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 ...
4
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1answer
35 views
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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1answer
53 views
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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1answer
9 views
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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1answer
78 views
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 ...
7
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1answer
55 views
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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0answers
28 views
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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1answer
46 views
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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1answer
32 views
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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71 views
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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1answer
27 views
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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1answer
205 views
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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0answers
51 views
$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 $$
The question is to compute $H_k(T^n)$ without using Kuneth formula.
So far I've noticed that $...
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2answers
64 views
Do homotopy pullbacks always compose?
Classical pullbacks compose, as is easily checked with the universal property. More precisely, if $\require{AMScd} \begin{CD}
A @>>> B\\ @VVV @VVV\\
C @>>> D
\end{CD}$
and $\require{...
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1answer
128 views
What is the shape group?
I'm familiar with topology and category theory a little bit as well as inverse limits of inverse systems. I would like to understand the shape group better.
For example, one could describe the ...
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Construct homotopy using illustration
I am currently studying simplicial topology and got stuck with a homotopy construction: $$ \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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1answer
18 views
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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1answer
34 views
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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1answer
64 views
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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1answer
42 views
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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32 views
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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2answers
55 views
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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0answers
43 views
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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0answers
56 views
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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2answers
55 views
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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1answer
37 views
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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1answer
29 views
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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2answers
65 views
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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1answer
71 views
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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60 views
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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141 views
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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29 views
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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1answer
55 views
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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1answer
58 views
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
$$(\eta_1\sqcup\eta_2)^*:[\mathbb{S}^1\vee\mathbb{S}^1,\mathbb{S}^1\vee\...
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0answers
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When does a fibration $f:X\rightarrow Y$ in a model category admit a section?.
If we have a fibration $f:X\rightarrow Y$ in a model category $C$, where $Y$ is cofibrant and both $X, Y$ are fibrant.
Does f admit a section (right inverse)?. If it does not work in general, ...
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1answer
40 views
Singular Cohomology satisfies the Mittag-Leffler Condition on CW complexes?
Is this true?
Let $H$ be singular cohomology. On an arbitrary CW complex $X$, given a filtration $X^0 \subseteq X^1 \subseteq \cdots X^n \subseteq \cdots \subseteq X$. Then
$$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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2answers
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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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0answers
45 views
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
54 views
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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0answers
32 views
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
29 views
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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0answers
34 views
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 ...
3
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1answer
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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1answer
38 views
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 $0 \in S^1$, ...
