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Steenrod Coalgebras Justin R. Smith 5 Department ofMathematics 1 DrexelUniversity 0 Philadelphia, PA19104 2 r a M 6 Abstract ] T This paper shows that a functorial version of the “higher diago- A nal” of a space used to compute Steenrod squares actually con- . h tains far more topological information — including (in some cases) t a the space’s integral homotopy type. m [ 1. Introduction 6 1 v It is well-known that the Alexander-Whitney coproduct is func- 8 torial with respect to simplicial maps. If X is a simplicial set, C(X) 1 6 is the unnormalized chain-complex and RS is the bar-resolution of 2 3 Z (see [12]), it is also well-known that there is a unique homotopy . 2 1 class of Z -equivariant maps (where Z transposes the factors of 0 2 2 4 the target) 1 ξ :RS ⊗C(X) → C(X)⊗C(X) X 2 : v and that this extends the Alexander-Whitney diagonal. We will call i X such structures, Steenrod coalgebras and the map ξ the Steenrod X r a diagonal. In his construction of cup-i products, Steenrod defined a kind of dual of this map in [24]. With some care (see appendix B), one can construct ξ in a X manner that makes it functorial with respect to simplicial maps although this is seldom done since the homotopy class of this map is what is generally studied. Essentially, [18, 20, 21] show that C(X) Email address: [email protected](Justin R. Smith) URL: http://vorpal.math.drexel.edu(Justin R. Smith) Preprintsubmitted to Elsevier March 9, 2015 1 INTRODUCTION possesses the structure of a functorial coalgebra over an operad S (see example 2.9) and that the arity-2 portion of this operad-action is a functorial version of ξ . Throughout this paper, we will assume X this functorial version of ξ . X It is natural to ask whether ξ encapsulates more information X about a topological space than its cup-product and Steenrod squares. The present paper answers this question affirmatively for degeneracy-free simplicial sets. Roughly speaking, these are simplicial sets whose degeneracies do not satisfy any relations other than the minimal set of identities all face- and degeneracy-operators must satisfy — see definition A.4 and proposition A.5. Every simplicial set is canonically homotopy equivalent to a degeneracy-free one (see proposition A.6). The only place degeneracy-freeness is used in this paper is lemma 5.3. Theorem. 5.9LetX andY bepointed, reduceddegeneracy-free(see definition A.4 in appendix A) simplicial sets with normalized chain- complexes N(X) and N(Y), let R = Z for some prime p or a subring p of Q, and let f:N(X)⊗R → N(Y)⊗R be a (purely algebraic) chain map that makes the diagram RS ⊗N(X)⊗R 1⊗f //RS ⊗N(Y)⊗R (1.1) 2 2 ξX⊗1 ξY⊗1 (cid:15)(cid:15) (cid:15)(cid:15) N(X)⊗R ⊗N(X)⊗R //N(Y)⊗R ⊗N(Y)⊗R f⊗f commute exactly (i.e., not merely up to a chain-homotopy). Then f induces a simplicial map f :R X → R Y ∞ ∞ ∞ where R denotes the R-completion, (see [1] or [6] for this concept) ∞ 2 1 INTRODUCTION that makes the diagram X Y φX φY (cid:15)(cid:15) (cid:15)(cid:15) R X f∞ //R Y ∞ ∞ qX qY (cid:15)(cid:15) (cid:15)(cid:15) R˜X //R˜Y Γf commute. If f is surjective, then f is a fibration, and if f is also a ∞ homology equivalence, then f is a trivial fibration. ∞ Corollary. If f is a surjective homology equivalence, R = Z, and X and Y are nilpotent then there exists a homotopy equivalence ¯ f:|X| → |Y| of topological realizations. Corollary 5.12 states that nilpotent, degeneracy-free spaces are homotopy equivalent if and only if there exists a homology equivalence of their chain-complexes that make diagram 1.1 commute. Here, R˜∗ is a pointed version of the R-free simplicial abelian group functor — see definitions 4.2 and 4.5. Becauseofthecanonicalhomotopyequivalencebetweenallsim- plicial sets and degeneracy-free ones, the result above implies: Corollary. 