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Questions tagged [homotopy-theory]

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 algebraic topology.

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A question about the homotopy.

Recently, I learned the definition of homotopy, and had a question. In all continuous curves(in complex plane) with $a$ as the starting point and $b$ as the end point, if the two curves are homotopic ...
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Detecting $\eta^3$ in stunted projective spaces.

Consider the stunted complex projective space $\mathbb{C}P^{n+2}_n:=\mathbb{C}P^{n+2}/\mathbb{C}P^{n-1}$ which is a three-cell complex of the form $$\mathbb{C}P^{n+2}_n\simeq S^{2n}\cup_{\alpha_n} e^...
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Construction of a K(pi_1)- space?

I was advised to post my question here. A colleague suggested a proof of a fact which I have hard time to believe. Since I am not a topologist by training I wonder if this can be true in such a ...
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Milnor construction and deloopings

To construct a classifying space (and universal bundle) of a topological group $G$ one can use the well-known Milnor construction based on the infinite join of $G$. On the other hand one can (at ...
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Is there a quick way to distinguish between a wedge of spheres and a suspended projective space?

I remember reading about the following example a while back in one of Steenrod's papers on cohomology operations. If we look at $S^3\vee S^5$ and $\Sigma \mathbb{C} P^2$, these spaces have isomorphic ...
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Constructing a Universal Cover--Proving Injectivity

Here is a quote from Hatcher's Algebraic Topology: Given a set $U \in \mathcal{U}$ and a path $\gamma$ in $X$ from $x_0$ to a point in $U$, let $$U_{[\gamma]} = \{[ \gamma \cdot \eta ] \mid \eta \...
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68 views

Higher inductive type: what for?

The typical example of higher inductive type (HIT) is the circle $S^1$ that is nicely described here. I understand HITs are convenient if you want to do homotopy theory within type theory. But what ...
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Homotopy groups of split orthogonal group

What are the homotopy groups of $O(d,d)$? Is it possible to compute them somwhow by using the fact that $O(d)\times O(d)$ is the maximal compact subgroup, and the homotopy groups of $O(d)$ are known?
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Space homeomorphic to Mobius Strip? [on hold]

I have a space $G$ of distinct pairs of points that are not ordered on $S^1$ with metric: $D = min(d(a,b) + d(a', b') + d(a, b') + d(a', b))$ Is $G$ homeomorphic to a mobius strip? This has been ...
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Showing $χ$ for a three manifold?

How would I show that the Euler number for $ (S^1 × S^1 × S^1) $ is $0$? Would it be different if we considered $S^2 × S^1 $ or just $S^3$? If so, how? Thanks for the help.
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A question about homotopy-type of a point

I´ve a question about how you can talk about contractible. I suppose I´m wrong in something with the definition. We say a space $X$ is contractible if it has the homotopy-type of a point, that is, ...
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$X \rtimes Y \simeq X \vee (X \wedge Y)$ for $X$ a Co-H-Space?

Let $X$ and $Y$ be topological spaces (pointed CW complexes), where $X$ is a Co-H-Space with co-multiplication $\mu$. I believe that the following identity holds. My first question is: Does it hold? ...
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Why is $S^1\times \{1\}$ homotopy equivalent to the solid torus $T^2 = D^2 \times S^1$? (see attached picture)

I am currently self-studying the basics of algebraic topology and i just learned the definitions of retract, deformationretract and homotopy equivalence. Now in my book there is an example of a ...
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Explicit model of the $E_{n}$-operad in simplicial sets

The space of $k$ little $n$-disks, denoted $E_{n}(k)$, is usually constructed in the category of topological spaces as the space of $k$-tuples $(c_{d_{1},p_{1}},\dots, c_{d_{k},p_{k}})$ of disjoint $n$...
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If $A$ has a neighbourhood in $X$ that deformation retracts onto $A$ then does $(X,A)$ have HEP?

Let $(X,A)$ be a topological pair. Assume that $A$ has a neighbourhood $V$ such that $V$ deformation retracts onto $A$. Then can we say that $(X,A)$ has the homotopy extension property? Edit: I know ...
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Suppose $A \subset V$. How does a deformation retraction of $V$ onto $A$ induces a deformation retraction of $V/A$ onto $A/A$?

Suppose $A \subset V$. If there is a deformation of $V$ onto $A$, then there exists maps $i: A \hookrightarrow V$ and $r: V \to A$ such that $ri=Id_A$ and $ir \simeq Id_V$. How does this ...
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Chain complexes, homology and homotopy type.

