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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63 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
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2answers
30 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 ...
0
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0answers
18 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
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0answers
36 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
174 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
30 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
82 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
62 views
+50

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
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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
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1answer
22 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
37 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
39 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
52 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
94 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
56 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
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0answers
55 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
152 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
45 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
31 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 ...
3
votes
1answer
62 views

Fundamental groupoid of a contractible space

I read that the fundamental groupoid of a contractible space is indiscrete. How can one show this? I found this as an exercise here.
0
votes
0answers
26 views

Winding number of $S^1$ vector fields with $|u| > |v|$

Let $u$ and $v$ are nonvanishing vector fields on $\mathbb{S}^1$ and $|u(z)| > |v(z)|$ at every point of $\mathbb{S}^1$. Prove that $deg(u) = deg(u + v)$. My idea is to take a homotopy $h_t(z) = ...
2
votes
1answer
40 views

question about “Homology fibrations and the group completion theorem”

In the paper Homology fibrations and group completion theorem, McDuff-Segal, page 281 line 17-line 18: we have a fibre bundle $M_\infty\to (M_\infty)_M\to BM$ with $(M_\infty)_M$ constractible. In ...
1
vote
0answers
28 views

Lifting of certain $S^1$ valued maps

It's well-known that every path $s(t): I \to S^1$ has a lifting, i.e. mapping $\widetilde{s(t)}: I \to \mathbb{R}$, so that $e^{i\widetilde{s(t)}} = s(t)$. The main idea of constructing ...
1
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0answers
53 views

homology of mapping telescope of a monoid

Let $M$ be a monoid with multiplication $\cdot$, $\pi_0(M)=\mathbb{N}$, and $m\in M$ in the component $1\in \mathbb{N}$ . We form a mapping telescope $$ M\overset{ {\cdot m}}\longrightarrow M\overset{ ...
2
votes
1answer
85 views

A map $h:S^1\to X$ Induces a Trivial Homomorphism of Fundamental Groups Iff it is Nullhomotopic.

I recently started reading Algebraic Topology from Part II of Munkres' book Topology(Second Edition). A part of Lemma 55.3 in the book proves the following: Let $h:S^1\to X$ be a continuous ...
1
vote
1answer
73 views

The winding number (topological definition) is well-defined

can someone please tell me if my proof of the lemma below is legit. At first I like to give a definition so you know what the lemma is about. Definition: Let $g:[0,1] \to S^{1}$ be a closed path in ...
1
vote
0answers
38 views

configuration space model for classifying space of monoid

Let $M$ be a monoid and $BM$ be its classifying space. There is a model for $BM$ based on labelled configuration spaces of the line $[0,1]$. Points of the configurations are labelled by elements of ...
2
votes
0answers
43 views

Different groups with the same classifying space.

Let $G$ be a topological group and $BG$ its classifying space. From the LES of the universal bundle, we get $\pi_i(BG)\cong\pi_{i-1}(G)$, so given the classifying space, we know all homotopy groups of ...
2
votes
1answer
29 views

explicit equivalent relation in the expression of the classifying space of a monoid

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. (It's also called the internal nerve.) The ...
12
votes
2answers
126 views

Is $\mathbb{R}^n$ properly homotopy equivalent to $\mathbb{R}^m$ if $n \neq m$?

$\DeclareMathOperator{\id}{id} \newcommand{\R}{\mathbb{R}}$ If $f,g : X \to Y$ are two maps (all maps considered are continuous here), a homotopy between $f$ and $g$ is a map $H : [0,1] \times X \to ...
1
vote
1answer
35 views

what means 'the realization of a topological category'

In the paper Homology Fibrations and the "Group-Completion" Theorem. page 280 bottom line 10-bottom line 12, what means 'the realization of a topological category'?
1
vote
1answer
85 views

Homotopy equivalence and chain complexes

This is from the book: Hilton and Stammbach, A Course in Homological Algebra, Chapter IV, Derived Functors, exercise 4.2. Let $\varphi:C \to D$ be a chain map of the projective complex $C$ into ...
0
votes
1answer
135 views

canonical map of a monoid to its classifying space

Every monoid $M$ is a category with one object $M$ and morphisms the elements of $M$. [Martin Brandenburg.] Every small category $C$ has a classifying space $BC$, defined as the geometric realization ...
5
votes
1answer
84 views

Ideas for basic application of homotopy theory to homological algebra?