5.13 If X and Y are pointed reduced simplicial sets and f:C(X) → C(Y) is a morphism of Steenrod coalgebras — over unnormalized chain- 3 1 INTRODUCTION complexes — then f induces a commutative diagram X Y OO OO gX gY d◦f(X) d◦f(Y) φ(d◦f(X)) φ(d◦f(Y)) (cid:15)(cid:15) (cid:15)(cid:15) R (d◦f(X)) f∞ //R (d◦f(Y)) ∞ ∞ q(d◦f(X)) q(d◦f(Y)) (cid:15)(cid:15) (cid:15)(cid:15) R˜(d◦f(X)) //R˜(d◦f(Y)) Γ˜f whereg andg arehomotopyequivalencesifX andY areKancom- X Y plexes— and homotopyequivalencesof their topologicalrealizations otherwise. In particular, if X and Y are nilpotent, R = Z, and f is an integral homology equivalence, then the topological realizations |X| and |Y| are homotopy equivalent. Here, f and d are functors defined in definition A.2 in appendix A. Singular simplicial sets are always Kan complexes. The reader might wonder how the Steenrod diagonal can con- tain any information beyond the structure of a space at the prime 2. The answer is that it forms part of an operad structure that contains information about all primes — and the only part of this complex operad structure needed to compute, for instance, Steen- rod pth powers is the Steenrod diagonal. For example, let X be a simplicial set with functorial higher diagonal h:RS ⊗C(X) → C(X)⊗C(X) 2 let ∆ = h([]⊗∗):C(X) → C(X)⊗C(X) — the Alexander-Whitney diagonal — and let ∆ = h([(1,2]⊗∗):C(X) → C(X)⊗C(X). A straight- 2 forward calculation shows that (1⊗∆)◦∆ :C(X) → C(X)⊗3 2 4 1 INTRODUCTION has the property that ∂{(1⊗∆)◦∆ } = (1⊗∆)◦∂∆ 2 2 = (1⊗∆)◦{(1,2)−1}∆ = (1,2,3)(∆⊗1)◦∆−(1⊗∆)◦∆ = {(1,2,3)−1}(1⊗∆)◦∆ (1.2) where (1,2,3) is a cyclic permutationof the factors. It follows that ∆ and ∆ incorporate information about X at the prime 3. Although 2 the argument in equation 1.2 is elementary, the author is unaware of any prior instance of it. This paper’s general approach to homotopy theory is the end result of a lengthy research program involving some of the 20th cen- tury’s leading mathematicians. In [15], Daniel Quillen proved that the category of simply-connected rational simplicial sets is equivalent to that of commutative coalgebras over Q. In [25], Sullivan analyzed the algebraic and analytic properties of these coalgebras, developing the concept of minimal models and relating them to de Rham cohomology. That work was dual to Quillen’s and had the advantage of being far more direct. Since then, a major goal has been to develop a similar theory for integral homotopy types. In [17], Smirnov asserted that the integral homotopy type of a space is determined by a coalgebra-structure on its singular chain-complex over an E -operad. Smirnov’s proof was somewhat ∞ opaque and several people known to the author even doubted the result’s validity. In any case, the E -operad involved was complex, ∞ being uncountably generated in all dimensions. In [21], the author showed that the chain-complex of a space was naturally a coalgebra over an E -operad S and that this could ∞ be used to iterate the cobar construction. The paper [19] applied those results to show that this S-coalgebra determined the integral homotopy type of a simply-connected space. In [13]1, Mandell showed that the mod-p cochain complex of a p-nilpotent space had a algebra structure over an operad that 1Based on Mandell’s 1997 thesis. 5 2 DEFINITIONS determined the space’s p-type. In [14], Mandell showed that the cochains of a nilpotent space whose homotopy groups are all fi- nite have an algebra structure over an operad that determined its integral homotopy type. The paper [18] showed that the S-coalgebra structure of a chain-complex had a “transcendental” structure that determines a nilpotent space’s homotopy type (without the finiteness conditions of [14]). It essentially reprised the main result of [19], using a very different proof-method. The present paper shows that this transcendental structure even manifests in the sub-operad of S generated by its arity-2 component, RS . 2 I am indebted to Dennis Sullivan for several interesting discus- sions. 2. Definitions Given a simplicial set, X, C(X) will always denote its unnormalized chain-complex and N(X) its normalized one (with degeneracies divided out). Remark 2.1. Throughout this paper R denotes a fixed ring satisfy- ing Z for some prime p or R = p (R ⊂ Q Definition 2.2. We will denote the category of R-free chain chain- complexes by Ch and ones that are bounded from below in dimen- sion 0 by Ch . 0 We make extensive use of the Koszul Convention (see [8]) re- garding signs in homological calculations: Definition 2.3. If f:C → D , g:C → D aremaps, anda⊗b ∈ C ⊗C 1 1 2 2 1 2 (where a is a homogeneous element), then (f ⊗ g)(a⊗ b) is defined to be (−1)deg(g)·deg(a)f(a)⊗g(b). Remark 2.4. If f , g are maps, it isn’t hard to verify that the Koszul i i convention implies that (f ⊗g )◦(f ⊗g ) = (−1)deg(f2)·deg(g1)(f ◦f ⊗ 1 1 2 2 1 2 g ◦g ). 