1. Higher order Whitehead products in Quillen’s models
José M. Moreno-Fernández
Joint with: F. Belchí, U. Buijs, A. Murillo
Rabat – July 2016

2. + Whitehead product −,− 𝑊
Some motivation
𝑋↦ 𝐻 ∗ 𝑋;𝑅 = 𝑛≥0 𝐻 𝑛 (𝑋;𝑅)
+ cup product ⌣
Graded 𝑅-algebra
𝑋↦ 𝜋 ∗ (𝑋)= 𝑛≥1 𝜋 𝑛 𝑋
+ Whitehead product −,− 𝑊
Whitehead algebra

3. I. M. James, J. H. C. Whitehead (53)
𝐻 ∗ 𝕊 4 × 𝕊 5 ; ℤ ≅ 𝐻 ∗ (𝑉 𝕊 5 ;ℤ) as graded rings
𝜋 𝑛 𝕊 4 × 𝕊 5 ≅ 𝜋 𝑛 (𝑉( 𝕊 5 )) for every 𝑛≥1
BUT!
−,− 𝕊 4 × 𝕊 5 ≠ −,− 𝑉( 𝕊 5 )

4. Toric Topology: J. Grbić, S. Theriault (2015)
Some more motivation
Toric Topology: J. Grbić, S. Theriault (2015)
𝒵 𝐾 ↪𝐷 𝐽 𝐾 → 𝑖=1 𝑛 ℂ 𝑃 ∞ 𝑤
𝑤 is shown to be sum of higher and iterated Whitehead products

5. Configuration spaces: P. Salvatore (2001)
Last motivation!
Configuration spaces: P. Salvatore (2001)
The configuration space 𝐹 𝑘 ( ℝ 𝑛+1 ) admits a minimal CW-decomposition of the form
{∗} ⊆ 𝑋 𝑛 ⊆ 𝑋 2𝑛 ⊆⋯⊆ 𝑋 𝑚𝑛 = 𝐹 𝑘 ( ℝ 𝑛+1 ),
in which the attachings are done via higher order Whitehead products.

6. = Let 𝕊 𝑝 and 𝕊 𝑞 be spheres. The Universal Whitehead Element
𝑤∈ 𝜋 𝑝+𝑞−1 ( 𝕊 𝑝 ∨ 𝕊 𝑞 )
is the homotopy class of the attaching map in
𝕊 𝑝 × 𝕊 𝑞 = (𝕊 𝑝 ∨ 𝕊 𝑞 ) ∪ 𝑤 𝑒 𝑝+𝑞
𝑝=𝑞=1
𝑒 2 =
Φ 𝑤
𝕊 1 ∨ 𝕊 1
𝕊 1 × 𝕊 1 =𝑇

7. −,− 𝑊 : 𝜋 𝑝 𝑋 × 𝜋 𝑞 𝑋 → 𝜋 𝑝+𝑞−1 (𝑋)
The Whitehead product is the operation
−,− 𝑊 : 𝜋 𝑝 𝑋 × 𝜋 𝑞 𝑋 → 𝜋 𝑝+𝑞−1 (𝑋)
𝑓,𝑔 ↦ 𝑓,𝑔 𝑊 ≔ 𝑓∨𝑔 ∘𝑤
H. Uehara, W. S. Massey (51) proved identities similar to those of a graded Lie algebra… But not quite… Notwithstanding,
Path space fibration
Ω𝑋→𝑃𝑋→𝑋
+ long exact sequence
of homotopy groups
𝜋 𝑛 𝑋 ≅ 𝜋 𝑛−1 (Ω𝑋)

8. −,− : 𝜋 𝑝 Ω𝑋 ⊗ℚ× 𝜋 𝑞 Ω𝑋 ⊗ℚ→ 𝜋 𝑝+𝑞 Ω𝑋 ⊗ℚ
𝑓,𝑔 ↦𝜕 [ 𝜕 −1 (𝑓), 𝜕 −1 (𝑔)] 𝑊
𝐿 𝑋 ≔ (𝜋 ∗ Ω𝑋 ⊗ℚ, [−,−]) is a graded Lie algebra: the rational homotopy Lie algebra of 𝑋.
Milnor-Moore (65)

