FUNCTORIAL DECOMPOSITIONS OF LOOP SPACES ... - Maths, NUS

FUNCTORIAL DECOMPOSITIONS OF LOOP SPACES OF SUSPENSIONS, AND THEIR RELATION TO REPRESENTATION THEORY FRED COHEN, PAUL SELICK AND JIE WU

0.1. Product decompositions of certain topological spaces. Example 0.1. ΩCP n ' ΩS 2n+1 × S 1 . Methods: Consider the fibre bundle S 1 → S 2n+1 → CP n . One gets a splitting fibre sequence ΩS 2n+1 → ΩCP n → S 1 .

Fred:

You might want to give a leisurely exposition of why this is correct as the idea is central to your stuff, and other interesting mathematics. Namely, point out that there is a principal fibration obtained by looping, and there is a section obtained by considering π1 . There is a splitting for any principal bundle with section. Thus ΩCP n is homotopy equivalent to the product S 1 × ΩS 2n+1 . This idea is useful in several places. One beautiful application is P. Selick’s product decomposition for the homotopy theoretic fibre of the p-th power map on Ω2 S 2p+1 . Fred: Applications: From this decomposition, one gets Ω2 CP n ' Ω2 S 2n+1 × ΩS 1 and so the base-point path connected component Ω20 CP n is homotopy equivalent to Ω2 S 2n+1 . By using this, people have given a configuration space model for Ω2 S 2n+1 using complex rational functions. Geometry and analysis techniques therefore apply to these “function spaces”. Product decompositions of loop spaces are important in homotopy theory. Cohen, Moore and Neisendorfer have successfully solved the exponent problem on spheres and the Moore spaces. Roughly speaking, they give a “complete” decomposition of 1

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FRED COHEN, PAUL SELICK AND JIE WU

certain loop spaces and an analysis of the smallest factors. So finding the factors of a loop space was very useful. Fred: Perhaps give more motivation ?: (i) Selick’s splitting giving the right exponent for S 3 . (ii) Another possible example is the paper on mod-2 Moore spaces of C/W. Namely, people( Mark ) were only able to find elements of order 8 in the homotopy of P n (2) for 3/4 of the values of n. A splitting allowed one to do the last case where the elements start occurring in roughly 30 times the connectivity of the mod-2 Moore space ( with top cell in dimension 3mod4 .....This might be off and is possibly 1mod4. ) (iii) Please be careful about trying to put in too much ( as the above suggestion might be too much ). The current lecture seems to be very informative, and interesting. Fred:

0.2. loop suspensions. Let X be a pointed space. The James construction J(X) is the (reduced) free monoid generated by X, where “reduced” means that one considers the base-point ∗ as the identity element. So, for example, one gets x∗y = ∗xy = xy. Theorem 0.2 (James). Let X be a path-connected pointed CW -complex. Then J(X) is homotopy equivalent to ΩΣX. For example, J(S 1 ) ' ΩΣS 1 = ΩS 2 . The homotopy groups of S 1 is easy: πn (S 1 ) = 0 for n 6= 1 and π1 (S 1 ) = Z. But the (general) homotopy group π∗ (ΩS 2 ) = π∗+1 (S 2 ) is not yet well understood. 0.3. James filtration. The James filtration of J(X) is actually the word filtration: Jn (X) = {w ∈ J(X)| the word length of w ≤ n}. For example, J0 X = ∗, J1 X = X. Theorem 0.3 (James). Jn X/Jn−1 X ∼ = X (n) the n-fold self smash product of X. The smash product X ∧ X = X × X/X ∨ X = X × X/∗ × X ∪ X × ∗. For example, (n) Jn (S 1 )/Jn−1 S 1 = S 1 = S n . The James construction tells the suspension splitting of ΩΣX.

FUNCTORIAL DECOMPOSITIONS OF LOOP SUSPENSIONS

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Theorem 0.4 (James). ΣΩΣX ' ΣJ(X) '

∞ _

ΣX (n) ,

n=1

the wedge of self smash products. The James construction tells the homology of ΩΣX. Let H∗ (−) denote the homology with coefficients in a field k. Theorem 0.5 (James). H∗ (ΩΣX) is isomorphic to the tensor algebra generated by ¯ ∗ (X), where the tensor length filtration corresponds to the the reduced homology H ¯ ∗ (X)) as Hopf algebras, James filtration. If X is a suspension, then H∗ (ΩΣX) ∼ = T (H where T (V ) is a Hopf algebra by saying that elements in V are primitive. Problem 0.6. How to decompose ΩΣX as a product of “smaller” spaces? We assume that X itself is a suspension, i.e., X = ΣY , has these have some useful features for technically easier in analysis the homology. Furthermore, we are looking for a “general” machine for product decompositions of ΩΣX. Thus the question can be re-asked as What are functorial product decompositions of ΩΣX, where X runs over all suspensions? “functorial” means the decompositions of the functor ΩΣ. For example, assume ' - A(X) × B(X) for any suspension X. Let that there is a decomposition ΩΣX f : X → Y be any map. Then there is a commutative diagram up to homotopy ΩΣX

' -

A(X) × B(X) A(f ) × B(f )

ΩΣf ?

