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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 ...

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Algebraic K theory linearization reference request

In Waldhausen's paper Algebraic K theory of Spaces(the long one) he proves the following: $$A(X)\simeq \mathbb{Z}\times B\widehat{Gl}(\Omega^{\infty}\Sigma^\infty |G|)$$ Where $|G|$ is a loop group ...
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$\mathbb C P^1\cong (D^2\times\{1\}+ D^2\times \{-1\})\big /_\sim $

I need to proove a bigger result that $\mathbb C P^1$ is homeomorphic to $S^2$. For that I have already showed $$S^2\cong (D^2\times\{1\}+ D^2\times \{-1\})\big /_\sim$$ where $(z,1)\sim (z,-1) \iff z ...
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Conley index as a subset of an isolated invariant set

In Sec. 7 (p. 60) of Conley's $\textit{Isolated Invariant Sets and the Morse Index (1976)}$, the following passage appears: (In fact any isolated invariant set in $S$ is isolated in $\Phi$. It ...
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Relation between transport functor of a fibration and a Hurewicz connection on it

Let $A\overset{\alpha}{\rightarrow}B$ be a (Hurewicz) fibration. The homotopy lifting property w.r.t a fiber $\alpha ^{-1}(b)$ furnishes for each path $b\to b^\prime$ in the base a continuous map $\...
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Topological space with binary operation has abelian fundamental group

I have been given this problem: Let $\textit{X}$ be a topological space and $\mu : \textit{X} \times \textit{X} \rightarrow \textit{X}$ a binary operation. Show that if $\mu$ is continuous and $\...
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37 views

Prove “Contractible implies simply connected” using tools in Munkres Topology. Context is theta-space.

I've read this online, but I haven't seen this proved in Munkres Topology. Has it been? If so, where? In any case, here is my attempt to show it using the tools given in the book. Please verify. The ...
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Undestand the proof of Cauchy Integral Formula using homotopy of curves.

https://math.berkeley.edu/~vvdatar/m185f16/notes/Lecture-16_CIF_Analyticity.pdf I'm trying to understand the proof of Theorem 0.1. Only the following part is missing: "Consider the contour in the ...
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44 views

Calculate homotopy of $S^2$ relative to great circle using exact sequence

Let $A\cong S^1$ be a great circle of $S^2$. I would like to calculate the relative homotopy $\pi_2(S^2,A,x_0)$. I know I have a long exact sequence of relative homotopies of pairs: $$\pi_2(A,x_0)\to\...
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35 views

Is connectedness needed in this statement?

Suppose $M$ is a arcwise connected metric space, $A_1$ and $A_2$ are mutually separated in $M$. Let $f_i:A_1\cup A_2\to M$ for $i=$1,2 be two continuous functions such that $f_0|$$A_i$ ($f_0$ ...
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When is commuting in $\pi_1$ equivalent to commuting up to homotopy?

It seems to me that commuting up to homotopy certainly implies commuting in $\pi_1$, but when is the converse also true? I am reading something on surfaces, and it says in that case commuting in $\...
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Explicit construction of map from $T^2\rightarrow S^2$ with lowest non-zero degree of map.

This question is related to my another question:Homotopy class of map from torus to sphere? I want to construct all different homotopy class of map from $T^n \rightarrow S^n$. From above question, ...
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Classifying map for the hyperplane bundle

Let $G$ be a topological group and $X$ be a CW complex. Then principal $G$-bundles on $X$ is classified by the classifying space $BG$ in the sense that, given a principal $G$-bundle $P \to X$, there ...
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55 views

Homotopy class of map from torus to sphere?

In algebraic topology, we use homotopy group $\pi_n(M)$ to classify the homotopy class of continuous map $f:S^n\rightarrow M$. My question: What's the mathematical object to classify the homotopy ...
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understanding Thom's theorem for $\mathrm{MU}$ and Milnor-Novikov's result.

Sorry if this has been asked before. $\newcommand{\MU}{\mathrm{MU}}$ I'm trying to understand the complex cobordism spectrum $\MU$, but I don't fully understand Thom's theorem, namely, that $$ \MU_\...
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How to show that S^1 is not contractible just by definition without going into Fundamental group.

