X) \to (Y' \rightarrow Z' \leftarrow X')$. If the maps $X \to X'$, $Y \to Y'$ and $Z \to Z'$ are homotopy equivalences, does it ...
0
votes
0answers
27 views

Is the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ isomorphic to $\frac{SO(3)}{H}$?

I have heard many times that the homotopy group of the coset space $\frac{SO(3)\times Z_2}{H \times Z_2}$ and of the space $\frac{SO(3)}{H}$ are identical. I.e., $\frac{SO(3) \times Z_2}{H \times Z_2} ...
1
vote
2answers
53 views

Action of the fundamental group

Suppose that $M$ is a smooth manifold. Is it true that the fundamental group $\pi_1(M)$ always acts on $M$? If so, how this action is defined? EDIT: Of course I want my action to be nontrivial, say ...
2
votes
1answer
39 views

Higher homotopy groups of wedge of circles.

Using van-kampen theorem, Fundamental group of wedge of n-circles is free group on n-generator. But I don't know how to calculate higher homotopy groups of wedge of spaces, in particular circles. I ...
4
votes
1answer
50 views

Fiber bundle with null-homotopic fiber inclusion

It is an exercise from Hatcher (exercise 31, page 392): For a fiber bundle $F \to E \xrightarrow{p} B$ such that the inclusion $F \hookrightarrow E$ is homotopic to a constant map, show that the long ...
4
votes
1answer
85 views

Fundamental group of open subsets of $\mathbb{R}^n$

Suppose that $U$ is an open subset of $\mathbb{R}^n$. What can be said about its fundamental group? I'm sure that the answer should be well known, since this is rather natural question.
1
vote
0answers
42 views

special kinds of homotopies

Let $X$ and $Y$ be two homotopy-equivalent topological spaces. That is, there exists maps $f:X \to Y$ and $g:Y \to X$ such that $g \circ f \simeq 1_X$ and $f \circ g \simeq 1_Y$. $1_X$ and $1_Y$ are ...
1
vote
0answers
34 views

Show that there exist no retraction from $RP^n$ to $RP^k$ if n>k.

I am trying this problem from Hatcher's algebraic topology book(4.2.1). If r:X$\rightarrow$A is retraction then I know that this induces injective map in the fundamental group level through inclusion ...
2
votes
1answer
47 views

Zero in the Grothedieck group of the derived category

I have a problem. I was wondering whether there is a precise answer to the following question. Let $\mathcal{A}$ be an abelian category and $\mathcal{D}^b(\mathcal{A})$ its bounded derived category. ...
0
votes
0answers
25 views

how to calculate relative homotopy groups?

I am studying nth relative homotopy groups from Hather.For a pair (X,A) where A$\subset$X nth-relative homotopy groups is defined by homotopy class of maps$(I^n,\delta I^n,J^{n-1})$ ...
2
votes
1answer
35 views

Equivalence between derived categories preserve distinguished triangles

I have a problem: Is it true that every equivalence between derived categories preserve their distinguished triangles? Thanks very much!
0
votes
2answers
52 views

Let $ \gamma $ be the unit circle then $ \int_\gamma \frac {dz}{z^2 − 2z} = -\pi i$

Definition: If $ f $ is holomorpic in $G$ and gamma $\gamma$ is $G$-homotophic to a point then gamma is G-contratible and if gamma is G-contractible then $ \int_\gamma f = 0 $. By splitting the ...
2
votes
0answers
31 views

definition of homology via spectra

Let $K(\mathbb{Z}, n)$ denote a Eilenberg-Mac Lane space, characterized by $H^n(X, \mathbb{Z})=[X, K(\mathbb{Z}, n)]$ for all spaces $X$. In stable homotopy theory, the corresponding homology theory, ...
1
vote
1answer
23 views

configuration-spaces and iterated loop-spaces

In the paper Configuration-Spaces and Iterated Loop-Spaces. Graeme Segal, page 213-214, it is obtained that the labelled configuration space $C_n$ is homotopy equivalent to a topological monoid ...
5
votes
2answers
73 views

actions of $\mathbb{Z}_2$ on spheres

Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. Why $F(S^m,2)/\mathbb{Z}_2$ is ...
0
votes
0answers
72 views

Precisely what is meant by “$\pi_1(M)$ is torsion”?

I am reading a paper where one of the conditions for a Theorem to hold is "the group $\pi_1(M)$ is torsion", where here $M$ is a compact differentiable manifold. What is meant by the first homotopy ...
2
votes
0answers
59 views

Homotopic maps between connected spaces inducing the same homomorphism between the fundamental groups

This is Problem 7-9 in Lee's Introduction to Topological Manifolds: Suppose $X$ and $Y$ are connected topological spaces, and the fundamental group of $Y$ is abelian. Show that if $F,G: X ...
3
votes
0answers
36 views

Is wedge sum for finite CW complexes cancellative in the homotopy category?

Let $X,Y,Z$ be finite pointed CW complexes. Is it possible that $X\vee Z$ and $Y\vee Z$ are homotopy equivalent, but $X$ and $Y$ are not? Remark 1: Without the finiteness assumption on $Z$, there are ...
1
vote
1answer
33 views

Definition of homotopy of slope fields

I can't come up with a correct definition of homotopic slope fields (on $\mathbb{R}^2$). Idea is clear - almost the same as vector field homotopy, but problem with defining slope as a function (case ...
1
vote
1answer
31 views

The covering map lifting property for simply connected, locally connected spaces

I wish to prove the following statement: Let $X$ be a simply connected and locally connected space, and let $p:Y\to Z$ be a covering map. Then given $f:X\to Z$ continuous, $x_0\in X$, $y_0\in Y$ ...
9
votes
0answers
98 views

Is a pathwise-continuous function continuous?