9. Rational models capture this structure:
𝐿,𝜕 model of 𝑋
As graded Lie algebras, 𝐻 ∗ 𝐿,𝜕 ≅ 𝜋 ∗ Ω𝑋 ⊗ℚ
(Λ𝑉, 𝑑) minimal Sullivan model of 𝑋
As graded Lie algebras,
𝑠𝑉 # ≅ 𝜋 ∗ Ω𝑋 ⊗ℚ
where:
< 𝑑 2 𝑢 ;𝑠𝑥,𝑠𝑦> = −1 |𝑠𝑦| <𝑢;𝑠[𝑥,𝑦]>

10. 𝕊 𝑛 1 ∨…∨ 𝕊 𝑛 𝑟 𝑋 𝕊 𝑁−1 𝑇(𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 )
For 𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 𝑟≥2 simply connected spheres, there ∃ a space 𝑇=𝑇( 𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 ) (the fat wedge) with the property that
𝕊 𝑛 1 ×…× 𝕊 𝑛 𝑟 =𝑇 ∪ 𝑤 𝑒 𝑁 , 𝑁=∑ 𝑛 𝑖
for a homotopy class 𝑤: 𝕊 𝑁−1 →𝑇.
[Porter (65)]
Let 𝑋 be a space, 𝑟≥2, and let 𝑥 1 ∈ 𝜋 𝑛 1 𝑋 ,…, 𝑥 𝑟 ∈ 𝜋 𝑛 𝑟 𝑋 , where each 𝑛 𝑗 ≥2. The higher Whitehead product of order 𝑟 of 𝑥 1 ,…, 𝑥 𝑟 is the (possibly empty) set 𝑥 1 ,…, 𝑥 𝑘 𝑊 ={ 𝑓 ∘𝑤∣ 𝑓 extends 𝑓 } ⊆ 𝜋 𝑁−1 (𝑋)
𝑓= 𝑥 1 ∨…∨ 𝑥 𝑟
𝕊 𝑛 1 ∨…∨ 𝕊 𝑛 𝑟 𝑋 𝑤
𝕊 𝑁−1
𝑇(𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 )

11. Problems: 𝑥 1 ,…, 𝑥 𝑟 𝑊 can be empty,
𝑥 1 ,…, 𝑥 𝑟 𝑊 usually has lots of non-homotopic classes,
There is no canonical way of picking one among these.
The rational homotopic interpretation of these products in Quillen’s language is original of C. Allday (73), but was completed and very well written in D. Tanré’s book (83).

12. Spaces under consideration
𝜕 𝑢 1234 =± 𝑢 1 , 𝑢 234 ± 𝑢 12 , 𝑢 34 ± 𝑢 13 , 𝑢 24 ± 𝑢 14 , 𝑢 23 ± 𝑢 123 , 𝑢 4 ± 𝑢 124 , 𝑢 3 ± 𝑢 134 , 𝑢 2
The free graded Lie algebra
𝕊 𝑁−1
𝐿 𝑢 ,0
𝑢 =𝑁−2
𝕊 𝑛 1 ∨…∨ 𝕊 𝑛 𝑟
𝐿 𝑢 1 ,…, 𝑢 𝑟 ,0
𝑢 𝑖 = 𝑛 𝑖 −1
𝑇(𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 )
(𝐿 𝑢 𝑖 1 … 𝑖 𝑠 ,𝜕)
1≤ 𝑖 1 <…< 𝑖 𝑠 ≤𝑟 𝑠<𝑟 𝑢 𝑖 1 … 𝑖 𝑠 = 𝑛 𝑖 1 +…+ 𝑛 𝑖 𝑠 −1
𝜕 𝑢 1 =…=𝜕 𝑢 𝑟 =0
𝜕 𝑢 𝑖 1 … 𝑖 𝑠 = 𝑝=1 𝑠−1 𝑆(𝑝,𝑠−𝑝) 𝜎 1 =1 𝜖 𝜎 [ 𝑢 𝑖 𝜎(1) … 𝑖 𝜎(𝑝) , 𝑢 𝑖 𝜎(𝑝+1) … 𝑖 𝜎(𝑠) ]
𝑟=3
𝑢 1 , 𝑢 2 , 𝑢 3 ,
𝑢 12 , 𝑢 13 , 𝑢 23

13. Quillen functor ℒ 𝕊 𝑛 1 ∨…∨ 𝕊 𝑛 𝑟 𝑋 𝕊 𝑁−1 𝑇(𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 )𝑓
𝕊 𝑛 1 ∨…∨ 𝕊 𝑛 𝑟 𝑋 𝑤
𝕊 𝑁−1
𝑇(𝕊 𝑛 1 ,…, 𝕊 𝑛 𝑟 )
Quillen functor ℒ 𝜙
𝐿 𝑢 1 ,…, 𝑢 𝑟 ,0
(𝐿,𝜕) 𝑤
𝐿 𝑢 ,0
(𝐿 𝑢 𝑖 1 … 𝑖 𝑠 ,𝜕)