Are they examples of easy chain complexes... that have the same homotopy type but are not isomorphic? That have the same homology groups but haven't got the same homotopy type?
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Showing that $\mathbb{C}$ minus a point is homotopy equivalent to $S^1$

Intuitively this is clear (I think)? However I struggled to construct the homotopy. Here's what I attempted: Define $M:=\mathbb{C} - p$ where $p$ is a point in $\mathbb{C}$. Define also, $\alpha:S^1\...
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377 views

Connections between Algebraic Topology and Set Theory

(Co) Homology functors are dependent on the homotopy type of the objects they act on and so a lot of results only care about the "loose" classification of spaces (including the use of co-final spectra ...
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Mapping spaces for pro-objects in a simplicial model category

If $C$ is a simplicial model category (I have simplicial commutative rings in mind), then one can simplicially enrich pro-$C$ over sSet by taking $$\lim_j \mathrm{colim}_i \underline{\mathrm{Hom}}_C(...
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Cohomology of colimit is limit of cohomology ? (group cohomology)

In Homotopy theoretic methods in group cohomology, Henn's part, section 1.2, the example following definition 1 has the following sentence "the cohomology $H^*(G,\mathbb{F}_p)$ of a group $G$, which ...
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Four sheeted cover of $\mathbb{R}P^2$ wedged with circles

Let: $$ X_n = \underbrace{S^1 \vee S^1 \vee \dots \vee S^1}_n \vee \mathbb{R}P^2 $$ Problem says: By computing a homology group of a suitable 4-sheeted covering space of $X_n$ show that it's not ...
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Graded $*$-homomorphisms where are homotopic to $0$

This is an Exercise 3.15, pg 42. Show that the map between $\Bbb Z/2 \Bbb Z$- graded $C^*$-algebras $$C_0(\Bbb R) \rightarrow M_2(\Bbb C)$$ $$ \varphi: f \mapsto \begin{pmatrix} f(0) & 0 \...
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Proof of why $S^\infty$ is contractible using Hilbert space

In John Baez's blog post http://www.math.ucr.edu/home/baez/week151.html, Baez gives a short proof of why $S^\infty$ is contractible, using that $S^\infty$ is the unit sphere in a infinite dimensional ...
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When one can extend $X\times \{0,1\}\cup A\times[0,1]\rightarrow Y$ to full homotopy?

I'm wondering if there is some general criteria/theorems to answer the following question: Question: Let $X$ and $Y$ be topological spaces (even metric spaces if that is necessary). Let $A\subseteq X$...
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How to interpret the structure of elements in a path-connected topological group.

I am working on solving the following question: Let $G$ be a path-connected topological group. Let $\alpha,\beta:[0,1]\rightarrow G$ be two loops based at $e$ in $G$, and consider the map $$F:[0,...
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58 views

Homotopy fixed point on the fibre of a fibration over BG

I am currently reading the paper Homotopy Fixed Point Methods for Lie Groups and Finite Loop Spaces by Dwyer and Wilkerson (can be found here http://www.math.purdue.edu/~wilker/papers/analysis.pdf for ...
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Whitehead product $[i_2, i_2]_W$

Let $i_2$ be the generator of $\pi_2(S^2)$ and $\eta$ be the Hopf fibration from $S^3$ to $S^2$. How would one go about showing that $[i_2,i_2]_W=2\eta$ (up to a sign)? This is one of the exercises ...
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Explicit definition of the čech nerve in this paragraph

In the picture below, how is the čech nerve $C(U)$ defined? In the first hom-set, it is taken to be a simplicial sheaf. It would actually make sense if it was a simplicial simplicial sheaf, however. ...
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Degeneracies for semi-simplicial sets

Consider the category of semi-smplicial sets, ie simplicial sets without degeneracies. It is a preseaf category over $\Delta_{inj}$, the category of ordinals together with injective maps. The usual ...
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Are fundamental groups (at any basepoint) of a commutative H-space abelian?

Let $X$ be a commutative simplicial monoid and let $s\in X$. Is $\pi_1(X,s)$ abelian? (obviously $s$ not necessarily in the same component of the identity element). The direct route would be to know ...
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on paths homotopic to a point

Let $U\subset \mathbb{C}$ be a domain and $\Gamma:[0,1]\to U$ be a closed Jordan path. Denote $G$ the interior of $\Gamma$, i.e. $G$ is a bounded set such that $\Gamma=\partial G$. We assume that for ...
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The torus cannot cover $\Bbb S^2$

I know that $S^2$ cannot cover the torus $\Bbb T^2 \cong \Bbb S^1\times \Bbb S^1$ because $S^2 \not \cong \Bbb R^2$ and both are universal coverings of $\Bbb T^2$ Conversely, is there a covering $p:\...
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In what sense is $S^n$ contractible in $S^{n+1}$?