1 2 6 2 DEFINITIONS The set of morphisms of chain-complexes is itself a chain complex: Definition 2.5. Given chain-complexes A,B ∈ Ch define Hom (A,B) Z to be the chain-complex of graded R-morphisms where the degree of an element x ∈ Hom (A,B) is its degree as a map and with differ- Z ential ∂f = f ◦∂ −(−1)degf∂ ◦f A B As a R-module Hom (A,B) = Hom (A ,B ). Z k j Z j j+k Remark. Given A,B ∈ ChSn, weQcan define Hom (A,B) in a corre- ZSn sponding way. Recall the concept of algebraicoperad in [11] or [10]: a sequence of ZS chain-complexes {V(n)} for n ≥ 0 with structure maps n γ :V(n)⊗V(i )⊗···⊗V(i ) → V(i +···i ) i1,...,in 1 n 1 n for n,i ,...,i ≥ 0. 1 n Definition 2.6. We will call the operad V = {V(n)} Σ-cofibrant if V(n) is ZS -projective for all n ≥ 0. n Remark. The operads we consider here correspond to symmetric operads in [22]. The term “unital operad” is used in different ways by different authors. We use it in the sense of Kriz and May in [10], mean- ing the operad has a 0-component that acts like an arity-lowering augmentation under compositions. Here V(0) = R. The term Σ-cofibrant first appeared in [3]. We also need to recall compositions in operads: Definition 2.7. If V is an operad with components V(n) and V(m), define the ith composition, with 1 ≤ i ≤ n ◦ :V(n)⊗V(m) → V(n+m−1) i 7 2 DEFINITIONS by V(n)⊗V(m) V(n)⊗Zi−1 ⊗V(m)⊗Zn−i 1⊗ηi−1⊗1⊗ηn−i (cid:15)(cid:15) V(n)⊗V(1)i−1 ⊗V(m)⊗V(1)n−i γ (cid:15)(cid:15) V(n+m−1) Here η:Z → V(1) is the unit. Remark. Operads were originally called composition algebras and defined in terms of these operations — see [5]. It is well-known that the compositions and the operad structure-maps determine each other — see definition 2.12 and proposition 2.13 of [22]. A simple example of an operad is: Example 2.8. For each n ≥ 0, S (n) = ZS , with structure-map a 0 n Z-linear extension of γ :S ×S ×···×S → S α1,...,αn n α1 αn α1+···+αn defined by γ (σ ×θ ×···×θ ) = T (σ)◦(θ ⊕···⊕θ ) α1,...,αn 1 n α1,...,αn 1 n with σ ∈ S and θ ∈ S where T (σ) ∈ S is a permutation n i αi α1,...,αn Pαi that permutes the n blocks {1,...,α },{α +1,α +α },..., 1 1 1 2 {α +···+α +1,α +···+α } 1 n−1 1 n via σ. See [21] for explicit formulas and computations. Another important operad is: 8 2 DEFINITIONS Example 2.9. The operad, S, defined in [21] is given by S(n) = RS n — the normalized bar-resolution of Z over ZS . This is well-known n (like the closely-related Barrett-Eccles operad defined in [2]) to be a Hopf-operad, i.e. equipped with an operad morphism δ:S → S⊗S and is important in topological applications. See [21] for formulas for the structure maps. For the purposes of this paper, the main example of an operad is Definition 2.10. Given any C ∈ Ch, the associated coendomorphism operad, CoEnd(C) is defined by CoEnd(C)(n) = Hom (C,C⊗n) Z Its structure map γ :Hom (C,C⊗n)⊗Hom (C,C⊗α1)⊗···⊗Hom (C,C⊗αn) → α1,...,αn Z Z Z Hom (C,C⊗α1+···+αn) Z simply composes a map in Hom (C,C⊗n) with maps of each of the n Z factors of C. This is a non-unital operad, but if C ∈ Ch has an augmentation map ε:C → R then we can regard ε as the generator of CoEnd(C)(0) = R ·ε ⊂ Hom (C,C⊗0) = Hom (C,R). Z Z We use the coendomorphism operad to define the main object of this paper: Definition 2.11. A coalgebra over an operad V is a chain-complex C ∈ Ch with an operad morphism α:V → CoEnd(C), called its structure map. We will sometimes want to define coalgebras using the adjoint structure map, α:C → Hom (V(n),C⊗n) (2.1) ZSn n≥0 Y 9 2.1 Coalgebrasoveroperads 2 DEFINITIONS where S acts on C⊗n by permuting factors or the set of chain-maps n α :C → Hom (V(n),C⊗n) n ZSn for all n ≥ 0 or even β :V(n)⊗C → C⊗n n It is not hard to see how compositions (in definition 2.7) relate to coalgebras Proposition 2.12. If the maps β :V(n)⊗C → C⊗n for all n ≥ 0 define n a coalgebra over an operad V, for any x ∈ V(n) and any n ≥ 0 define ∆ = β (x⊗∗):C → C⊗n x n If x ∈ V(n) and y ∈ V(m), then ∆ = 1⊗···⊗1⊗∆ ⊗1⊗···⊗◦∆ y◦ix y x ithposition | {z } Proof. Immediate, from definitions 2.7 and 2.10. 2.1. Coalgebras over operads Example 2.13. Coassociative coalgebras are precisely the coalgebras over S (see 2.8). 0 Definition 2.14. Comm is an operad defined to have one basis element {b } for each integer i ≥ 0. Here the arity of b is i and the i i degree is 0 and the these elements satisfy the composition-law: γ(b ⊗b ⊗···⊗b ) = b , where K = n k . The differential of this n k1 kn K i=1 i operad is identically zero. The symmetric-group actions are trivial. P Example 2.15. Coassociative, commutative coalgebras are the coalgebras over Comm. We can define a concept dual to that of a free algebra generated by a set: 10