I'm taking a first course in homological algebra. As a project, the lecturer suggested each student find a topic, presentable in an hour, relating to the material studied in the course. The material ...
0
votes
1answer
40 views

No retraction of $B^{n+1}$ onto $S^n$.

Use the fact that $\text{id}:S^n\to S^n$ is not homotopic to a constant to show that there is no retraction of $B^{n+1}$ onto $S^n$. I tried to look up online but most of the solutions use the ...
1
vote
1answer
76 views

what is the classifying space of a monoid

In the paper Homology Fibrations and the "Group-Completion". Theorem. McDuff, D.; Segal, G., 1976, the first line: A topological monoid $M$ has a classifying space $BM$. I do not understand this ...
1
vote
1answer
48 views

If $(X,A)$ has homotopy extension, then $X \times I$ def. retracts to $X \times \{0\} \cup A \times I$

Exercise 0.26 in Hatcher's Algebraic Topology is Use Corollary 0.20 to show that if $(X,A)$ has the homotopy extension property, then $X \times I$ deformation retracts to $X \times \{0\} \cup A ...
4
votes
1answer
53 views

Homotopy equivalence between finite, discrete topological spaces.

How would one go about proving that if a discrete topological space with m elements is homotopy equivalent to a discrete topological space with n elements, then m=n?
0
votes
2answers
47 views

if two space are homotopy equivalent and one is connected, prove that the other is connected as well

I've tried using the definition of homotopy equivalent spaces which states that X and Y are homotopy equivalent if: There are continuous functions $f:X \rightarrow Y,g:Y \rightarrow X$ such that $f ...
0
votes
1answer
39 views

iterated loop spaces and configuration spaces

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map $$ \phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y) $$ is defined. And a map $$ ...
5
votes
1answer
107 views

Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$

Show that $\mathbb{R}^m$ is not homeomorphic to $\mathbb{R}^n$ if $m\ne n$. You may assume that $S^m$ and $S^n$ are different homotopy type if $m\ne n$. My attempt: Suppose $\mathbb{R}^m$ is ...
2
votes
2answers
60 views

Counting singularities

It is well known that a smooth vector field on a 2-sphere must vanish twice. What is the general technique for counting singularities of a smooth map between manifolds? For example, how many ...
1
vote
1answer
39 views

complex integral of 1/z independent of choice of ellipse?

Can Someone please help me with the following. complex integral of 1/z over an ellipse is independent of choice of ellipse centered at zero. Why is this the case. Is it due to homotopy ...
2
votes
2answers
41 views

Meaning of n-connected pairs

A topological space $X$ is $n$-connected if the homotopy groups $\pi_r(X)$ for $0 \leq r \leq n$ are trivial groups. This means (let's say geometrically), $X$ is $0$-connected if it is non-empty and ...
1
vote
2answers
34 views

Homotopy equivalence between circles

I'm wondering about one thing: let's consider a plane with one hole $ \mathbb{R}^2 \setminus \{0\} $. I'm wondering whether the two subsets: $$ S^1 = \{(x,y) \in \mathbb{R}^2 \setminus \{0\}: x^2 + ...
0
votes
2answers
21 views

Convex Homotopy

Suppose $f , g : X \to U \subset \mathbb R^2$ are two mappings from a topological space $X$ to a convex set $U$. Prove that $f$ and $g$ are homotopic, using only the definition of the product ...
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0answers
34 views

Simplicial function space and homotopy colimits

I am currently reading the book by Bousfield and Kan, in particular Ch. XII, par. 2, and would like to understand why the functor $hocolim: Top_{+}^{I} \rightarrow Top_{+}$ is left adjoint to $hom(I ...
3
votes
1answer
60 views

cup product in cohomology ring of a suspension

Let $X$ be a CW-complex. Let $\Sigma$ be suspension. Let $R$ be a commutative ring. Is the cup product of $$ H^*(\Sigma X;R)$$ trivial? How to prove? Where can I find the result?
2
votes
0answers
53 views

When the free loop space fibration splits?

Let $X$ be a (nice) connected topological space. Let $LX=Map(S^1,X)$ be the free loop space and $\Omega X = Map_*(S^1,X)$ the subspace of based loops (with some choice of base point for X). Now, there ...