Suppose $X$ and $M$ be separable metric spaces, these are some of the definitions: Two maps $f_0:X\rightarrow M$ and $ f_1:X\rightarrow M$ are said to be homotopic in M if there is a continuous ...
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1answer
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Let $A ∈ SO(n+1)$ and write $τ(A)$ for the induced map $S^n → S^n$. Prove that $τ(A)\simeq id$

Let $A ∈ SO(n+1)$ and write $τ(A)$ for the induced map $S^n → S^n$. Prove that $τ(A)\simeq id$ . My attempt: So $\tau(A)$ is the map $v \mapsto Av$. So have to construct a map $H:S^n \times [0,1] \...
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Retract of Compact $2$-manifold

Consider by $S_g := T \# T \# ... \# T$ the $g$ times connected sum of tori $T$. Obviously since it is a compact $2$-manifold in light of the famous classification of compact $2$-manifolds $S_g$ is ...
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35 views

Example of a fibration that is not a fibre bundle

Let $X$ be any topological space. Is it possible to construct a certain subspace $E\subseteq X\times X$ of the Product space, such that the restriction of the trivial bundle map $\pi\colon X\times X\...
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What is the fundamental group action on universal covering?

Given a topological space $X$ with a base point $x_0$, which is locally path-connected, path-connected, and semi-locally simply connected. So, by Hatcher's book, its universal covering can be ...
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How to prove (rigorously) that these spaces are homotopy equivalent?

I wonder how to prove rigorously that the figure eight, or a pair of circles intersecting at one point (looking like OO), is homotopy equivalent to a disjoint pair of circles joined by a straight line ...
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Prove that $λ(A)^+ : S^n → S^n$ (the map induced on one-point compactifications) is homotopic to identity if $\det(A) > 0$

Let $A ∈ GL(n,\Bbb R)$. a) Prove that the induced map $λ(A)$ from $\Bbb R^n$ to $\Bbb R^n$ corresponding to the associated linear transformation is a proper map. b) Prove that $λ(A)^+ : S^n →...
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What is a low-brow reason symmetric spectra admit smash products?

This question is similar to this one, though it did not receive much attention. In his book project Symmetric Spectra, Stefan Schwede mentions why we care for having a model of the stable homotopy ...
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Prove that $S^n /\{p,q\}$ is homotopy equivalent to $S^n \vee S^1$.

Prove that $S^n /\{p, q\}$ homotopy equivalent to $S^n ∨ S^1$ My attempt: I can see the picture quite clearly: But how to write explicit homotopies ? Thanks in Advance for help!
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Wedge of aspherical spaces

I‘d need a reference for the following fact: the one-point union of (nice) aspherical spaces is aspherical. I.e., from $\pi_kX=0$ and $\pi_kY=0$ follows $\pi_k(X\vee Y)=0$. EDIT: Let‘s assume that the ...
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Why Homotopy so interesting in Mathematics and what is its relationship with Manifolds?

Homotopy Theory is one of the essential topics in Mathematics. It helps to study and classify curves, surfaces, and manifolds of any dimension. One of the major issues tackled by Homotopy Theory is to ...
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1answer
16 views

Is $Cat_{\Delta}$ enriched over itself?

Question is just as in the title. Is the category of simplicially enriched categories enriched over itself? If not, is it enriched over another relevant category, e.g., simplicial sets?
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106 views

A proof of $\pi_1(S^1)=\mathbb{Z}$ without universal covers

I'm reading "A Guide to the Classification Theorem for Compact Surfaces" by Jean Gallier adn at page 45 is this proposition: I don't understand how $\beta_k$ and $\delta_k$ are defined. Can someone ...
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Replacing $H$-space multiplication with a fibration

Let $X$ be an $H$-space with multiplication $\mu: X \times X \to X$. Does there exist a space $\overset{\sim}{X}$ homotopy equivalent to $X$ such that the induced map $\overset{\sim}{\mu}: \overset{\...
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1answer
27 views

Bergner homotopy category of simplicially enriched caterories is cartesian closed

Let $Cat_{\Delta}$ be the model category of simplicially enriched categories with the Bergner model structure. In a paper I am reading, they state without proof that $Ho(Cat_{\Delta})$ of this model ...
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344 views

what is an ∞-group?

I was reading on nLab and I found the term infinity group. The definition is awfully abstract: An ∞-group is a group object in ∞Grpd. Equivalently (by the delooping hypothesis) it is a pointed ...
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54 views

Must one space be the loop space of other

If X and Y be two topological spaces such that n-th homotopy group of X and (n+1)-th homotopy group of Y are isomorphic for all natural number n. Does it imply that X is homotopy equivalent to the ...
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What is the third homotopy group of $SO(5)$?

I am trying to compute the $\pi_3(SO(5))$, but not getting any idea to compute it. As in the case of first and second homotopy groups of $SO(5)$, which can easily be computed using the long exact ...
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Contractability of topological space

Let X and Y be two topological spaces such that X and X x Y are homotopy equivalent. Does it imply that Y is contractable? As I am trying, it seems to be true if X and Y are CW complexes, in that ...
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Like a cell complex but characteristic maps send n-cell boundary to other n-cells

Recall that a cell complex is a topological space $X$ that satisfies the following definition: $X$ decomposes as a union of open cells of varying dimensions. For each cell $C$ of dimension $n\geq ...
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Give examples of two $C_2$ actions on $S^n$ such that the orbit spaces are not homotopy equivalent.