ΩΣY

' -

?

A(Y ) × B(X).

For any individual suspension X, the real decomposition on ΩΣX will mean the evaluation of the general decomposition machine (the decomposition of the functor ΩΣ) on the particular space X. 0.4. Cohen Groups. Observation: suppose that there is a functorial decomposition ΩΣX ' A(X) × B(X), then one gets a self functorial idempotent map ΩΣX

proj

- A(X)

incl

- ΩΣX.

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FRED COHEN, PAUL SELICK AND JIE WU

So the first question is: What is the set of all functorial self map: ΩΣX → ΩΣX? This was known by Fred Cohen six years ago.

Fred: a year after we met in Tianjin, and a year after Sarah was born... Fred:

Let H(∞) be the set of the homotopy classes of all functorial self map on ΩΣX. Theorem 0.7 (Cohen). H(∞) is a progroup: H(∞)

- · · · · · · H(n)

- H(n − 1)

- ···

with Ker(H(n) → H(n − 1)) = LieZ (n). The group H(n) is the set of the homotopy classes of all functorial maps: Jn (X) → ΩΣX. The group LieR (n) is as follows: Let V be a free R-module generated by letters x1 , · · · , xn . Then LieR (n) is the sub R-module of the n-fold tensor product V ⊗R n generated by the commutators [[xσ(1) , xσ(2) ], · · · , xσ(n) ] for σ ∈ Sn the symmetric group. The symmetric group action on LieR (n) is given by permuting letters. LieR (n) is the usual Lie representation of the symmetric group Sn . A relation between the Cohen groups and the Goodwillie’s tower of the functor ΩΣ has been studied by Bill Dwyer and others. 0.5. The Algebraic Problem. The functorial decompositions of loop suspensions can be reduced to a problem in algebra in the following sense. Let the ground field k = Z/pZ. Let V be a vector space and let T (V ) be the tensor algebra generated by V . T (V ) is a Hopf algebra by saying that elements in V are primitive. Theorem 0.8 (Geometric Realization Theorem). Let T (V ) ∼ = A(V ) ⊗ B(V ) be a functorial coalgebra decomposition of T (V ). Let X be a p-completed suspension. Then there is a functorial decomposition ΩΣX ' A(X) × B(X)

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¯ ∗ (X)) and H∗ (B(X)) ∼ ¯ ∗ (X)). such that the mod p homology H∗ (A(X)) ∼ = A(H = B(H Fred: Perhaps you should say why this result is natural, and how you might expect it to give further applications. Also, you have some examples which might be useful to point out. ( I believe that your results are quite important. They have been very useful in understanding some recent stuff on Barratt’s conjecture, at least for me. ) One other point: did you point out that each factor is non-trivial, and that A(X) is the smallest functorial retract that factors the inclusion of X in J(X) ? Also, it might make sense to say more about the emphasis on functoriality like: the history of the subject suggests that constructing enough useful examples together with functoriality lies at the foundation of gaining further understanding. Fred:

The importance here is: 1) the decomposition of T (V ) must be functorial and 2) the decomposition must be a coalgebra decomposition. Roughly speaking, “functorial” means that the decomposition is independent on the choice of a basis for V . By thinking dually, roughly speaking, the dual statement of “coalgebra decomposition” of T (V ) is to decompose the divided algebra T (V ) as a tensor product of subalgebras. The proof of this theorem is given by: 1) determining the set of all functorial coalgebra map: T (V ) → T (V ) and 2) showing that any functorial coalgebra map: T (V ) → T (V ) is geometrically realizable by using Cohen groups. 0.6. Functorial version of Poincar´ e-Birkhoff-Witt Theorem. Let the ground ring is a field k and let V be a vector L space. Let Λ(V ) be the free commutative algebra generated by V . Let L(V ) = ∞ n=1 Ln (V ) be the free Lie algebra generated by V , where Ln (V ) is the set of homogeneous Lie elements of degree n. The usual PBW theorem is as follows. Theorem 0.9 (PBW Theorem). Let V be a vector space. Then there is an isomorphism of coalgebras ∞ O ∼ T (V ) = Λ(Ln (V )) n=1

The PBW isomorphism is given by the following steps: Step 1. Choose a ordered basis for V ;

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FRED COHEN, PAUL SELICK AND JIE WU

