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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Hatcher 4.2.19 homotopy group of skeleton of $K(G,1)$
Let $X$ be a $K(G,1)$ which is a CW complex. We want to show that $\pi_n(X^n)$ is free (abelian when $n\geq 2$ where $X^n$ is the $n$ skeleton of $X$.
I thought I could choose any model for a $K(G,1)$ ...
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No $p$-torsion in $\pi_{2p+1}(S^3)$?
For a prime $p$, it is well-known that the first $p$-torsion in $\pi_i(S^3)$ appears at $i=2p$.
Naturally, one is curious about the next homotopy group:
Is there any $p$-torsion in $\pi_{2p+1}(S^3)$?
...
- 43
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70
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Homotopic maps induce the same maps on homotopy groups given path connected spaces [duplicate]
Given pointed maps $f,g: (X, x_0) \longrightarrow (Y, y_0)$ that are pointed homotopic then $f_* = g_* : \pi_n(X, x_0) \longrightarrow \pi_n(Y, y_0)$.
Now let $X$ and $Y$ be path connected. Then if $f,...
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A certain map involving Eilenberg-MacLane spaces is a well-defined homomorphism (towards a universal coefficient theorem)
I'm working through Arkowitz's Introduction to Homotopy Theory for self-study. In the beginning of chapter 5 section 2, we are aiming for a proof of (a simplified version of) the Universal Coefficient ...
- 466
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1
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96
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Map induced by inverse on homotopy groups of SO(n)
Let $SO(n)$ be the special orthogonal group. There is a self-diffeomorphism $\phi:SO(n)\to SO(n)$ taking an element $A$ to $A^{-1}$. I am interested in the induced automorphism $\phi_\ast:\pi_i(SO(n))\...
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Uniqueness of comultiplication for sufficiently connected spaces of restricted dimensions
I'm working through an introductory homotopy theory book (Arkowitz) for self-study, and I'm a bit stuck on the following exercise. Not a lot of machinery is available at this point in the book. I'm ...
- 466
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Bilinear pairing on homotopy groups
Let $X,Y,Z$ be pointed spaces, and $f:X \wedge Y \rightarrow Z$ a map.
Then, $f$ induces a bilinear pairing $\pi_n(X) \times \pi_m(Y) \rightarrow \pi_{n+m}(Z)$ ($n,m \geq 1$).
I see what the pairing ...
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Geometric characteristics or combinatorics analog for homotopy groups of spheres other than $\pi_n(S^n)$
The fact that $\pi_1(S^1) = \mathbb Z$ is geometrically intuitive.
It is linked to a wealth of concepts and results, notably the winding number, residue theorem on $\mathbb C$, etc.
There is a ...
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Computation of the LES of homotopy groups associated with compact symmetric spaces
I am looking for an efficient way to compute the homotopy groups, as well as morphisms between them, of certain matrix groups and compact symmetric spaces. To be specific, I want to determine the long ...
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Showing that inducded maps to relative homotopy groups is 0
Let $X, Y, Z$ be based spaces and define
$F_2(X, Y, Z) = \{(x, y, z) \in X \times Y \times Z|$ at least one of $x, y, z$ is $= * \}$
Prove that the inclusion $F_2 (X, Y, Z) \to X \times Y \times Z$ ...
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The fundamental group of the loop space of $(X,x_0)$ with the base point chosen not to be the constant loop in $x_0$
My question is pretty simple although I have not been able to find an answer yet:
Let $c_{x_0}\in\Omega(X,x_0)$ denote the constant loop in $x_0$. Then, by standard homotopy theoretical arguments, we ...
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Inverse of a homotopy class in the $n$-th homotopy group
I just started to study higher homotopy groups in the book Introduction to topology by V.A. Vassiliev. In such book, the author defines the product of two homotopy classes in $\pi_{n}(X,x_0)$ as ...
