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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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 ...
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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 ...
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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 ...
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35 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 ...
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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 ...
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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. $S^2 / S^0$, $S^2 / S^1$, $S^n / S^1$, ...
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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 ...
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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*} ...
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Spaces $X$ and $Y$ with $[Z, X]_{\bullet} \cong [Z, Y]_{\bullet}$ for all cogroup objects $Z$ in $\mathsf{hTop}_{\bullet}$

Throughout, $(X, x_0)$ and $(Y, y_0)$ will be connected pointed topological spaces. If $f : (X, x_0) \to (Y, y_0)$ is a continuous map and $f_* : \pi_n(X, x_0) \to \pi_n(Y, y_0)$ is an isomorphism ...
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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 ...
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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 ...
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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 ...
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1answer
61 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.
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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) = ...
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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 ...
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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 ...
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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{ ...
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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 ...
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72 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 ...
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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 ...
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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 ...
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28 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 ...
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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 ...
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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'?
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1answer
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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 ...
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1answer
132 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 ...
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78 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 ...
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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 ...
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74 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 ...
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47 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 ...
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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?
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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 ...
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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 $$ ...
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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 ...
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57 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 ...
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1answer
38 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 ...
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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 ...
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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 + ...
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2answers
20 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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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 ...
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1answer
58 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?
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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 ...
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$\mathbb{R}^{2}$ and $\mathbb{R} \times [0, +\infty]$ are homotopy equivalent, but not homeomorphic

So, let's consider $M=\mathbb{R}^{2}$ and $N= \mathbb{R} \times [0, +\infty]$ - two topological spaces. Since $\pi_{1}(M)=\pi_{1}(\mathbb{R}) \times \pi_{1} (\mathbb{R}) = \{0 \}$ (since $\mathbb{R}$ ...
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2answers
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Group homomorphism on unit circle

For $n\in \mathbb{Z}$, define the map $f_n:S^1\to S^1$ as $f_n(z)= z^n$, where the unit circle $S^1$ is observed as the subspace $\{z\in\mathbb{C}|\ |z|=1\}$. How would one compute the induced group ...
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1answer
36 views

What is the relationship between the path-loop space fibration and path induction?

I have an intuition that I can't quite put into words that the path-loop space fibration $\Omega X \rightarrow PX \rightarrow X$ and the based path induction axiom are related, but I don't know enough ...
6
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Homotopical perspective on the long exact sequence in homology and Mayer-Vietoris

For a while I have been wondering whether the long exact sequence in homology and the Mayer-Vietoris sequence can be phrased in homotopical terms. Recently, I heard that both may be reformulated via ...
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114 views

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map?

Why is the identity map from $S^1$ to $S^1$ not homotpic to the constant map? I get the picture that if the identity map $id$ is homotopic to the constant map then as the circle transforms ...
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1answer
28 views

$|\beta(s)-\alpha(x)|<d$ implies that $\beta$ is homotopic to $\alpha$.

Let $D$ be an open subset in $\mathbb{R}^n$. Let $\alpha$ be a path in $D$ from $x$ to $y$, and set $d=\inf\{|\alpha(s)-w|:w\in \partial D, 0\le s\le 1\}$. Show that if $\beta$ is any path in $D$ ...
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1answer
40 views

Homotopy of a pair

Let $X$ be a topological space and $A \subset X$ a subspace. For $x \in A$, the homotopy group $\pi_n(X,A,x)$ of the pair $(X,A)$ is by definition the set of homotopy classes of n-cells relative to ...
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1answer
70 views

Understanding cofibration sequence

Recently I'm studing some basic homotopy theory. An important brick of the exact sequence of cofibration is the following sequence: $$X \stackrel{f}{\longrightarrow} Y \stackrel{i}{\longrightarrow} C ...