THE BAKER-RICHTER SPECTRUM AS COBORDISM OF QUASITORIC MANIFOLDS 2 1 JACK MORAVA AND NITU KITCHLOO 0 2 Abstract. BakerandRichterconstructaremarkableA∞ring-spectrum n MΞ whose elements possess characteristic numbers associated to qua- a sisymmetric functions; its relations, on one hand to the theory of non- J commutative formal groups, and on the other to the theory of omnior- 5 iented (quasi)toric manifolds [in the sense of Buchstaber, Panov, and 1 Ray], seem worth investigating. ] T A . Introduction: Most of this paper is a draft for a talk JM wishes he had h t given at the August 2011 conference on toric manifolds [10] at Queen’s Uni- a m versity, Belfast(as opposedtothetalk heactually gave). ThankstoThomas Hu¨ttemann for organizing thatvery interesting meeting, and to Tony Bahri, [ Martin Bendersky, Fred Cohen, and Sam Gitler for helpful conversations 1 there. Those notes are little more than a collage of conversations, sugges- v tions, and howler-preventing interventions courtesy of Andy Baker, Michiel 7 2 Hazewinkel, Birgit Richter, and Taras Panov, in the course of the last few 1 years; he is deeply indebted to them all. 3 . 1 He is also extremely grateful to Nitu Kitchloo, for permission to include 0 2 an appendix by the latter, which outlines some work in progress. 1 : v i X 1. Conjectures about the spectrum MΞ r a 1.1 This is the Thom spectrum defined by A Baker and B Richter [2, with slightly different notation] constructed by pulling back the canonical bundle ξ → BU along the abelianization map ΩΣBT= BΩ2ΣBT→ BU. [An analytic construction of the associated representation Ξ := Ω2ΣBT→ U might be very interesting.] Date: 14 January 2012. 2000 Mathematics Subject Classification. 05E05, 14M25, 55N22. 1 2 JACKMORAVAANDNITUKITCHLOO They show that MΞ is an A ring-spectrum, with MΞ torsion-free and ∞ ∗ concentrated in even degrees, and that the Hurewicz homomorphism MΞ∗ → H∗(ΩΣCP∞)∼= NSymm∗ takes values in the (graded) ring of noncommutative symmetric functions [6 §4.2, 8]; it is injective, and becomes an isomorphism after tensoring with Q. Finally, and most striking of all, they show that MΞ⊗Z is a wedge of p copies of BP. 1.2 A (unital) S-algebra A defines a cosimplicial algebra A• : n→ A∧n with maps built from its unit and multiplication. In good cases [1] this is a resolution, in a suitable sense, of the sphere spectrum, and its homotopy groups define a cosimplicial algebra A∗ //// A∗A⊗AA∗A ////// ... which leads to the construction of an Adams spectral sequence. If A= MU, then π (MU∧MU) = MU ⊗ S ∗ ∗ Z ∗ is the product of the Lazard ring with the algebra S∗ ∼= H∗(BU,Z) of functions on the group of formal power series under composition, and the resulting cosimplicial ring can be interpreted as a presentation MU → MU ⊗S → (MU ⊗S )⊗ (MU ⊗S ) → ... ∗ ∗ ∗ ∗ ∗ MU ∗ ∗ of (the graded algebra of functions on) the moduli stack of one-dimensional formalgrouplaws. TheclassicalSteenrodaugmentation MU → Zclassifies ∗ the additive group law, and its composition MU → MU ⊗S → Z⊗S = S ∗ ∗ ∗ ∗ ∗ with the coaction represents inclusion SpecS → SpecMU ∗ ∗ of the orbit of the additive group, under coordinate changes, in the moduli stack of formal groups. It is also the (injective) Hurewicz map π∗(MU) → H∗(MU,Z) ∼= Hom(H∗(BU),Z) whichassignstoamanifold,itscollection ofChernnumbers;andfromeither point of view it is a rational isomorphism. The cosimplicial algebra π MU• ⊗Q is thus a resolution of πS ⊗Q = Q; ∗ ∗ it is a cosimplicial presentation of the stack over Q defined by the action of the group of formal diffeomorphisms on itself1. 