Suppose that $X$ is a locally connected and simply connected space and $f:X\to Y$ is a function such that for every path $\phi:[a,b]\to X$ the composition $f\circ\phi$ is continuous. Does it follow ...
0
votes
2answers
42 views

A trivial fundamental group

I am reading fundamental groups from Munkres book. As stated in the definition, I understand a fundamental group relative to a base point $x_0$ includes all the loops based at point $x_0$. Later ...
1
vote
0answers
22 views

How do the direct and inverse image sheaf functors interact with homotopy?

The direct image sheaf functor $f_\ast$ and inverse image sheaf functor $f^\ast$ (here I mean the usual inverse image sheaf functor often denoted by $f^{-1}$) form a well-known adjunction for ...
0
votes
0answers
41 views

Extend vector fields from several $S^1$ to $D^2$

Let's take a disk $D^2 \subset \mathbb{R}^2$ with $n$ holes ($n = 0, 1, ...$). In case $n = 0, 1$ it's clear how to extend any non-zero (i.e. with no singular points) vector field from $S^1$ to disk ...
6
votes
1answer
195 views

Map between $SO(n)$ is homotopic to the identity?

I'm given an exercise, in a differential geometry class, where I need to detemine wether or not the smooth map between manifolds: \begin{align} f \colon\ &SO(n) \rightarrow SO(n)\\ & A \mapsto ...
1
vote
1answer
32 views

Prove (in the example) that being homotopic depends on the range of the Homotopy

Question: Define $F : [0,1]\times [0,1] \rightarrow X$ by $F(x, t) = (cos(\pi x), (1 - 2t) sin(\pi x))$. Take a straight-line homotopy between $F(x, 0)$ and $F(x, 1)$. Show that they are ...
7
votes
1answer
90 views

General relationship between braid groups and mapping class groups

I just finished correcting my answer on visualizing braid groups as fundamental groups of configuration spaces, and in the process became interested in the other pictorial definition of the braid ...
6
votes
1answer
84 views

For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic

PROBLEM: For any smooth n-manifold $M$, construct a smooth map $f:M\to S^n$ which is not null-homotopic ,or even of degree 1 The following is my idea: First, choosing an arbitrary open coordinate ...
1
vote
0answers
31 views

natural map to the homotopy fibre

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 280, paragraph 4, line 2-line 3 and Configuration spaces of positive and negative particles, McDuff, page 105, line ...
0
votes
1answer
24 views

action of a monoid on a mapping telescope

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281, line 14-line 15: For a topological monoid $M$, if $\pi_0(M)=\{0,1,2,3,......\}$, then the action of $M$ on ...
0
votes
2answers
41 views

Lift of $z^2$ map $S^1 \to S^1$

$\newcommand{\id}{\operatorname{id}}$It can be proved that identity map $\id: S^1 \to S^1$ does not lift to $\widetilde{\id} : S^1 \to \mathbb{R}$ such that $e^{\widetilde{\id(z)}i} = z$. The ...
2
votes
1answer
40 views

homomorphism of $H$-spaces between a monoid and loop space of its classifying space

Let $M$ be a topological monoid. $M$ can be considered as a category internal to topological spaces and has a simplicial space $N_\bullet(M)$ as its nerve. The geometric realization ...
3
votes
2answers
57 views

Homotopy equivalence of $S^{2} \vee S^1$ to $S^{2} \cup A$ where A is a line segment joining noth and south poles

I have some problems trying to show homotopy equivalence of $S^{2} \vee S^1$(one-point union) to $S^{2} \cup A$ where $A$ is a line segment joining north and south poles of a sphere. I understand the ...
6
votes
2answers
98 views

Homotopy equivalences between some sphere-based spaces (quotient of spheres, bouquet of spheres, difference of spheres)

I'd like to prove the following equivalences ($k < n$): $S^n / S^k \sim S^n \vee S^{k+1}$; $S^n \backslash S^k \sim S^{n-k-1}$. Low-dimension cases (e.g. $S^2 / S^0$, $S^2 / S^1$, $S^n / S^1$, ...
2
votes
0answers
60 views

Example of topological spaces with continuous bijections that are not homotopy equivalent

In one of the books on algebraic topology (I don't remember exactly which one) there was an exercise to build an example of two topological spaces having two continuous bijections between them which ...
1
vote
0answers
57 views

the geometric realization of a simplicial set is contractible

Let $M$ be a monoid up to homotopy. The simplicial set $WM$ is defined by setting $$ WM_n=M^{n+1}=\{(g_0, g_1,\cdots,g_n)\mid g_i\in M\} $$ with faces and degeneracies given by \begin{eqnarray*} ...
7
votes
1answer
164 views

Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$

Edit: As there are many comments and an answer already, I have left the original question below. I was unaware that there are different ways one could try to define $\mathsf{hTop}_{\bullet}$, 'the' ...
2
votes
0answers
46 views

Why are chains topologically analoguous to distributions?

This question is related to my other question here but is different enough that I thought I might ask separately. At the nLab page on rational homotopy theory it is stated that chains are ...
2
votes
0answers
33 views

What is the connection between $\widehat{\mathbb Q G}$ and distributions near the identity of $G$?

I'm studying Quillen's rational homotopy theory and trying to understand this MathOverflow description of Quillen's functor provided by Hiro Lee Tanaka. When discussing connections between how ...
2
votes
0answers
22 views

pontrjagin ring of the homology of iterated loop suspension

In The homology of C n+1 spaces, n>=0, F. Cohen, proof of Theorem 3.1 and proof of Theorem 3.2 (p. 228 - 243) I totally do not understand the proofs of these two theorems from page 228 to page 243 ...