Skip to main content

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 $\gamma:(S^n,s_0)\rightarrow(X,x_0).$

Filter by
Sorted by
Tagged with
0 votes
0 answers
34 views

How do we describe the 'isomorphism subspace' of the mapping space $Map_{C}(x,y)$ for an infinity category C?

When $E$ is a locally free sheaf of rank n on a classical scheme $X$, there is a sheaf $Isom$ on the category $Sch_{X}$ defined as $(S\rightarrow X)\rightarrow Isom_{O_{S}}(O_{S}^{n},E)$. And this ...
Yang's user avatar
  • 95
0 votes
0 answers
22 views

A model category whose Hammock localization gives the infinite category of infinite functors from differential graded algebras to simplicial sets

I have been reading a little bit of Homotopical Algebraic Geometry I: Topos theory by Toën and Vezzosi (HAG 1) (https://arxiv.org/pdf/math/0207028) in the search of a specific result I am trying to ...
Tomas Fernandez's user avatar
3 votes
1 answer
61 views

What goes wrong when we try to define Eilenberg-MacLane space $K(G,n)$ with non-abelian $G$ for $n>1$?

The Eilenberg-MacLane space $K(G,n)$ is defined for non-abelian $G$ for $n=1$ and abelian $G$ for $n>1$. I know there is a theorem that states that the higher homotopy group $\pi_n(X)$ is abelian ...
Leo L.'s user avatar
  • 358
3 votes
1 answer
57 views

Composing sums of elements in $\pi_{n}(S^m)$

Suppose $m,n,r\geq 1$ and that we have homotopy group elements $f,g\in \pi_{n}(S^m)$ and $\alpha,\beta\in \pi_r(S^n)$. Certainly, it is the case that $f\circ (\alpha+\beta)=(f\circ\alpha)+(f\circ\beta)...
J.K.T.'s user avatar
  • 1,568
1 vote
0 answers
21 views

suspension map between homotopy groups of spheres

Let $p$ be an odd prime. For $n\geq 3$, suspension map $\Sigma:\pi_{n+2p-3}(S^n;p)\to\pi_{n+2p-2}(S^{n+1};p)$ is an isomorphism on $p$-primary part and $\pi_{2p}(S^3;p)=\mathbb{Z}/p$. Meanwhile, there ...
Visible Wings's user avatar
0 votes
0 answers
41 views

Associativity of a spectrum with relation to complex bordism spectrum

I am reading Lurie's notes about chromatic homotopy theory. In Lecture 22, he considers the p-local complex bordism spectrum $\operatorname{MU}^p$ with $\pi_*(\operatorname{MU}^p)=\mathbb{Z}_p [t_1,...
Runner's user avatar
  • 145
3 votes
0 answers
54 views

Is openness crucial in Puppe's proof of the Blakers-Massey Theorem?

Suppose a space $Y$ is given along two subspaces $Y_1,Y_2$ s.t. $Y=Y_1^{\circ}\cup Y_2^{\circ}$ and the intersection $Y_0=Y_1\cap Y_2$ is non-empty. If $(Y_1,Y_0)$ is $p$-connected and $(Y_2,Y_0)$ is $...
Thorgott's user avatar
  • 12.7k
1 vote
1 answer
52 views

Homotopy sequence of a weak fibration with a section

Suppose that $p:E \to B$ is a weak fibration with a section $s:B \to E$ ($p \circ s=id_B$). Also here $F=p^{-1}(b_0)$. I need to show that the sequence $0 \to \pi_n(F,s(b_0)) \xrightarrow {i_*}\pi_n(E,...
givememeth's user avatar
2 votes
1 answer
66 views

What is the difference between an E∞-Space and an H-Space

An H-space produces a commutative monoid in the homotopy category of based CW-complexes, but so does an E∞-space. In my approach to higher mathematics, I intend to use one of these or perhaps both, ...
user avatar
0 votes
1 answer
54 views

S : ∞-Grpd₀ ⭢ ∞-Grpd₀ such that Homology of X is the homotopy of SX?

Let $X$ be a based CW-complex, and write ∞-Grpd₋₁ for the category of based CW-complexes. I am wondering if there is some operation S or functor on spaces, remaining in the homotopy category, which ...
user avatar
4 votes
1 answer
149 views

Higher Homotopy Groups of Spheres: Motivation and State of the Art

I know that the higher homotopy groups of spheres are extremely challenging to compute and have driven a lot of research in algebraic topology. I was under the impression that many such computaions ...
Mithrandir's user avatar
1 vote
0 answers
54 views

Proving that $\pi_q(T(S^m)\setminus f^{-1}(A))\cong \pi_q(S^m\setminus A)$ for $q\ge 1$

Consider the tangent fiber bundle $(T(S^m), f, S^m)$ and the finite set $A\subseteq S^m$. Here $f:T(S^m)\to S^m, f(x, v)=x$ just to have everything defined. I am first asked to prove that the ...
math_is_hard's user avatar
3 votes
0 answers
50 views

For an explicit mapping $S^m\to S^6$ for $m<12$, how can we understand whether it is homotopy zero?

