Questions tagged [higher-homotopy-groups]
For questions related to higher homotopy groups. A higher homotopy group, $\pi_n(X,x_0)$, is the set of based homotopy classes of based maps $\gamma:(S^n,s_0)\rightarrow(X,x_0).$
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Where can I get an alternative proof of this characterisation of weak equivalences?
Brayton Gray's book Homotopy Theory is the only book I've seen that states and proves the following theorem.
Lemma 16.17. Suppose $f: X \to Y$ is a base point preserving map. $f$ is a weak homotopy ...
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Eckmann–Hilton Argument and magma homomorphisms
The Eckmann-Hilton result is as follows:
Let $X$ be a set equipped with two binary operations $\circ$ and $\otimes$, and suppose
$\circ$ and $\otimes$ are both unital, meaning there are identity
...
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Modification of a CW-Complex $X$ with control over Homology groups
Let $X$ be a finite dimensional CW-complex, ie we endow it with CW structure such that there exist a $n_0>0$ such that there are no $n$-simplices for $n>n_0$. Let $[\beta] \in H_k(X, \mathbb{Z}),...
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Surjectivity of J homomorphism
Can somebody say something about surjectivity of the $J$ homomorphism $J_{7,7}$ : $\pi_{7}$SO($7$) $\rightarrow$ $\pi_{14}(S^7)$ ? Husemoller in Fibre Bundles says this :
Where $M_{n}^{1}$ denotes ...
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Hurewicz homomorphism and generators of the cohomology ring
Let $X$ be a compact manifold of dimension $\geq k$. Denote by
\begin{equation}
h: \pi _k(X) \rightarrow H_k(X,\mathbb{Z})
\end{equation}
be Hurewicz homomorphism and by $\Gamma _k(X)\subset H_k(X,\...
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Is $\pi_2 (X)$ a projective $\mathbb{Z}\pi_1 (X)$-module when $\pi_1 (Z)$ is free?
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...
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Defining the sheaf of bigraded homotopy groups in motivic homotopy theory.
I have been learning motivic homotopy theory from these notes and on page 154 (page 8 of the pdf), the author defines $\pi_{p,q}(E)$ where $E$ is an $(s,t)$-bispectrum. He defines it as the sheaf of ...
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What do the homotopy classes of higher homotopy groups mean?
I am new to studying homotopy and the groups associated with them (as I come from a physics background, so I might be missing a few rigorous details), but when I am trying to apply them to objects ...
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Shankar Monopole belonging to the class $-1$ of $\pi_3(\mathbb{R}P^3)$
I am having a lot of trouble trying to understand how the classes of homotopy groups relate to point-defects in physics (and how they can be used/represent in general). This is a problem from Nakahara'...
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CW-complex $X$ has the homotopy type of a finite wedge of spheres
For a given map $\phi :X\longrightarrow Y$, the mapping cylinder of $\phi$ is defined by $M_{\phi}:=Y\bigcup_{\phi} (X \times \{ 1\})$. Denote $\pi_n (M_{\phi},X \times \{ 1\} )$ by $\pi_n (\...
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Are there interesting $H$-cogroups (other than spheres)?
I know that if a pointed topological space $(Q,\ast)$ is an $H$-cogroup, then we can define a functor:
$$[(Q,\ast),-]:\mathbf{Top}_\ast\to \mathbf{Groups}$$
that acts similarly to the homotopy groups ...
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The natural map $\operatorname{colim}_i\pi_n(X_i)\to \pi_n(X)$ is an isomorphism for each $n$.
One of the lemma in May's Concise course in algebraic topology is the following:
If $X$ is the colimit of a sequence of inclusions $X_i\to X_{i+1}$ of based spaces, then the natural map
$$\...
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The cartesian product of contractible spaces is contractible [duplicate]
Let $X_i$, $i\in I$ be contractible spaces. Is the Cartesian product $\prod_iX_i$ contractible, too?
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Inverses in the spherical interpetration of higher homotopy groups
I have just started studying the higher homotopy groups $\pi_n(X, x_0)$ in more detail than I have before and I am getting confused about the inverse operation which makes $\pi_n(X, x_0)$ a group. I'...
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Does the proof that the homotopy groups are abelian in Switzer's book only work for hausdorff spaces?
The proof that the homotopy groups of a pointed space, $\pi_n(X,x_0)$, are abelian for $n\geq 2$, given by Robert P. Switzer in his book "Algebraic Topology - Homology and Homotopy", relies ...
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Loop-Suspension adjunction unit is stable equivalence
Let $X$ be a sequential spectrum, then the unit of the $(\Sigma,\Omega)$-adjunction yields a map $\eta : X \to \Omega \Sigma X$. The authors of the book Foundations of Stable Homotopy Theory (p. 48) ...
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The homotopy fiber $F$ of the map $p_2 : S^2 \to \mathbb{C} P^\infty$ is weakly homotopy equivalent to $S^3$
I'm stuck with the following question:
Consider the Postnikov tower $S^2 \to \ldots \to P_3S^2 \to P_2S^2 \to P_1S^2$ of the $2$-sphere.
