Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Homotopy theory of A n-spaces in Lie groups
Mitsunobu Tsutaya (Kyushu Univ.)

Kyoto University
June 26, 2019

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Outline1. Background – A n-space
2. Higher homotopy commutativities
3. Full higher homotopy commutativity of Lie groups
4. Samelson products in Lie groups

Background – A n -space

Higher homotopy commutativitiesFull higher commutativity of Lie groups

1. Background – A n-space
I H -space
II

Homotopical associativity
Higher homotopy associativity

Samelson products in Lie groups

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

H-space

G: pointed space
m: G × G → G continuous map
Definition(G, m) is a (homotopy unital) H-spacedef⇐⇒ x 7→ m(∗, x) and x 7→ m(x, ∗) are homotopic to id : G → G.
ExampleA topological group is an H-space.
Example

X: pointed spaceΩX = {` : [0, 1] → X | `(0) = `(1) = ∗}: based loop space=⇒ ΩX is an H-space with m: concatenation of loops.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Homotopical associativity

G = (G, m): H-space
Definition
G is homotopy associativedef⇐⇒ (x, y, z) 7→ x(yz) and (x, y, z) 7→ (xy)z are homotopic.
ExampleTopological groups and based loop spaces are homotopy
associative.
ExampleS7 admits a structure of an H-space (cf. unit vectors in octonions),
but no homotopy associative H-space.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher homotopy associativity

G: homotopy associative H-spaceM(t; x, y, z): “associating homotopy” such that
M(0; x, y, z) = x(yz),M(1; x, y, z) = (xy)zx((yz)w)xM(t;y,z,w)

M(t;x,yz,w)(x(yz))wx(y(zw))M(t;x,y,zw)(xy)(zw)

M(t;x,y,z)w
M(t;xy,z,w)((xy)z)w

There are two canonical way to associate x(y(zw)) and ((xy)z)w. If
these two homotoies are homotopic, we say G is an A4 -space.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher homotopy associativity→ Generalized to A n-spaces ( n = 1, 2, . . . , ∞).
n=2

n=4

xy

n=3
x(yz)

x(y(zw))(xy)z(xy)(zw)

x((yz)w)n=5(x(yz))w((xy)z)w

The parameter spaces are called associahedra K n (Stasheff,1963).
ExampleTopological groups and based loop spaces are A∞ -spaces.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher homotopy associativity

A n-maps (morphisms between A n-spaces) are higher homotopies
parametrized by multiplihedra J n (Stasheff, Boardman–Vogt,Iwase).
n=1

f (x)

n=2
f (xy)

n = 3 f (x(yz))
f (x) f (yz)f (x) f (y) f (x)( f (y) f (z))

f ((xy)z)n=4

f (xy) f (z)( f (x) f (y)) f (z)ExampleHomomorphisms between topological groups are A∞ -maps.
ExampleA topological group G and an A∞ -space ΩBG are A∞ -equivalent(i.e. ∃G → ΩBG: A∞ -map, homotopy equivalence).

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

2. Higher homotopy commutativities
Homotopy commutativity
Higher homotopy commutativities
I p-localizatonII

Samelson products in Lie groups

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groupsSamelson products in Lie groups

Homotopy commutativity

G: H-space
Definition
G is homotopy commutativedef⇐⇒ two maps (x, y) 7→ xy and (x, y) 7→ yx are homotopic.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher homotopy commutativities

Higher homotopy commutativities:
Williams Ck-space
xyz
xyyx

Sugawara Ck-space
xzyyxzzxyyzxzyx

xy

x1x2y1y2

y1x1x2y2

x1y1y2x2

x1y1x2y2yx

The higher homotopies for Williams C k -space are parametrized bypermutohedra P k (convex hull of (σ(1), . . . , σ(k)) ∈ R k , σ:permutation).
G: A n-space
Definition
G is a Sugawara C k -spacedef⇐⇒ the binary operation G × G → G is an A k -map.
cf. Γ: group, Γ × Γ → Γ: homomorphism ⇒ Γ: commutative

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher homotopy commutativities

Fact:
Sugawara C2 -space ⇔ Williams C2 -space ⇔ homotopy
commutative
Fact:
Sugawara C k -space =⇒ Williams C k -space.
Question
What can we say about (higher) homotopy commutativity of Liegroups?
RemarkA connected Lie group G is A∞ -equivalent to the maximal compact
subgroup K by the inclusion K → G.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher homotopy commutativities

G: compact connected Lie group
Theorem (Araki–James–Thomas, 1960)

G is homotopy commutative⇒ G is a torus.
Theorem (Bott, 1960)

SU(s) and SU(t) are not homotopy commutative in SU(s + t − 1).
In these examples, there is no difference between the usual
commutativity and the homotopy commutativity.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

p-localizaton

p: prime number or 0 ∈ Z≥0∃L p : Spaces → Spaces: p-localizaton functor∃η : X → L p X: natural map (We denote X(p) = L p X.)
I

X: simply connected⇒ π n(X(p) )  π n(X)(p) = π n(X) ⊗ Z(p) ,η∗ : π n(X) → π n(X(p) ) is the canonical homomorphismof the p-localization.

Some “obstructions” vanish after localization.
Fact
G: A∞ -space⇒ G(p) : A∞ -space.

→ What can we say about (higher) homotopy commutativity ofp-localized Lie groups?

