# 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 $\pi_{2p+1}(S^3)$? ...
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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 ...
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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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### 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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### 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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### 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'...
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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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### 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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### 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 ...
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 ...