THE T-FUNCTOR OF J. LANNES

CLARENCE W. WILKERSON, JR.

The calculation of the homotopy type of the space of continuous maps Map(X, Y) is a fundamental problem of homotopy theory. The set of path components, π_{0}Map(X, Y) = [X, Y] corresponds to the homotopy classes of such maps. There are relatively few cases for which this information is explicitly known. A major impact of Jean Lannes’ work [1] on unstable modules and the T functor has been to expand this knowledge to include many cases in which the sources and targets are classifying spaces of finite and compact Lie groups.

The work of Steenrod and others assigns in a natural way to each topological space X and each prime p an algebraic model consisting of a graded algebra H^{∗}(X, 𝔽_{p}) = R^{∗} over 𝔽_{p} and an algebra A_{p} of natural operations called the Steenrod algebra. Each f : X → Y induces an element f^{∗} ∈ Hom_{alg}(H^{∗}(Y, 𝔽_{p}), H^{∗}(X, 𝔽_{p})) that commutes with the action of A_{p}. A_{p} is a connected graded Hopf algebra acting on the graded algebra R^{∗}.

The hypothesis that R^{∗} is the cohomology of a space imposes an additional “unstable” condition. This is most simply stated if p = 2 . A_{2} is generated as an (non-commutative) algebra by the Steenrod operations {Sq^{i}, i ≥ 0}, with relations forced by its actions of the cohomology of all topological spaces. For example, Sq^{0} = Id and Sq^{1} = β, the mod 2 Bockstein. The unstable condition is then that Sq^{i}x_{n} = 0 for i > n and Sq^{n}x_{n} = x_{n}^{2}. The algebraic category K of unstable algebras {R∗} over A_{p} is thus an approximation to the homotopy category of topological spaces. The larger category U of unstable modules over A_{p} has also proved useful.

For p > 2, the structure of A_{p} and unstable actions are similar, but slightly more involved. However, in all cases, the set of relations in the Steenrod algebra and the unstable condition are derivable from the known action of A_{p} on the cohomology of products of copies of Bℤ/pℤ. In the following, we omit explicit references to the coefficients.

The relationship of π_{0}Map(X, Y) to its model Hom_{K}(H^{∗}(Y, 𝔽_{p}), H^{∗}(X, 𝔽_{p})) is of particular interest. The equivalence

Map(X × Z, Y) → Map(X, Map(Z, Y))

raises the hope that in very favorable cases the map

Hom_{K}(H^{∗}Map(Z, Y), H^{∗}X) → Hom_{K}(H^{∗}Y, H^{∗}X ⊗ H^{∗}Z)

might be an isomorphism. That suggests that in the category K, H^{∗}Map(Z, Y) should be approximated by the left adjoint functor to tensoring on the right by H^{∗}Z. This motivates Lannes’s definition of T:

Definition. If E is a finite dimensional 𝔽_{p}-vector space, then the T-functor T_{E} : U → U is the left adjoint in U of the functor (( ) ⊗_{𝔽_{p}} H^{∗}BV ) : U → U. In the topological case, there is a natural map

λ_{X} : T_{E}H^{∗}X → H^{∗}Map(BE, X).

For general Z, the adjoint to (( ) ⊗_{𝔽_{p}} H^{∗}Z) accounts for only part of the starting page of a Bousfield-Kan unstable Adams spectral sequence for Map(Z, Y). Lannes provides the basic connection to topology by blending the algebraic properties of T_{E} and K with the Bousfield-Kan spectral sequence:

Theorem. (Lannes) For many interesting spaces X,

H^{∗}Map(BE, X) ≈ T_{E}H^{∗}X.

In particular,

π_{0}Map(BE, X) = [BE, X] = Hom_{K}(H^{∗}X, H^{∗}BE).

For f : X → Y, one has the path component Map(X, Y)_{f} of functions homotopic to f. The analogous T-construct is

Definition. Each φ ∈ Hom_{K}(R^{∗}, H^{∗}BE) induces a T_{E}^{0} module structure on 𝔽_{p} and

T_{E,φ}R^{∗} = T_{E}R^{∗} ⊗_{T_{E}^{0}} 𝔽_{p}.

The most striking features of T_{E} are summarized below. To some extent, these were presaged by work of Carlsson and Miller establishing that the {H^{∗}BV } are injectives in U.

Theorem (Lannes).

(a) T_{E} is exact.

(b) T_{E} respects tensor products, i.e T_{E}(M ⊗_{𝔽_{p}} N) = T_{E}M ⊗_{𝔽_{p}} T_{E}N.

(c) T_{E} commutes with the p-th power operations in a suitable sense.

(d) T_{E} maps K to K.

In principle, T_{E}M^{∗} can be calculated by using the exactness property and a resolution of M^{∗} by free unstable A_{p}-modules. In practice, other methods are often more effective:

Examples.

1. If M^{∗} is finite, T_{E}M^{∗} = M^{∗} .

2. If R^{∗} = H^{∗}BV, then T_{E}R^{∗} = ∏_{Hom_{grp}(E,V)} H^{∗}BV , for E and V finite dimensional 𝔽_{p} vector spaces.

3. If τ : R^{∗} → H^{∗}BE in K is an inclusion, then T_{E,τ} R^{∗} is the smallest subHopf algebra of H^{∗}BE that contains τ(R^{∗}).

4. If X is a finite E-complex with fixed point set X^{E} and H_{E}^{∗}X the mod p cohomology of the Borel construction, then T_{E,id}H_{E}^{∗}X = H^{∗}BE ⊗_{𝔽_{p}} H^{∗}X^{E} in K.

5. If G is a compact Lie group, then

T_{E}H^{∗}BG = ∏_{φ ∈ Hom_{grp}(E,G)/G−conj} H^{∗}BC_{G}(im φ(E))

These examples each have powerful topological consequences. For example, the first and fourth lead to new proofs of the Sullivan conjecture, orignally proved by Miller and Carlsson. The last leads to a new view of the homotopy theory of classifying spaces. Most of the above is referenced in the book of Schwartz [2]. One should also consult the many articles in the past decade by authors such as Lannes, Schwartz, Henn, Zarati, Kuhn, Farjoun-Smith, Morel, Jackowski-McClure-Oliver, Notbohm, Dwyer-Wilkerson, and others.

References

[1] Jean Lannes. Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math., (75):135–244, 1992. With an appendix by Michel Zisman.

[2] Lionel Schwartz. Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture. University of Chicago Press, Chicago, IL, 1994.

Department of Mathematics, Purdue University, West Lafayette, IN, 4907-1395

   E-mail address: wilker@math.purdue.edu