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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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Filtered colimits in Adams's category

I am currently reading Part 3 of Adams's book "stable homotopy and generalized cohomology", and I got stuck when following his argument. In Proposition 5.4, he states that when $W$ and $X$ are finite ...
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When do elements of $\operatorname{Hom}(G,G)$ correspond to invertible self maps of $K(G,n)$?

Suppose we pick a natural isomorphism between $H^n(-;G)$ and $\langle -, K(G,n)\rangle$, when does an element of $H^n(K(G,n),G)=\operatorname{Hom}(G,G)$ correspond to a self map of $K(G,n)$ that has a ...
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Computing $[\mathbb{R}P^2, S^k]$

I am trying to compute $[\mathbb{R}P^2,S^k]$ for $k\geq 0$, via the cofiber sequence associated to $f:S^1\to S^1$ given by $z\mapsto z^2$, where we get the mapping cone $C_f \cong \mathbb{R}P^2$. The ...
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Reference for the cohomology of SU

Let SU be the infinite special group. Where can I find the following fact (state in part III of the Adam's blue book): $H^{6}(SU,Z)=0$. Thank you.
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Relationship between homotopy pushout and ordinary pushout

I'm trying to understand the homotopy pushouts and currently looking at the homotopy cofiber. For two maps $f \colon C \to A$ and $g \colon C \to B$ we defined the homotopy pushout to be the regular ...
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Corestricting a weak homotopy equivalance

Let $X$ and $Y$ be topological spaces. Let $f: X \to Y$ be a continuous map. Recall that $f$ is a weak homotopy equivalence iff $f$ induces group isomorphisms on the homotopy groups, i.e.: $$\...
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Functoriality of parallel transport of a Hurewicz connection on a fiber bundle

Let $A\overset{\alpha}{\rightarrow}B$ be a Hurewicz fibration. Any Hurewicz connection defines parallel transport along curves in the base. In general, such parallel transport maps $\alpha^{-1}(b)\to \...
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Map of H-spaces inducing zero on homologies

If a map of $H$-spaces $f:X\rightarrow Y$ induces zero on the homology groups at dimensions greater than zero does it necessarily induce zero map on the homotopy groups? It is definitely true for $\...
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Does there exist a homotopy between identity function and any continuous function?

(My question is related to the Brouwer fixed-point theorem.) Let $B$ be a closed ball of $\mathbb{R}^n$. Q 1. If $f : B \rightarrow B$ is a continuous function, then is there a homotopy between $...
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Derived $p$-completion : injective in the category of diagrams?

I'm reading a set of notes that has an interlude about derived $p$-completions of abelian groups. My first question stems from the fact that this interlude, although interesting, is not very ...
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Possible error in proof of PL-approximation (Hatcher)

I am reviewing some technical results from the fourth chapter of Hatcher's Algebraic Topology. In the proof of PL-approximation (Lemma 4.10), we let $B_1,B_2\subset e^k$ denote the balls of radius 1 ...
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Higher homotopy group and fundanmental group

I am a new learner in topology and manifold and get some problems. The lemma that higher homotopy group of a pathwise connected topological space are commutative can be proved by using Eckmann-...
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Milnor's proof that a smooth manifold has the homotopy type of a CW complex

I have some questions about the proof of Theorem 3.5 of Milnor’s “Morse Theory”: At the end of the proof of this theorem, Milnor addresses the case when $f$ has infinitely many critical points: ...
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$k^{th}$ homology of $n$-torus via Mayer–Vietoris sequence

Let $T^n$ denote $n$-torus defined as: $$ T^n = \underbrace{S^1\times S^1 \times \dots \times S^1}_n $$ The question is to compute $H_k(T^n)$ without using Kuneth formula. So far I've noticed that $...
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Do homotopy pullbacks always compose?

Classical pullbacks compose, as is easily checked with the universal property. More precisely, if $\require{AMScd} \begin{CD} A @>>> B\\ @VVV @VVV\\ C @>>> D \end{CD}$ and $\require{...
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What is the shape group?

I'm familiar with topology and category theory a little bit as well as inverse limits of inverse systems. I would like to understand the shape group better. For example, one could describe the ...
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Construct homotopy using illustration

I am currently studying simplicial topology and got stuck with a homotopy construction: $$ \alpha(s) = \begin{cases} e^{4\pi is} \\ e^{4\pi i(2s-1)} \\ e^{8\pi i(1-s)} \end{cases} \simeq \beta(s) = e^{...
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Compact space homotopy equivalent to a CW complex

Assume that $X$ is a compact Hausdorff space homotopy equivalent to some CW complex. Does it follow that it is homotopy equivalent to a compact CW complex?
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Is there an operad whose algebras are homotopy commutative $E_1$-algebras?

