Higher homotopy normalities in topological groups
Mitsunobu Tsutaya (Kyushu University)

Classifying spaces in homotopy theory: in honour of Ran Levi’s 60th Birthday
6 September 2022

Mitsunobu Tsutaya (Kyushu University)

Higher homotopy normalities in topological groups

Classifying spaces in homotopy theory

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Contents

1. Higher homotopy associativity and commutativity
2. Higher homotopy normality
3. Results

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Higher homotopy normalities in topological groups

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1. Higher homotopy associativity and commutativity

Mitsunobu Tsutaya (Kyushu University)

Higher homotopy normalities in topological groups

Classifying spaces in homotopy theory

3 / 23H-map▶ A map f : H → G between topological groups is said to be an H-map iff ◦ µ ≃ µ ◦ (f × f ).
▶ H-map is much far from homomorphism. There exists an H-map f : H → G not

H-equivalent to a homomorphisms as in the following diagram:fG≃

G′/H

≃/ H′a homomorphism▶ So, what is the obstruction to a map being a homomorphism up to H-equivalence?

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A∞ -map▶ Sugawara (1960) (with Milnor’s result) answered this question.

An H-map f : H → G is homotopy equivalent to a homomorphism if and only if it admits an
A∞ -form {fi : I i−1 × H i → G }i≥1 which describes how the associativity is preserved throughf.
▶ f2 is the homotopy between f ◦ µ and µ ◦ (f × f ).
▶ f : [0, 1]2 × H 3 → G is depicted as follows.
3▶

f(h1h2)f(h3)f(h1)f(h2)f(h3)f3

f(h1h2h3)▶

f(h1)f(h2h3)

An H-map equipped with an A∞ -form is called an A∞ -map.

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Projective space and An -map
▶ The previous result actually follows from the following result:▶

A pointed map f : H → G admits an A∞ -form if and only if the suspension Σf : ΣH → ΣG
extends to a map between the classifying space BH → BG :
ΣH

BHf

∃

/ ΣG

/ BG

▶ Stasheﬀ (1963) generalized this result.
▶

The classfying space has the natural filtration:∗ = B0 G ⊂ ΣG = B1 G ⊂ B2 G ⊂ · · · ⊂ Bn G ⊂ · · · ⊂ BG .

▶

Bn G is called the n-th projective space of G .
He proved that Σf : ΣH → ΣG extends to Bn H → BG if and only if f admits an An -form{fi : I i−1 × H i → G }1≤i≤n .
▶ This result is basic in the present work.
▶

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Homotopy commutativity▶ A topological group G is said to be homotopy commutative if the Samelson productG ∧ G → G,(x, y ) 7→ xyx −1 y −1

is null-homotopic.
▶ Through the isomorphism[G ∧ G , G ] ∼= [G ∧ G , ΩBG ] ∼= [ΣG ∧ G , BG ],the Samelson product corresponds to the Whitehead product [ι, ι] of the inclusionι : ΣG → BG .
▶

So, G is homotopy commutative if and only if [ι, ι] = 0.

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Higher homotopy commutativity▶ Cn -space in the sense of Williams is defined by some higher homotopy condition using

permutohedra Pn .
▶ An equivalent condition is as follows: the wedge sum of the inclusion(ΣG )∨n → BG
extends over the product(ΣG )×n → BG .

▶ This is just the vanishing of the higher Whitehead products.

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Higher homotopy commutativity (continued)▶ The higher homotopy commutativity of Lie groups and their p-localizations is wellstudied. Here is a typical argument to show the non-commutativity.
ExampleLet G = SU(2) = S 3 and p an odd prime. Suppose the wedge sum of the inclusion
p+1(S 4 )∨ 2 → B SU(2) = HP ∞extends to a map
p+1

f : (S 4 )× 2 → B SU(2).
p+1

We know P 1 c2 = ac2 2 with a ̸= 0 in H 2p+2 (B SU(2); Fp ). Then f ∗ P 1 c2 ̸= 0. But P 1 must
p+1
be trivial in H ∗ ((S 4 )× 2 ; Fp ) by the Cartan formula. This contradicts to f ∗ c2 ̸= 0. Therefore,
SU(2) is not p-locally a C p+1 -space.
2▶ What about “higher homotopy normality”?
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2. Higher homotopy normality