Give examples of two $C_2$ actions on $S^n$ such that the orbit spaces are not homotopy equivalent. My attempt: For $n \ge 2$. I considered the following actions of $C_2$ on $S^n$ : (i) mapping to ...
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Prove that for any point $p ∈ \Bbb RP_n$ , $\Bbb RP_n − p \simeq S^{n−1}$

Hope this isn't a duplicate. I was trying the following Problem: Prove that for any point $p ∈ \Bbb RP_n$ , $\Bbb RP_n − p \simeq S^{n−1} \text{ (homotopy equivalent)}$ But I am not quite sure ...
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when are the winding number of homotopic curves not equal?

For what cases do we have it that even though two closed rectifiable curves are homotopic, that they do not share the same winding number ?
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Mapping Cylinder Fibration

Let $p:E \rightarrow B$ be a Serre fibration (i.e. it has homotopy lifting property with respect to all CW-complexes or equivalently cubes $I^n$) and $M_f$ be the mapping cylinder of $p$. We can ...
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Classifying the spaces that Eilenberg-Steenrod axioms determine the cohomology of

We know that for sufficiently nice spaces, (e.g. spaces with the homotopy type of a CW complex) the Eilenberg-Steenrod axioms determine the ordinary cohomology of the space. One can construct nasty ...
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closed cells form a covering of the CW complex.

A CW complex $X$ is the union of open cells $\{e^i_\alpha\}$ and these open cells are disjoint subsets of $X$, so these open cells form a partition of $X$. Now if we take the closed cells $\{\bar e^...
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On the notion of 'winding numbers' of maps $\mathbb{C} \setminus \{0\} \to \mathbb{C} \setminus \{0\}$

In complex analysis, the winding number (around the origin) of a continuous loop $\gamma: [0,1] \to \mathbb{C} \setminus \{0\}$ is the number of times the loops "winds" around zero, which is given by ...
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Symmetric powers of odd spheres

Given a sufficiently nice space $X$, say a connected and compact polyhedron, one has a nice formula for the Poincaré polynomial of the orbit space $SP^n X:= X^n/S_n$ of the $n$-fold Cartesian product ...
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Homotopy type of the diffeomorphism group of the sphere.

I've seen in several places the claim that $$\mathrm{Diff}(S^n) \approx O(n+1) \times \mathrm{Diff}(D^n\,\text{rel}\,\partial D^n),$$ where: $\mathrm{Diff}(S^n)$ is the group of $C^{\infty}$ ...
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Space with semi-locally simply connected open subsets

A topological space $X$ is semi-locally simply connected if, for any $x\in X$, there exists an open neighbourhood $U$ of $x$ such that any loop in $U$ is homotopically equivalent to a constant one in $...
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the Euler characteristic of $K\#K$

I'm trying to find the Euler characteristic of $K\#K$, here $K$ is the Klein bottle. I tried to use the fact that $\chi(A\cup B)=\chi(A)+\chi(B)-\chi(A\cap B)$ but it seems not working hence I'm ...
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What is the difference between $D(\mathbb Z)$ and $Spectra$ in terms of $t$-structures?

I'm trying to see my way around the following False Claim: Bounded spectra are the same as bounded chain complexes (as a triangulated category, say). Dubious Proof: Consider the standard $t$-...
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If $X$ is the retract of its cone $CX$ it is contractible

I'm trying to prove this statement, but find that "retract" only implies a continuous map from the cone of the underlying space to itself but doesn't offer information about contractibility of the ...
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If $A$ is a retract of $X$, how to prove $X \times I \times I$ has $X \times I \times \{0\} ∪ (X \times \partial I ∪ A \times I) \times I$ as retract?

The original problem is to prove: If $(X,A)$ has HEP (homotopy extension property) then so has $(X \times I, X \times \partial I ∪ A \times I)$. I found a proof which is too abstract to me: https:...
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Fundamental group of a plane united with a circle in 3D space

Let $X=\{(x,y,z)\in\mathbb{R}^3|x^2+y^2+z^2=9\}\cup\{(x,y,0)\in\mathbb{R}^3|(x-4)^2+y^2=1\}\subset\mathbb{R}^3$ Find the fundamental group $\pi_1(X,p_0)$ where $p_0=(3,0,0)$ Possible solution: I'm ...
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The identity homotopy on the circle can create an arc of any length

Let $f:S^1\to S^1$ be continous. Show that if $f$ is a homotopy to the Constant function, there exists a point $x\in S^1$ such that the length between $x$ and $f(x)$ is $\frac{\pi}{6}$. Hint: You can ...