I might guess that the Boardman-Vogt tensor product of the $E_1$ operad and the $A_2$ operad might do the trick. That is, I would guess that an $A_2$ object in $E_1$ algebras, or equivalently an $E_1$ ...
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Intuition behind the injectivity part of Hurewicz Theorem

The surjectivity part of Hurewicz Theorem is easy to understand: under the inductive hypothesis that all homotopy groups (of a CW-complex) up to dimension $n$ are trivial, it is clear (I believe) how ...
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Taking homotopy fixed points preserves fibrations

I'm reading a paper where they have an appendix about homotopy fixed point sets of a $G$-space, and at some point they claim that if $f:X\to Y$ is a $G$-map that is an ordinary (non-equivariant) ...
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Question about loops(closed paths) in $\mathbb{S}^{1}$

Let $$\alpha \left(s \right) =\left( \cos{2\pi s},\sin{2\pi s}\right)$$ and $$\beta \left(s \right) =\ \left( \alpha \land\left( \alpha \land \overline{\alpha} \right)\right)\left( s \right)$$ with $...
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Evaluating a symplectic form on $\pi_2$ or its image through the Hurewicz map

Let $(M,\omega)$ be a symplectic manifold. There are a priori two ways of evaluating $\omega$ on an element $A \in \pi_2(M)$: we can integrate $\omega$ on any representative $u : S^2 \to M$ of the ...
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Homotopy Invariance of the Homotopy (Co)limit

I'm trying to understand why the following proposition is true: Let $J$ be a small category and $F, G : J \to \textbf{Top}$ functors. If $\tau : F \Rightarrow G$ is a pointwise homotopy equivalence,...
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Center of fundamental group

Let $f_t: X \rightarrow X$ be a homotopy of maps such that $f_0 = f_1 = \mathrm{id}_X$. For any $x_0 \in X$, the map $t \mapsto f_t(x_0)$ is a loop based at $x_0$. To prove: $[f_t(x_0)]$ is ...
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A stable system of vector bundles can be obtained by pulling back the universal stable system

I am trying to solve the problem 180 on Davis-Kirk (p.269). However, the authors give some special definitions, so-called "stable system of vector bundles", which I have never seen on other ...
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Serre fibration and Mittag Leffler condition

The proof I am concerned is 2.2.5 pg 37 Kochman Stable Homotopy. Let $R$ be a commutative ring. Let $S^n \rightarrow E \xrightarrow{p} B$ be Serre fibration. $B$ a CW complex, $B$ simpliy ...
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If $\operatorname{deg}(f) \ne 0$ then $f(S^1) = S^1.$

Let $f : S^1 \rightarrow S^1$ be a continuous map such that $\operatorname{deg}(f) \ne 0.$ [Note : I have the degree defined as in this question.] I'm trying to prove that the hypotheses above ...
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What is relation between the path fibration of classifying map and the Borel construction

I am trying to solve exercise 179 on Davis-Kirk: Show that given a principal $G$-bundle $E\to B$, there is a fibration $$E\hookrightarrow EG\times_G E\to BG$$ where $EG\times_G E$ denotes ...
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The cohomology $H^*(BU(n); R)$

I am reading this post, Prop. 2.1. It seem that none of the argument is dependent on the we are working with coefficient in $\Bbb Z$. Hence, let $R$ be a unital commutative ring. (I) Do the ...
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Do the homotopy groups of a space determine the homotopy groups of its suspension?

I am trying to work out an answer to my latest question I know that there is not a known formula for the homotopy groups of the suspension in terms of the homotopy groups of the space being ...
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What is the fiber of the map $Y\to \Omega SY$

I am trying to write down explicitly the fiber of the map $$Y\to \Omega SY,$$ where $\Omega$ is the loop space functor and $S$ is the suspension functor. The map is the adjoint map of identity map $...
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Non-existence of pushout in homotopy category

I want to show that $S^1_{(0)}\leftarrow *\to S^1_{(1)}$ has no pushout in the homotopy category without using Eilenberg–MacLane spaces. In a first step, I want to show that if there is such a ...
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Is a finite dimensional CW complex with the same homotopy groups as $S^n$ homotopy equivalent to $S^n$

After you introduce the homotopy groups and upon restricting to CW complexes, you ask the question: "Do homotopy groups determine a space up to homotopy equivalence?" With the answer being "No, $\...
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Fiber integration is independent of the operations involved.