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Crossed module▶ In the rest of this talk, let H and G be topological groups of homotopy types of CWcomplexes.
▶ A normal subgroup H ⊂ G is a subgroup stable under the inner automorphisms.
▶ Crossed module is a generalization of normal subgroup to non-inclusions.
Definition (MacLane, J.H.C.Whitehead 1940s)
A (topological) crossed module consists of homomorphisms f : H → G and ρ : G → Aut(H)satisfying the conditions▶ ρ(f (h))(x) = hxh−1 for any x, h ∈ H,
▶ f (ρ(g )(x)) = gf (x)g −1 for any x ∈ H and g ∈ G .
▶ Farjoun and Segev (2010) proved that the Borel construction EH ×H G of a crossed

module f : H → G naturally inherits a group structure. This should be considered as “thehomotopy quotient group of a homotopically normal subgroup”.

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Nk (ℓ)-map▶ Aℓ (H, G ) denotes the space of Aℓ -maps.

▶ conjH : H → Aℓ (H, H) denotes the conjugation conjH (h)(x) = hxh−1 .

Definition (T.)
A homomorphism f : H → G is an Nk (ℓ)-map if an Ak -map ρ : G → Aℓ (H, H) is given andthe following conditions hold:▶ ρ◦fis homotopic to conjH as an Aℓ -map,▶ the map ∗ → Aℓ (H, G ), ∗ 7→ f

is Ak -equivariant with respect to the action of G ,▶ the higher homotopy in the second condition coincides with the first condition on H.
▶ This is a higher homotopy analogue of crossed module.
▶ N1 (1)-map is equivalent to homotopy normal map introduced by McCarty (1964).
▶ James (1967) defined another homotopy normality which is slightly weaker thanMcCarty’s.
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N1 (1)-map

Definition (McCarty 1964)
A homomorphism f : H → G is homotopy normal (an N1 (1)-map) if there exists a mapγ̃ : G ∧ H → H making the diagram
H ∧Hf ∧idγH∃γ̃G ∧H/;Hγf/G

commute up to homotopy and the homotopies comapatible with the stationary homotopy ofthe outer square.

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13 / 23Immediate consequences▶ If f : H → G is an Nk (ℓ)-map and k ≥ k ′ and ℓ ≥ ℓ′ , then f▶ If f : H → G is a crossed module, then fis an Nk ′ (ℓ′ )-map.

is an N∞ (∞)-map.

▶ The homomorphism f : H → ∗ is an Nk (ℓ)-map if and only if conjH : H → Aℓ (H, H) is

homotopic to the constant map as an Ak -map.
▶

The latter condition is equivalent to being a C (k, ℓ)-space (T. 2016), which is a higher
homotopy commutativity introduced by Kishimoto and Kono (2010).
▶ C (∞, ∞)-space and Sugawara C -space are known to be equivalent. Then we conclude that∞H → ∗ is an N∞ (∞)-map if and only if BH is an H-space.
▶ This is analogous to the fact that H → ∗ is a crossed module if and only if H is commutative.

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3. Results

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Equivariant and fiberwise homotopy theory▶ The Borel construction defines the correspondencea G -space X

7→

a fiberwise space EG ×G X → BG .

This provides an “equivalence” between the equivariant homotopy theory and thefiberwise homotopy theory in an appropriate sense.
▶ The idea of the main theorem is based on this equivalence.

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Main theorem
Theorem (T.)
Let f : H → G be a homomorphism and F : Ek H ×H H → Ek G ×G G denote the induced mapof f . Then f is an Nk (ℓ)-map if and only if there exists a fiberwise Aℓ -space E → Bk G and Ffactors asϕψ
Ek H ×H H −→E −→ Ek G ×G Gup to homotopy over Bk f : Bk H → Bk G such that the following conditions hold:▶ ϕ covers Bk f

and ψ covers the identity on Bk G ,▶ ϕ and ψ restrict to Aℓ -maps on each fiber,▶ ϕ is a weak homotopy equivalence on each fiber,▶ the restriction of ψ ◦ ϕ to the fiber over the basepoint is homotopic to f

as an Aℓ -map.