14. 𝑥 1 ,…, 𝑥 𝑟 𝑊 ={[ 𝜙 𝑤 ]∣ 𝜙 extends 𝜙}⊆ 𝐻 ∗ (𝐿).
𝐿 𝑢 1 ,…, 𝑢 𝑟 ,0
(𝐿,𝜕) 𝑤
𝐿 𝑢 ,0
(𝐿 𝑢 𝑖 1 … 𝑖 𝑠 ,𝜕)
𝑤 𝑢 ≡𝑤=𝜕 𝑢 1…𝑟
D. Tanré (83)
Let 𝑥 1 ∈ 𝐻 𝑛 1 𝐿 ,…, 𝑥 𝑟 ∈ 𝐻 𝑛 𝑟 𝐿 , 𝑟≥2 be homology classes. Define the 𝑟𝑡ℎ Whitehead bracket set 𝑥 1 ,…, 𝑥 𝑟 𝑊 as the (possibly empty) set of homology classes
𝑥 1 ,…, 𝑥 𝑟 𝑊 ={[ 𝜙 𝑤 ]∣ 𝜙 extends 𝜙}⊆ 𝐻 ∗ (𝐿).

15. Homotopy transfer of algebraic structures
A retract, denoted by 𝐿,𝐻 ,
is the datum of
𝐿,𝜕
(𝐻,0)
(𝐻,{ 𝑙 𝑘 } 𝑘 ) is an 𝐿 ∞ -algebra
Chain complexes maps satifying some conditions
𝑙 1 :𝐻→𝐻 𝑙 2 : 𝐻 ⊗2 →𝐻 ⋮ 𝑙 𝑘 : 𝐻 ⊗𝑘 →𝐻
𝑙 1 =𝜕=0
𝑙 2 = [−,−]
With this structure, 𝐻 retains all information of (𝐿,𝜕) up to 𝐿 ∞ -isomorphism

16. Let (𝐿,𝜕) be model of a space 𝑋.
Main questions
Let (𝐿,𝜕) be model of a space 𝑋.
Endow 𝐻 with an 𝐿 ∞ -structure. What is the relationship between −,…,− 𝑊 and the operation 𝑙 𝑘 ?
When do we recover higher Whitehead products with the evaluation 𝑙 𝑘 ( 𝑥 1 ,…, 𝑥 𝑘 ) ? ⋮
Not enough time for more questions!

17. Some results we show
Proposition: If 𝑥 1 ,…, 𝑥 𝑘 𝑊 ≠∅, then, for any 𝑥∈ 𝑥 1 ,…, 𝑥 𝑘 𝑊 and for any homotopy retract 𝐿,𝐻 , we have that
± 𝑙 𝑘 𝑥 1 ,…, 𝑥 𝑘 =𝑥+ 𝑖=1 𝑘−1 𝐼𝑚 𝑙 𝑖 .
In particular,
𝑙 1 =…= 𝑙 𝑘−1 =0⟹± 𝑙 𝑘 𝑥 1 ,…, 𝑥 𝑘 =𝑥.

18. Theorem: Let 𝑥∈ 𝑥 1 ,…, 𝑥 𝑘 𝑊 and let 𝐿,𝐻 be an adapted retract
Theorem: Let 𝑥∈ 𝑥 1 ,…, 𝑥 𝑘 𝑊 and let 𝐿,𝐻 be an adapted retract. Then, ±𝑙 𝑘 𝑥 1 ,…, 𝑥 𝑘 =𝑥
Corollary:
𝑥 1 , 𝑥 2 , 𝑥 3 𝑊 ≠∅ ⟹ ±𝑙 3 ( 𝑥 1 , 𝑥 2 , 𝑥 3 )∈ 𝑥 1 , 𝑥 2 , 𝑥 3 𝑊
Example:
We find a space 𝑋 with the property that 𝑥 1 ,…, 𝑥 4 𝑊 ≠∅, but for every retract 𝐿,𝐻 , the induced operation never falls into that set:
𝑙 4 𝑥 1 ,…, 𝑥 4 ∉ 𝑥 1 ,…, 𝑥 4 𝑊

19. Thank you for attending
More info:
“Higher order Whitehead products and 𝐿 ∞ structures on the homology of a DGL” on Arxiv.org