1If G is a group object, then the category [G/G] defined by its translation action is equivalentto the category with one object and its identity morphism. THE BAKER-RICHTER SPECTRUM AS COBORDISM OF QUASITORIC MANIFOLDS3 1.3 It would be nice to have a similar description for A = MΞ, when π∗(MΞ∧MΞ) = MΞ∗MΞ ∼= MΞ∗⊗ZNSymm∗ , but we don’t yet have a good description of the coaction maps in the cosim- plicial ring it defines. I am indebted to Michiel Hazewinkel for suggesting the following possibility: Conjecture: The Hurewicz homomorphism MΞ∗MΞ → H∗(MΞ∧MΞ)∼= NSymm∗⊗ZNSymm∗ is a homomorphism of Hopf algebras, with target the Novikov double [9] of the Hopf algebra defined by the diagonal ∆ Z(t) = res Z(u)⊗(u−Z(t))−1 BFK u=0 [4 §2.4] on the ring NSymm∗ = ZhZi|i≥ 1i, Z(t)= XZiti+1 i≥0 of noncommutative symmetric functions. 1.4 In further work Baker and Richter construct [3] an injective homomor- phism λ BR z 7→ c+Xzici+1 : MΞ∗BT∼= MΞ∗hhzii→ H∗MΞ[[c]] ∼= NSymm∗[[c]] i>0 of Hopf algebras, where c is a central element corresponding to the Chern class for line bundles in ordinary cohomology. The diagram MΞ // NSymm ∗ ∗ (cid:15)(cid:15) ww (cid:15)(cid:15) MΞ ⊗Q λBR⊗//QNSymm ⊗Q ∗ ∗ lets us regard the coefficients z of their logarithm as elements of MΞ ⊗Q. i 2i 1.5 Conjecture: With the left vertical homomorphism defined by the nat- ural coaction, the diagram MΞ∗BT λBR // NSymm [[c]] ∗ ∆BFK (cid:15)(cid:15) (cid:15)(cid:15) MΞ∗BT⊗ˆMΞ∗MΞ∗MΞ ∼= MΞ∗BT⊗ˆZNSym∆mB∗T // (NSymm∗⊗ZNSymm∗)[[c]] commutes; where ∆ (1⊗c) := 1⊗Z(c) : NSymm [[c]] → (NSymm ⊗ NSymm )[[c]] . BT ∗ ∗ Z ∗ Moreover, λ ⊗Q maps π MΞ•⊗Q isomorphically to the resolution BR ∗ Q // NSymm∗⊗Q //// (NSymm∗⊗QNSymm∗) ////// ... 4 JACKMORAVAANDNITUKITCHLOO 2. Characteristic numbers for quasitoric manifolds 2.1Acomplex-oriented 2m-dimensionalquasitoric manifold M hasasimple quotient polytope P, with an omniorientation [5 §5.31] defined by a char- acteristic map Λ [5 §5.10] from the ordered set of vertices of the simplicial (m−1)-sphere K bounding the dual simplicial complex P∗ [5 §1.10], to a P free abelian group Θ with generators θ ,...,θ . 1 m The cohomology of M can be naturally identified [5 §5.2.2, §6.5, 7] with a quotient k∗(KP)⊗P(Θ∗)Z , of the Stanley-Reisner face ring [5 §3.1, 3.4]; it is an algebra over the sym- metric algebra P(Θ∗) on the Z-dual of Θ via Λ∗ : P(Θ∗) → P(V∗)→ k∗(K ) P P (where V is the free abelian group generated by the vertices of K , and Z P P is a P(Θ∗)-algebra via augmentation). 2.2 The order on the vertex set of K embeds it as the initial segment of P the natural numbers N, identifying V with a subgroup of a free abelian P group V on a countable set of generators. A similar identification embeds Θ in another free abelian group Θ on a countable set of generators, defining an extension Λ: V → Θ of Λ by Λ([i]) = θ when i is not a vertex of K . The resulting homomor- i P phism P(Θ∗) → P(V∗)= P(V∗)⊗P(V∗ ) P ¬P makes k∗(K ) := k∗(K )⊗ P(V∗ ) P P Z ¬P ∗ into an algebra over the polynomial ring P(Θ ) generated by a countable sequence of variables, such that H2∗(M,Z) ∼= k∗(KP)⊗P(Θ∗)Z . 2.3 For an ordered partition m = m +···+ m 1 r of m, let [m] be the image in H2m(M,Z) of the formal sum P [m] := X xmi11···xnmrr , i1<···<ir THE BAKER-RICHTER SPECTRUM AS COBORDISM OF QUASITORIC MANIFOLDS5 where x is the polynomial generator corresponding to i ∈ V [6 §4]. If i is i sufficiently small, x corresponds to a vertex of K ; otherwise, it is a kind i P of dummy element, and is killed by −⊗ ∗ Z. P(Θ ) 2.4 More generally, if M and N are almost-complex quasitoric manifolds of dimension m,n respectively, with quotient polytopes P,Q, then the product k∗(KP)⊗Zk∗(KQ) ∼= k∗(KP ∗KQ) of face rings is naturally isomorphic to the face ring of the join KP ∗KQ ∼= KP×Q of the simplicial spheres K and K [5 §2.13]. P Q Claim: the corresponding isomorphism H∗(M)⊗H∗(N) → H∗(M ×N) sends [m] ⊗[n] to P Q [m+n] := [m +···+m +n +···+n ] . P×Q 1 r 1 s P×Q Proof: [m+n] is the image in H2(m+n)(M ×N) of P×Q X xm1···xns , i1 ir+s i1<···<ir+s summed over strings i < ··· < i of elements