▶ The last for conditions correspond to the compatibility required in the definition ofNk (ℓ)-map.
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Remark on main theorem▶ Roughly, this theorem states that f : H → G is an Nk (ℓ)-map if and only if the following“unusual” factorization of F : Ek H ×H H → Ek G ×G G exists:/GH




/E


/ Ek G ×G G/ Bk G

Bk G

Ek H ×H H

Bk H

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f

H

Bk f



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Remark on main theorem (continued)▶ The “usual” factorization is as follows. The middle column is induced from theconjugation action of H on G through f .
fH

Ek H ×H H


Bk H/GG/ Ek H ×H G/ Ek G ×G G/ Bk G

Bk H

Bk f▶ This factorization is possible for any homomorphism f .

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H-structure on Borel construction

Theorem (T.)▶ Let f : H → G be a homomorphism. Then the Borel construction X = EH ×H G is an

H-space if f is an Nk (k)-map and cat X ≤ k.

Example▶ Let H = K (Q, 2n − 1) and G = K (Q, 4n − 1). Consider the homomorphism f : H → G
with classifying map Bf : K (Q, 2n) → K (Q, 4n) corresponding to u 2 ∈ H 4n (K (Q, 2n); Q).

Then the Borel construction is2n
EH ×H G ≃ hofib(Bf ) ≃ S(0).
2n does not admit an H-structure and cat S 2n = 1, f is not an N (1)-map.
Since S(0)1(0)

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Preceding results on examples▶ There have been many results on homotopy normality of Lie groups.
▶ (James 1967)

The inclusion U(m) → U(n) is not (2-locally) homotopy normal in the sense of James for1 ≤ m < n. Similar results hold for O(m) → O(n) (2 ≤ m < n) and Sp(m) → Sp(n) for1 ≤ m < n.
▶ Other results include: McCarty (1964), James (1971), Kachi (1982), Furukawa (1985),Furukawa (1987), Furukawa (1995), Kudou–Yagita (1998), Kudou–Yagita (2003),
Kono–Nishimura (2003), Nishimura (2006), Kishimoto–T. (2018).
▶ These results suggest that H → G tends to fail to be p-locally homotopy normal for small

prime p.

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Higher homotopy normality of SU(m) → SU(n)▶ Main theorem provides an obstruction theory for Nk (ℓ)-map applying the fiberwise

projective space functor.
▶ Then, by a typical argument using Steenrod operations as mentioned before, we canobtain the following result.
Theorem (T.)▶ If max{kn − 2, (k − 1)n + 2} < p ≤ kn + 2(ℓ − 1) for some n ≥ 3 and k, ℓ ≥ 1, then theinclusion SU(2) → SU(n) is not a p-local Nk (ℓ)-map.

▶ If max{kn − m, (k − 1)n + 2} < p ≤ kn + (ℓ − 2)m for some 2 ≤ m < n and k, ℓ ≥ 1,then the inclusion SU(m) → SU(n) is not a p-local Nk (ℓ)-map.

▶ This result is not very sharp. For example, the normality is not determined whenkn + (ℓ − 2)m < p < kn + ℓm.
▶ A similar result is obtained for SO(2m + 1) → SO(2n + 1).
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Summary▶ Nk (ℓ)-map is a higher homotopical analogue of crossed module.
▶ Nk (ℓ)-map is characterized by fiberwise Aℓ -maps over the k-th projective spaces.
▶ The Borel construction EH ×H G of an Nk (k)-map f : H → G is an H-space if

cat EH ×H G ≤ k holds.

▶ Fiberwise projective space provides a method to detect obstructions to being Nk (ℓ)-maps.

Thank you!

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Classifying spaces in homotopy theory

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