Step 2. Construct the PBW isomorphism. In other words, the construction of the PBW map is NOT functorial. We will see that PBW isomorphism fails functorially in modular case. (We know that the PBW isomorphism holds functorially if k is of characteristic 0.) It turns out that the functorial version of PBW Theorem is a representation problem which isdescribed as follows. Let Lie(n) = Liek (n). Fred: Perhaps define Lie(n) slowly, and carefully ? By the way, this Sn -action extends to an Sn+1 -action as Lie(n) is the top homology group of the configuration space of (n+1) points in the 2-sphere ( after tensoring with the sign representation). I wonder whether/where this might fit in with your work with Paul. Fred: Remember that Sn acts on Lie(n) by permuting letters. Let Liemax (n) be a maximal projective Sn -submodule of Lie(n). Let V be a vector space and let Sn act on V ⊗n by permuting letters. Let max Lmax (n) ⊗k(Sn ) V ⊗n ⊆ Ln (V ). n (V ) = Lie

Lmax is a functor from k-modules to k-modules. Let B (n) (V ) be the sub algebra of n T (V ) generated by Lmax m (V ) for m ≥ n. Then one gets a functorial sequence · · · B (n+1) (V ) ⊆ B (n) (V ) ⊆ · · · ⊆ B (1) (V ) = T (V ). Let (n) Amin (V ; Lmax (V ). n ) = k ⊗B (n+1) (V ) B min This gives a functor Amin (−; Lmax (V ; Lmax n ). We show that A n ) is a functorial coalgebra retract of T (V ). Theorem 0.10 (Functorial Version of PBW Theorem). There is a functorial isomorphism of coalgebras ∞ O ∼ T (V ) = Amin (V ; Lmax n ). n=1 min

Furthermore, A (V contains Lmax n (V ).

; Lmax n )

is the smallest functorial coalgebra retract of T (V ) which

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) is the smallest retract of T (V ) which conFor example, Amin (V ) = Amin (V ; Lmax 1 tains V . Now good questions would be: 1) What is Liemax (n)? 2) What is Amin (V ; Lmax n )? In the rational case, the answer is: Theorem 0.11. If k is of characteristic zero, then Liemax (n) = Lie(n), Lmax n (V ) and Amin (V ; Lmax ) = Λ(L (V )). In particular, PBW Theorem holds functorially in the n n rational case. In the modular case, these problems seem much more complicated and only partial information is known. We list some information aleady known. Let k be of characteristic p. Theorem 0.12. 1) Liemax (n) = Lie(n) if n 6≡ 0 mod (p); max 2) Lie (n) is a proper submodule of Lie(n) if n ≡ 0 mod (p). This shows that the PBW theorem fails functorially in the modular case. Let B be the subalgebra of T (V ) generated by Ln (V ) for n not a power of p. Theorem 0.13. Λ(V ) < Amin (V ) < k ⊗B T (V ). This Theorem was conjectured by Fred Cohen. Example 0.14.

1) Lmax (V ) = L(L2 (V ), · · · , Lp−1 (V )) ∩ Lp (V ); p dim Liemax (p) = (p − 1)! − p + 1.

2) p = 5, Lmax (V ) = [L2 (V ), L3 (V )]. 5 3) p = 2 Liemax (2) = Liemax (4) = 0, Liemax (3) = Lie(3), Liemax (5) = Lie(5), Liemax (7) = Lie(7), dim Liemax (6) = 96. Applications: Let G be an abelian group and let expp (G) be the exponent of the p-torsion primary component of G. Theorem 0.15. Let X be a suspension such that πn (X) is a finite torsion group for each n. Then  expp (π∗ (Amin (X))) ≤ expp (π∗ (ΩΣX)) ≤ max expp (π∗ (Amin (X (n) )))|n ≥ 1 . Theorem 0.16. There exist infinitely many Z/8Z-summands in π∗ (Σn RP 2 ) if n ≥ 1. Some related references.

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FRED COHEN, PAUL SELICK AND JIE WU

References [1] F. Cohen, On combinatorial group theory in homotopy theory, Contemp. Math., 188 (1995), 57-63. [2] F. Cohen, J. Moore and J. Neisendorfer, Torsion in homotopy theory, Ann. Math. 109 (1979), 121-168. This is a classical work in unstable homotopy theory. [3] Fred Cohen, Paul Selick and Jie Wu, On groups Kn (k), preprint. [4] F. R. Cohen and J. Wu, A remark on the homotopy groups of Σn RP 2 , Contem. Math. 181 (1995) 65-81. [5] D. Ravenel, Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies 128 (1192), Princeton University Press. There are some applications of representation theory to stable homotopy theory in this book. [6] Paul Selick and Jie Wu, On Natural Decompositions of Loop Suspensions and natural coalgebra decompositions of tensor algebras, to appear in Memoirs AMS. [7] J. Wu, On fibrewise simplicial monoids and Milnor-Carlsson’s constructions, Topology 37(5) (1998), 1113-1134. [8] J. Wu, On combinatorial calculations for the James-Hopf maps, Topology 37(5) (1998), 1011-1023. [9] J. Wu, A product decomposition of Ω30 (ΣRP 2 ), Topology 37(5) (1998), 1025-1032. Department of Mathematics, University of Pennsylvania, Philadelphia, PA 19104, USA, [email protected]