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

3. Full higher commutativity of Lie groups
I Homotopy commutativity of p-localized Lie groups
IMethod to prove

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Homotopy commutativity of p-localized Lie groups

G: compact connected simple Lie group,H∗ (G; Q) = ∧Q (x1 , . . . , x` ) |xi | = 2ni − 1, n1 ≤ · · · ≤ n`
Theorem (McGibbon 1984 (C2 ), Saumell 1995 (Williams C k( k > 2), G , G2 ), Hasui–Kishimoto–T 2019 (Sugawara C k ,G = G2 ))1. If p > kn` , then G(p) is a Sugawara C k -space.
2. If p < kn` , then G(p) is not a Williams C k -space except in thecase (G, p) = (Sp(2), 3), (G2 , 5).
3. Sp(2)(3) (n` = 4) is homotopy commutative.
4. (G2 )(5) (n` = 6) is homotopy commutative but not a Williams
C3 -space.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Method to prove

G: A∞ -space⇒ ∗ = B0 G ⊂ B1 G ⊂ · · · ⊂ BG
BG: classifying space of G
B nG: n-th projective space of GExample

G = Z/2Z, S1 , S3 (= SU(2) = Sp(1))⇒ B nG = RP n, CP n, HP nSamelson products in Lie groups

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Method to prove

X, Y : pointed spaces
X ∨ Y = {(x, y) ∈ X × Y | x = ∗ or y = ∗} ⊂ X × Y
X∨n = {(x1 , . . . , x n) ∈ X×n | if x j , ∗, then xi = ∗ for i , j} ⊂ X×n
Fact
G is a Williams C k -space⇔ (B1 G)∨k → BG, (∗, . . . , x, . . . , ∗) 7→ x extends over (B1 G)×k .
Fact
G is a Sugawara C k -space⇔ B k G ∨ B k G → BG, (∗, x) 7→ x, (x, ∗) 7→ x extends overB k G × B k G.
Extendability follows from the results of homotopy groups ofspheres. Non-extendability follows from the computations ofcertain Steenrod operations and Chern characters.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

4. Samelson products in Lie groups
Samelson productSamelson products in Lie groups
I Higher Samelson product and another partial higher homotopy
commutativity
I A -triviality of adjoint bundlesnII

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Samelson product

G: topological group
α : A → G, β : B → G
Definition

hα, βi : A ∧ B → G, (a, b) 7→ aba−1 b−1 , is called the Samelson
product.

G is homotopy commutative⇔ all Samelson products are trivial.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Samelson products in Lie groups r ∈ π2r+1 (SU(n))  Z: generator ( r = 1, 2, . . . , n − 1)
Theorem (Bott, 1960)
If r + s = n + 1, then the order of h r ,  s i ∈ π2n(SU(n))  Z/n!Z isn!
.
r!s!
Determined: The Samelson products of free parts of homotopy
groups in G( p) for a compact connected simple Lie group G for
p > n` and “quasi- p-regular” case (Bott, Mahowald,
Hamanaka–Kono, Hasui–Kishimoto–Ohsita, Kishimoto–T,
Hasui–Kishimoto–Miyauchi–Ohsita).
For p > n` , all such Samelson products are detected by theSteenrod operation P1 in BG(p) (Kishimoto–T, 2018).

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Samelson products in Lie groups

α : S n−1 → Ghα, idG i appears as ∂ in the evaluation fiber sequence∂G(Pα ) → G −→ Map∗ (S n, BG)α → Map(S n, BG)α → BG.
G(Pα ): the gauge group of the principal G-bundle over S nclassified by α.
→ The homotopy type of G(Pα ) is determined by hα, idG i.
Such Samelson products are studied in many cases.Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher Samelson product and another partial higher homotopy commutativity∃ higher versions of Samelson product.
Little is known about higher Samelson products in Lie groups....
RemarkAll Samelson products of order k in G are trivial⇔ G is a Williams C k -space.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

Higher Samelson product and another partial higher homotopy commutativity

Another kind of “partial higher homotopy commutativity”:
for α : S n → BG, does the map(α, incl) : S n ∨ B k G → BG
extend over S n × B k G?
(cf. T α -space by Iwase–Mimura–Oda–Yoon)kRemark
G is a Sugawara C k -space⇒ it extends.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

A n -triviality of adjoint bundles

Pα : principal G-bundle over S n classified by α : S n → BG
ad Pα = Pα ×G G ((u g, x) ∼ (u, gx g−1 )): group bundleΓ(ad Pα )  G(Pα ) as topological groups
Proposition(α, incl) : S n ∨ B k G → BG extends over S n × B k G⇔ ad Pα is trivial as a fiberwise A k -space.

Background – A n -space

Higher homotopy commutativities

Full higher commutativity of Lie groups

Samelson products in Lie groups

A n -triviality of adjoint bundles

G = SU(2)π4 (B SU(2))  π3 (SU(2))  Za k : characterized by the following condition:ad P` and ad P`0 are equivalent as fiberwise A k -spaces
if and only if (a k , `) = (a k , `0 ).
Theorem

a1 = 12 = 22 31 , a2 = 180 = 22 32 51 (Crabb–Sutherland, 2000),
a3 = 15120 = 24 33 51 71 (T, 2018),
v3 (a k ) = k (T, 2012).
The growth of the number of the fiberwise A k -equivalence classes
kof ad P` (` ∈ π4 (B SU(2))) is at least e log k (T, 2012).