In my course of algebraic topology our professor said that although $S^n$ is not contractible, it is as a subspace of $S^{n+1}$, but he said it just as a comment and he gave the intuitive idea of why ...
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Loops: associativity for groups

How would you show that for any three loops $x, y, z$ that $(x*y)*z$ is equivalent to $x*(y*z)$. I want to show that $([x]*[y])*[z] = [x]*([y]*[z])$. I am terrible at this stuff, would appreciate a ...
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Homotopic Equivalence of the set of all continous maps from $X$ to $\Bbb R^n$

If we have the set of all continuous maps from $X$ to $\Bbb R^n$ denoted by $C(X,\Bbb R^n)$ and $\sim$ is the relation on this set, how could we prove that every two elements in $C(X,\Bbb R^n)$ is ...
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How to see that this space with two vertices and four edges is homotopy equivalent to $S^1 \vee S^1 \vee S^1$?

I am having trouble seeing why this space has fundamental group $\mathbb Z * \mathbb Z * \mathbb Z$. I have read that this space is homotopy equivalent to $S^1 \vee S^1 \vee S^1$, from which we can ...
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Factorisation into cofibration and trivial fibration

Suppose $M$ is a simplicial model category, $f: R \to T$ a morphism between fibrant objects in $M$. Is there a nice way to construct a factorization of $f$ into a cofibration followed by a trivial ...
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26 views

How to systematically collapse a $n$-simplex to a point via elementary collapses

An elementary collapse (https://en.wikipedia.org/wiki/Collapse_(topology)) is the removal of a pair of simplices $\sigma,\tau$, such that $\dim \tau=\dim\sigma-1$ and $\tau$ is a free face of $\sigma$....
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Implications of a model structure in which every object is fibrant and cofibrant?

If we are trying to study a category using model categories, is it a "good" or "bad" thing if our choices of fibrations and cofibrations make everything fibrant and cofibrant? Does it signal that our ...
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What are sufficient conditions for finitely many equivalence classes of slice contours of surfaces?

Apologies in advance for imprecision of the question. Thanks for improving it. Let M be a compact, connected, orientable surface in three dimensional Euclidean space without boundary and without self-...
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48 views

Homotopy Limit Intuition

I am currently reading through Emily Riehl's Categorical Homotopy Theory book, and have gotten a bit stuck on the section on calculating homotopy limits. The colimit case is fine, but for some reason ...
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59 views

$S^3\times S^5 \simeq \sum X$ for a finite CW-complex $X$

I want to know if $S^3\times S^5$ can be homotopy equivalent to the suspension $\sum X$ of a finite CW-complex $X$. I know the following properties for singular homology (but perhaps it is possible to ...
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41 views

Fundamental group as a product of normal subgroups

Let $A$ be a retract of a non-empty topological space $X$ and let $a \in A$. Let's denote $r : X → A$ the retraction and $i : A → X$ the inclusion. prove that $i_∗ (π_1 (A, a))\triangleleft π_1 (X, ...
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38 views

The proof of the homotopy extension and lifting property (HELP)

I am reading J. P. May's Book, the section about homotopy extension and lifting property (HELP) on page 75: I know this is true for $$(X,A)=(D^n,S^{n-1})\cong (CS^{n-1},S^{n-1}),$$ where $CS^{n-1}$ ...
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existence and uniqueness of the index isomorphism

Let $x_0 ∈ \Bbb C$ and $ S = \Bbb C \setminus \{x_0 \}$ $\forall x ∈ S$, prove existence and uniqueness of an isomorphism $ψ_{x_0, x} : π_1 (S, x) → \Bbb Z$ such that $ψ_{x_0 ,x} (α) = 1$, where $α$ ...
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32 views

homotopy equivalence between the cylinder of a map

Given a continuous map $f\colon X\to Y$ between two non-empty topological spaces, show that there is homotopy equivalence between the mapping cylinder $(X\times I)\sqcup _{f}Y$ and Y. Here we have I=$...
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Proof of $\pi_1(S^n)=0$ if $n \geq 2$.

Proposition $\pi_1(S^n)=0$ if $n \geq 2$. Let $f$ be a loop in $S^n$ at a chosen basepoint $x_0$. If the image of $f$ is disjoint from some other point $x \in S^n$ then f is nullhomotopic since $S^n -...
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Homotopy invariance of pullbacks of principal bundles

This is the proof of lemma 7.2 in a notes by Stephen Mitchell, on classifying spaces. Essentially one step of the proof claims that: Let $p:Y \rightarrow B \times I$ be a principal bundle. If $B$...
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What can we say about two topological spaces with the same fundamental group?

Let's consider two topological spaces. If they are homeomorphic, or homotopic equivalent, they have isomorphic fundamental groups, but the converse is not true. My question is: is there a (non trivial)...