Show that $P_2S^2$ and $\mathbb{C} P^\infty$ are weakly ...
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Construction of Eilenberg-Maclane Space
I have just learned about the construction of the Eilenberg-Maclane space $K(A,n)$ ($n \geq 1$). I am using Hatcher's text. He briefly describes the construction on p.365. I will recap the ...
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Order of the suspension of the Hopf map
I am working on the following problem from an old qualifying exam in algebraic topology:
The Hopf map $\eta:S^3 \rightarrow S^2$ can be defined as $\eta(z_0,z_1) = z_0/z_1$, with $S^3$ the unit ...
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Homotopy groups of infinite dimension lens space
Let $n > 1$. We define the infinite dimensional lens space $L$ as follows. Let $S^{\infty}$ be the unit sphere in the infinite dimensional complex vector space $C^{\infty}$, and let $\mathbb{Z}/n = ...
- 1,399
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Homotopy groups of wedge of spaces [duplicate]
Let $X$ and $Y$ be topological spaces with $\pi_{i}(X) = \pi_{i}(Y) = 0$ for $i \geqslant n$
Is it true that $\pi_{i}(X\vee Y) = 0$ for $i \geqslant n$?
If no, is it true for $CW$ complexes?
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Equivalence between two homotopy groups definition
Homotopy groups of a (pointed) topological space can be defined in multiple ways. In particular, I'm interested in proving rigourosly that the following two definitions are equivalent:
$\pi_n(X,x_0)=[...
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Examples of spaces with non-abelian $\pi_2(X, A)$
It is well known that $\pi_n(X)$ are abelian for all $n\geq2$, but this only follows in relative homotopy groups for $n\geq3$. I am writing some notes on higher homotopy groups, and was searching for ...
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Homotopy Lifting Property in Hatcher's Spectral Sequences
Let $\pi: X \to B$ be a fibration with $B$
path-connected CW complex filtered by
$p$-skeleta $B_0 \subset B^1 \subset ... \subset B^p \subset ...
B^{\dim(B)}=B$. This induces a filtration on $X$ via
$...
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A pair $(X,A)$ is $n$-connected iff the inclusion $A\rightarrow X$ is an $n$-equivalence.
This book says followings:
Definition. A continuous map $e:A\rightarrow Z$ is an n-equivalence if, for all $y\in Y$, the induced map $e_{*}:\pi_q(Y,y)\rightarrow \pi_q(Z,e(y))$ is an injection for $q&...
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Explicit description of $\pi_1(M)$-action on the relative homotopy groups $\pi_k(\Omega_0M,M)$
Let $M$ be a path-connected space (say, a connected manifold) and denote by $\Omega_0 M$ the based loop space component of the constant loop $x_0$. If $c \colon M \to \Omega_0 M$ denotes the obvious ...
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If we have a weak homotopy equivalence $f:X \rightarrow Y$, does there necessarily exist a weak homotopy equivalence $g:Y \rightarrow X$?
Suppose $X$ and $Y$ are topological spaces. Is it true that, if we have a weak homotopy equivalence $f:X \rightarrow Y$, then there exists a weak homotopy equivalence $g:Y \rightarrow X$?
I can't ...
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Why is this CW complex connected?
I am working on Exercise 4.1.11 in Hatcher's Algebraic Topology text, which is as follows:
Show that a CW complex $X$ is contractible if it is the union of an increasing sequence of subcomplexes $X_1 ...
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Homotopy group of pairs: equivalent descriptions
I'm reading May's A concise Course in Algebraic Topology and I have a question about the definition of the homotopy group of a pair. Given a pair $(X,A)$ of pointed topological spaces, the relative ...
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What is $\pi_3(S^2\times S^1)$?
What is the third homotopy group of $S^2\times S^1$? Here $S^n$ means the $n$-dimensional sphere.
I am a physicist and did not take courses on too mathematical topics. I just need a fact of this in a ...
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A map between connected $n$-dimensional CW complexes is a homotopy equivalence
Show that a map between connected $n$-dimensional CW complexes is a homotopy equivalence if it induces an isomorphism on $\pi_i$ for $i\leq n$. [Pass to universal covers an use homology].
I'm in the ...
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Action of $\pi_1(\Bbb RP^n)$ on $\pi_n(\Bbb RP^n)$
The following is one of Hatcher AT exercise problem.
Show the action of $\pi_1(\Bbb RP^n)$ on $\pi_n(\Bbb RP^n)\simeq\Bbb Z$ is trivial for $n$ odd and nontrivial for $n$ even.
Using the fact that ...
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Does there exist a smooth compact manifold whose homotopy groups are not finitely generated?
Does there exist a smooth compact manifold whose homotopy groups are not finitely generated?
I found a counter-example for topological manifolds, but I did not understand whether it is possible to ...
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Long exact sequence of homotopy group
$\pi_1(X,x_0)\xrightarrow{j_*}\pi_1(X,A,x_0)\xrightarrow{\partial}\pi_0(A,x_0)\xrightarrow{i_*}\pi_0(X,x_0)$ is exact.
Here, I interpreted $I^0 = \{1\}$.