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Can a simply connected manifold satisfy $M\simeq M\times M?$
Let $M$ be a simply connected, (finite dimensional) smooth manifold. Is it possible that $M$ is homotopy equivalent to $M\times M,$ without $M$ being contractible? This would imply $\pi_n(M)\times\...
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Question about relative homotopy Groups
I have a question about this passage of Hatcher book "Algebraic Topology":
A sum operation is defined in $\pi_n(X,A,x_0)$ by the same formulas as for $\pi_n(X,x_0)$, except that the ...
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Computing homotopy classes of maps between small finite CW complexes
Given finite CW complexes $X$ and $Y$ with $X$ connected, how can one go about computing the set of (baseless) homotopy classes of maps $[X, Y]$? Does a general procedure/algorithm exist?
I should say ...
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Nullhomotopicity of $\mathbb{S}/p \to^p \mathbb{S}/p$ for $p=2$ and $p \neq 2$?
For a given spectra $X$ we have $X/p$ defined as the cofiber $X \to^p X$ where the map is basically defined via defining it on the sphere spectrum $\mathbb{S}$! To define $\cdot p$ on the sphere ...
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Finite CW-complex with finite homotopy groups
Is it possible to find a finite (connected) CW complex $K$ such that $\pi_i(K)$ is finite $\forall i\geq 1$?
Can I ask for $K$ to be simpliy connected as well or to have only a finite number of non ...
- 145
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Map on homotopy groups induced by the swap map
Let $X$ be a pointed topological space and let $\tau:\Sigma^2X\to\Sigma^2X$ be the map swapping the two suspensions. I want to understand the map on homotopy groups induced by $\tau$.
I tried ...
- 1,554
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Reference for Jacobi identity for general Whitehead product
I know that for "regular" Whitehead product $[\cdot,\cdot]:\pi_k(X)\times\pi_l(X)\to\pi_{k+l-1}(X)$ there is Jacobi identity in the following form.
We have $\alpha\in \pi_k(X), ~\beta\in\...
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Complement of low dimension subsets and higher homotopy group
I suspect the following statement is wrong:
Let $X$ be a smooth connected manifold, and $V \subseteq X$ be an embedding submanifold of codimension $\geq d+2$ and $i: X-V \rightarrow X$ the inclusion ...
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Prove that $\pi_{2n+1}(\mathbb CP^n) \cong \mathbb Z$ using the mapping cone
Let $f : S^{2n+1} \rightarrow \mathbb C P^n$ be a continuous map and $X := \mathrm{Cone}(S^{2n+1}) \cup_f \mathbb C P^n$.
We have the following long exact sequence:
$$\cdots \rightarrow \pi_{2n+2}(X) \...
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Infinite symmetric product of an $H$-space and graded product structure on homotopy groups.
Assume $X$ is an $H$-space and let's look at homotopy groups of $SP(X)$ where $SP$ here is the infinite symmetric product. By Dold-Thom we have $\pi_i(SP(X))\cong \tilde{H}_i(X)$. Since $X$ is an $H$-...
- 1,354
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Kunneth and Dold-Thom
Let $X$ and $Y$ be connected CW complexes. By Kunneth's theorem we have a map of the form $H_i(X)\times H_j(Y) \rightarrow H_{i+j}(X\times Y)$. By Dold-Thom this translates to a map of homotopy groups ...
- 1,354
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Calculate homotopy groups of $\mathbb{Z}_2$-equivariant loop spaces of "complex" topological spaces
Let $X$ be a topological space such that complex conjugation is defined (e.g. $\mathbb{C}^n$) and let us define the set of maps $$S_d:= \left\{f: (I^d,\partial I^d)\to (X,x_0)\mid \overline{f(k)} = f(...
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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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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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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'...
- 147
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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 ...
- 1,459
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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 ...
- 1,459
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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 = ...
- 1,459
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108
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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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98
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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)=[...
- 3,593
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1
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95
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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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138
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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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1
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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&...
- 454
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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 ...
- 1,655
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2
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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 ...
- 1,459
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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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