In my work I have mappings from $S^m$ to $S^6$, where $m\leq 11$. People know the homotopy group of these mappings. However, is there some algorithm that helps to recognize when mappings (we can ...
Timur's user avatar
  • 31
0 votes
2 answers
92 views

"Homotopy" group with the torus $\mathbb{T}^2$ as a domain and the sphere $\mathbb{S}^2$ as codomain

I'm watching this lecture in Condensed Matter physics that concerns topological aspects of materials. In particular, the lecturer is considering the homotopy groups of various spaces. At one ...
Níckolas Alves's user avatar
4 votes
1 answer
156 views

What is the value of homotopy group $\pi_1(SU(2)\times U(1))$

Question background: In particle physics, the weak interaction symmetry group is described by $G=SU(2)\times U(1)$, which is spontaneously broken into $H=U(1)_{em}$. ($U(1)_{em}$ is the ...
Daren's user avatar
  • 225
7 votes
1 answer
142 views

A possible error in May's Concise Algebraic Topology (prespectra)

In chapter 22.1 of May's A Concise Course in Algebraic Topology, he claims that the prespectrum $\{T_n\}$ of spaces where each $T_n$ is $(n-1)$-connected yields a reduced homology theory by setting $\...
Emory Sun's user avatar
  • 1,650
4 votes
0 answers
107 views

Is the Whitehead product $[\iota_2, \eta] : \pi_4\mathbb{S}^2$ trivial?

Let $\iota_2 : \pi_2\mathbb{S}^2$ be a generator and let $\eta : \pi_3\mathbb{S}^2$ be the Hopf map. The Whitehead product $[\iota_2, [\iota_2, \iota_2]] : \pi_4\mathbb{S}^2$ must be trivial, because $...
Tom's user avatar
  • 41
0 votes
1 answer
104 views

Computing the homotopy groups $\pi_n(\mathbb RP^2 \times \mathbb S^3).$

How can I Compute the homotopy groups $\pi_n(\mathbb RP^2 \times \mathbb S^3)$ ? Which theorems will help me in this? Should I use Kunneth theorem? I have seen this post here Calculate $[\mathbb{S}^n,...
user avatar
1 vote
2 answers
114 views

Ranking topological invariants by "strength?"

Is it possible to rank the common topological invariants (homology, cohomology, homotopy) according to their "strength?" By this I mean, can spaces have different homotopy groups yet the ...
raynea's user avatar
  • 343
1 vote
1 answer
114 views

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 $K(G,1)$ ...
DevVorb's user avatar
  • 1,495
2 votes
1 answer
214 views

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 $\pi_{2p+1}(S^3)$? ...
Adam Shaw's user avatar
1 vote
0 answers
73 views

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,...
user avatar
1 vote
1 answer
156 views

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 ...
abstractnonsense's user avatar
2 votes
1 answer
116 views

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))\...
Sam Ballas's user avatar
1 vote
1 answer
211 views

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 ...
abstractnonsense's user avatar
2 votes
1 answer
51 views

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 ...
mathable's user avatar
  • 444
2 votes
0 answers
60 views

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 ...
Jean-Armand Moroni's user avatar
4 votes
0 answers
81 views

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 ...
Hyeongmuk LIM's user avatar
0 votes
0 answers
26 views

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 $F_2 (X, Y, Z) \to X \times Y \times Z$ ...
Subham Jaiswal's user avatar
1 vote
1 answer
64 views

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 ...
Mathematics enthusiast's user avatar
3 votes
0 answers
147 views

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 ...
ferolimen's user avatar
  • 630
11 votes
2 answers
453 views

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\...
JLA's user avatar
  • 6,534
1 vote
1 answer
102 views

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 ...
Horned Sphere's user avatar
1 vote
0 answers
66 views

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 ...
xzd209's user avatar
  • 335
5 votes
0 answers
71 views

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 ...
user135743's user avatar
5 votes
1 answer
111 views

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 ...
Sloth's user avatar
  • 145
2 votes
0 answers
58 views

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 ...
Tipping Octopus's user avatar
2 votes
0 answers
49 views

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\...
Haldot's user avatar
  • 830
0 votes
0 answers
62 views

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 ...
hyyyyy's user avatar
  • 350
1 vote
0 answers
55 views

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) \...
Luke's user avatar
  • 765
3 votes
0 answers
103 views

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 $H$-...
user127776's user avatar
  • 1,364
1 vote
0 answers
73 views

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 ...
user127776's user avatar
  • 1,364
5 votes
1 answer
107 views

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(...
Mathematics enthusiast's user avatar
2 votes
1 answer
68 views

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 ...
Jesus's user avatar
  • 1,798
0 votes
1 answer
94 views

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 ...
Mithrandir's user avatar
3 votes
0 answers
151 views

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}),...
user267839's user avatar
  • 7,663
1 vote
1 answer
177 views

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 ...
SAR's user avatar
  • 41
4 votes
2 answers
239 views

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,\...
Andrea Antinucci's user avatar
3 votes
1 answer
114 views

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 ...
LoneStar's user avatar
  • 912
3 votes
1 answer
213 views

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 ...
MathZilla's user avatar
  • 257