$\newcommand{\im}{\operatorname{im}}$...
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Why does the contractibility of $CX$ imply that $\pi_n(CX,X,x_0)\cong\pi_{n-1}(X,x_0)$?
My Algebraic Topology notes say that because the cone of $X$ is contractible (which is easy enough to see, $H([x,t],s)=[x,t(1-s)]$ does the job) the Long Exact Sequence of Relative Homotopy Groups ...
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Definition of weak homotopy equivalence
Recall a weak homotopy equivalence is a continuous map $f:X_1 \rightarrow X_2$ that induces a bijection $f_*:[Y,X_1] \rightarrow [Y,X_2]$ for any CW-complex $Y$, where $[Y,X]$ denotes the homotopy ...
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If higher homotopy groups are trivial, then the fundamental group is a complete invariant?
Let $X$ and $Y$ be two n-manifolds all of whose higher homotopy groups are trivial, and the first homotopy groups are isomorphic (but the existence of a mapping inducing an isomorphism is not assumed)....
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Showing an element of $\pi_n(S^n)$ is a sum of maps acting locally, invertibly near one point
By simplicial approximation, a continuous map $f:S^n \to S^n$ is homotopy equivalant to a simplicial map. Such a map must send vertices to vertices, from which the full map is determined by linearly ...
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Covering ${\Bbb R}^{3}$ by disoint circles.
Theorem. It is impossible to make ${\Bbb R}^{3} = \underset{\lambda \in \Lambda}{\bigcup} {(S_1)}_{\lambda}$, where each $(S_1)_{\lambda}$ never meets other $(S_1)_{\lambda'}$. That is any two circles ...
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Is an homotopy class represented by the homotopy class of inclusion as subspace?
The following question is related to this other question. In the question I have to show the exactness of $$\pi_1(A,x_0) \overset{i_*}{\longmapsto} \pi_1(X,x_0)\overset{j_*}{\longmapsto} \pi_1(X,A,x_0)...
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Understanding Theorem 6.3.1 Tom Dieck
I'd like to understand a part of the Theorem on Tammo Tom Dieck p.$130$. The theorem is the following:
I do understand that $T$ is homeomorphic to $J^n$ with the exchange of the last two coordinates,...
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Understanding $CW$ approximation theorem.
The statement is the following:
Approximation theorem: Let $A$ be a $CW$ and $k \in \mathbb{Z} \cup \left\lbrace-1\right\rbrace$. Let $Y$ be a topological space with $f:
A \longmapsto Y$ continuous ...
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Can the fundamental group $\pi_1(X)$ of topological space $X$ be non-abelian? or be continuous? [closed]
We are familiar with the homotopy group of spheres $S^n$: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres.
There we learn that
$\pi_d(S^n)$ must be abelian and discrete. They are direct sum ...
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Homotopy groups of projective Lie groups PO(N), PSO(N), and PSpin(N)
Previously, we have learned from Homotopy groups O(N) and SO(N): $\pi_m(O(N))$ v.s. $\pi_m(SO(N))$ that:
$\pi_m(SO(N))$: a table consisting of the groups $\pi_m(SO(N))$ for $1 \leq m \leq 15$ and $2 \...
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Question regarding the proof of Theorem 6.1.2 Tom Dieck
I'd like to understand the proof of the exact sequence of the pair in homotopy. I'm reffering to the one I found here: https://www.maths.ed.ac.uk/~v1ranick/papers/diecktop.pdf (p.123)
The implication ...
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Bijection $[((X,A)),((Y \star))]\simeq[X/A,Y]^0$
I'd like to prove the following, to well define homotopy classes and essentialy work with the most useful space, (i.e being able to switch between $(I,\partial I)$ and $\mathbb{S}^1$):
Lemma : A ...
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The n-sphere is not a deformation retract of the bouquet of k n-spheres
I'm trying to show that $S^n$ is not a deformation retract of $\bigvee^k S^n$ as a generalization of a proof (or an attempt) I made showing that $S^1$ is not a deformation retract of $S^1\vee S^1$, ...
user949426
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Reflection of $S^n$ is the inverse of the identity map in $\pi_n(S^n)$
Regard $S^n$ as a subspace of $\Bbb R^{n+1}$, and consider the reflection $f:S^n\to S^n$, $(x_1,\dots,x_{n+1})\mapsto (-x_1,x_2,\dots,x_{n+1})$. This defines an element of the group $\pi_n(S^n)$. How ...
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Five lemma for homotopy exact sequences of triple
Suppose we have topological spaces $B\subset A\subset X$ and $B'\subset A'\subset X'$, and a continuous map $f:X\to X'$ with $f(A)\subset A'$, $f(B)\subset B'$. Consider the homotopy long exact ...
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Homotopy cardinality of fiber bundles
Consider the Homotopy cardinality (or $\infty$-groupoid cardinality) $\chi(X):=\sum_{[x]\in\pi_0(X)}\prod_{i\geq0}|\pi_i(X,x)|^{(-1)^{i+1}}$ associated to a space $X$. Suppose we have a fibre bundle $...
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