of the disjoint union of the 1 r+s vertex sets of P∗ and Q∗. For a monomial of this sort to have a nontrivial image in the top-dimensional cohomology of M×N, the elements of the set {x ,...,x } must be vertices of K , and those of {x ,...,x } must i1 ir P ir+1 ir+s be vertices of K . The image of the sum is thus the product of the images Q of thesums[m] and [n] (modulothe identification of the top-dimensional P Q cohomology group of P (resp. Q) with the integers). (cid:3) This construction associates to a 2m-dimensional complex-oriented qua- sitoric manifold M, a homomorphism m7→ [m] :QSymmm → Z P ofabeliangroups(ieanoncommutativesymmetricfunctionM),whichsends M ×N to a noncommutative symmetric function M×N = M•N equalto the productof thenoncommutative symmetric functionsM andN. Problem: What is the noncommutative symmetric function CPn defined by complex projective n-space, with its usual toric structure (and the n- simplex as associated polytope)? 6 JACKMORAVAANDNITUKITCHLOO 2.5 This might be paraphrased as saying that the Davis-Januszkiewicz con- struction defines a ring homomorphism from the algebra generated by the monoid, under join, of certain omnioriented simplicial spheres, to the free graded associative algebra NSymm ; in other words, something like a coor- ∗ dinate patch for a noncommutative space of quasitoric manifolds. 3. Appendix, by Nitu Kitchloo: A quasitoric manifold M of dimension 2m admitting an action of a torus T of rank m is associated with a polytope P. Assume F = {f } is the i set consisting of the co-dimension one faces f of P. The data required to i construct M involves a collection of primitive characteristic weights λ ∈ i π (T), indexed on the set F. 1 Let Tˆ denote the torus (S1)F, of rank given by the cardinality of F, with a canonicalsetofgeneratingcirclesindexedbythefacesf . LetH ⊂Tˆ denote i the kernel of the map λ :Tˆ −→ T, defined by λ(exp(tf )) =exp(tλ ). i i The procedure for constructing M is described as follows: Notice that Tˆ acts on CF in a canonical way via Hamiltonian symplectomorphisms. This induces an action of H. Let H denote the Lie algebra of H. Let ϕ : CF −→ H∗ denote the moment map of the H action. Let the ”Moment angle complex” Z(P) denote the preimage of a regular value. The manifold M is defined as the orbit space: M = Z(P)/H. From this it follows easily that: 3.1 Claim: M = Z(P)/H has stable tangent bundle classified by the composite map: τ(M) :Z(P)/H −→ BH −→ BTˆ −→ BU(F), where BU(F) denotes the group of unitary transformations of CF, with maximal torus Tˆ. LetusnowtrytofindanaturalThomspectrumthatisthereceptacleforthe cobordism class of M. Firstly notice that there is a commutative diagram: BTˆ // ΩΣBU(1) ϕ (cid:15)(cid:15) (cid:15)(cid:15) BU(F) // BU where ϕ : ΩΣBU(1) −→ BU is the A extension of the inclusion map ∞ BU(1) −→ BU, and the map BTˆ −→ ΩΣBU(1) is the inclusion of the |F|-th James filtration. We conclude: THE BAKER-RICHTER SPECTRUM AS COBORDISM OF QUASITORIC MANIFOLDS7 3.2 Corollary: Let MΞ denote the Thom spectrum of −ϕ, then the cobor- dism class of M belongs to π MΞ. 2m 3.3 Remark: Notice that since ϕ is an A map, the Thom spectrum of ∞ −ϕ is equivalent to the Thom spectrum MΞ of ϕ, as seen easily from the following commutative diagram: ΩΣBU(1) −Id // ΩΣBU(1) ϕ −ϕ (cid:15)(cid:15) (cid:15)(cid:15) BU Id // BU In particular, the algebraic procedure described in Section 2 is indeed a method of computing the characteristic numbers of the tangent bundle of M. References 1. 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Surveys47 (1992) 198 - 199 10. http://toricmethodsbelfast.zzl.org/ DepartmentofMathematics,JohnsHopkinsUniversity,Baltimore,Maryland E-mail address: [email protected] DepartmentofMathematics,JohnsHopkinsUniversity,Baltimore,Maryland E-mail address: [email protected]