In Definition 2 of Fiber Integration nlab post, the author claimed that the operation is indpendent of the choices involved. How is this so? The post itself is quite long. I think it is easier ...
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Different definitions of the minimal Chern number and the monotonicity of symplectic manifolds

I am trying to understand the differences between several definitions used in many texts in symplectic topology. Let $(M,\omega)$ be a symplectic manifold, and $c_1 \in H_2(M,\mathbb{Z})$ be its first ...
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Homotopy classes of self-maps on $\mathbb{S}^1\vee\mathbb{S}^1$

Consider the two inclusions $\eta_i:\mathbb{S}^1\to \mathbb{S}^1\vee\mathbb{S}^1$. I claim that the following map is injective $$(\eta_1\sqcup\eta_2)^*:[\mathbb{S}^1\vee\mathbb{S}^1,\mathbb{S}^1\vee\...
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When does a fibration $f:X\rightarrow Y$ in a model category admit a section?.

If we have a fibration $f:X\rightarrow Y$ in a model category $C$, where $Y$ is cofibrant and both $X, Y$ are fibrant. Does f admit a section (right inverse)?. If it does not work in general, ...
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1answer
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Singular Cohomology satisfies the Mittag-Leffler Condition on CW complexes?

Is this true? Let $H$ be singular cohomology. On an arbitrary CW complex $X$, given a filtration $X^0 \subseteq X^1 \subseteq \cdots X^n \subseteq \cdots \subseteq X$. Then $$H^*(X) \cong \...
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1answer
46 views

Postnikov towers agreeing at some stage

Let $\cdots \to X_2\to X_1$ be a Postnikov tower for an $n$-dimensional CW-complex $X$. Given some $k<n$, is it possible to find a CW-complex $Y$, such that $\dim(Y)\leq k$ and with a Postnikov ...
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Do i imagine the linear (straight line) homotopy in a correct way?

Today i learned about the linear homotopy which says that any two paths $f_0, f_1$ in $\mathbb{R}^n$ are homotopic via the homotopy $$ f_t(s) = (1-t)f_0(s) + tf_1(s)$$ Am i right in imagining the ...
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1answer
72 views

What is Thom Isomorphism?

I am reading the following post on Thom Isomoprhism and I also have the Thom Isomoprhism from Hatcher's, Corollary 4.9,pg441 nlab's: Let $V \rightarrow X$ be a rank $n$ vector bundle over a simply ...
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Ring structure of $\Bbb CP^n$ and Chern class.

In this notes Prop 1.71 in nlab, the author aims to compute $H^*(\Bbb C P^n, \Bbb Z)$. I have two confusions. What makes it justified to use $c_1$ as the generator of $H^2(\Bbb CP^n, \Bbb Z)$? There ...
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1answer
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Pushout-product of anodyne extensions is again anodyne

Currently, I'm reading Simplicial Homotopy Theory by Jardine and Goerss and I got stuck in the proof of the theorem about pushout-products of anodyne extensions (corollary 4.6 in the book). Namely, I ...
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CW Structure on Self-Homeomorphism Groups

If $X$ is a finite CW complex then according to Milnor's article On Spaces Having the Homotopy Type of a CW-Complex, the mapping space $$Map(X,X)$$ (which we furnish with the compact-open topology) ...
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1answer
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A description of the map between Grassmanians $G_1^k \rightarrow G_k$,

We know that $G_k:=co\lim G_k(\Bbb C^n)$ is the classifying space for $k$ dimensional complex vector bundles. With total space $E_k = \{(x,v) \, :|, x \in G_k, v \in \Bbb C ^\infty \}$. So we may ...
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Definition of mapping telescope

In Kochman's stable homotopy theory, pg 121 prop 4.24 We let $X$ be a based CW complex. Let $X^n$ be an increasing sequence of subcomplex whose union equals $X$. We define $$TX = \bigcup_{n \ge ...
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88 views

Serre Spectral Sequence I

I am trying to follow the proof in Kochman's Introduction to Stable Homotopy Theory, page 59. This will be first part of a series of post. (Serre Spectral Sequence) Let $R$ be a commutative ring ...
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1answer
42 views

Prerequisite of understanding a topological construction in Spectral Sequences

This is a post on the construction of a spectral sequence. I am in fact lost in the first paragraph. Let $B$ be a CW complex and $\pi\colon X\to B$ a Serre fibration. Put $X^k=\pi^{-1}(B^k)$. A ...
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What are the maps in the long exact sequence of homotopy groups for the free loop space fibration?

Let $(X,x)$ be a pointed connected CW complex, and let $\mathcal L X = Map(S^1, X)$ be its free loop space. We have a fibration $\mathcal L X \to X$ given by evaluating at the basepoint $0 \in S^1$, ...