aa r X i v : . [ m a t h . A T ] A ug Steenrod operations on bar complex
Syunji Moriya ∗ Abstract
We define a chain map of the form E ( k ) ⊗ BA ⊗ k −→ BA , where E is a combinatorial E ∞ -operad called the sequence operad, and BA is the bar complex of an E -algebra A . We see thatSteenrod-type operations derived from the chain map are equal to the corresponding operationson the cohomology of the based loop space under an isomorphism. The bar complex is a model of cochain of a based loop space. J.R.Smith [9] and B.Fresse [11] enrichedthe bar complex of an E ∞ -algebra with an E ∞ -structure. In view of M.A.Mandell’s theorem [4,Main Theorem], this enrichment provides a complete algebraic model of p -adic homotopy type of abased loop space and enables them to iterate the bar construction. As another complete model, thecategorical bar complex is known but the (classical) bar complex has the advantage that one doesnot need to take a cofibrant replacement of an algebra to obtain the right answer with it. Fressedefined the E ∞ -structure, using systematically bar modules and model category structures on thecategories of modules and algebras over an operad.Our motivation is to find a combinatorial alternative of these E ∞ -structures on the bar complex.In this article, we prove the following theorem. Theorem 1.0.1 (Thm.2.4.3) . Let k be a positive integer. Let E denote the sequence operad and A be an E -algebra. Let BA be the bar complex of A . There exists a chain map Φ k : E ( k ) ⊗ Σ k BA ⊗ k −→ BA.
Here, Σ k is the k -th symmetric group. Here, the sequence operad is a small combinatorial model of E ∞ -operad introduced by J.E.McClureand J.H.Smith [5], which naturally acts on the normalized cochain. We warn the reader that thechain map Φ k does not define an operad action. So this result cannot be used for iteration.The utility of the map Φ k is that Steenrod operations can be derived from it. In fact, we canapply the framework of J.P.May [1] to the map Φ k in order to define operations P s and βP s . Weshow these operations are isomorphic to the corresponding operations on the cohomology of a basedloop space by a simple application of an argument of Fresse (see section 3).We shall mention preceeding works. H.J.Baues [2] defined a product on the bar complex ofnormalized cochains which is equivalent to the cup product. T.Kadeishvili [7] defined ∪ i -producton the bar complex, generalizing Baues’ construction (over a field of characteristic 2). Thm. 1.0.1 ∗ Corresponding address: Department of Mathematics, Faculty of Science, Kyoto University, Kyoto, 606-8502,Japan. E-mail adress: [email protected]
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1s considered as a generalization of them as the operations of Baues and Kadeishvili are equal toevaluations of Φ at some elements, see Prop.2.5.1. For another generalization, see Fresse [6].In the last section, we define a diagonal on the sequence operad. An immediate consequence isthat the tensor of two E -algebra has functorial E -algebra structure. For Barrat-Eccles operad [8],which is another combinatorial model of E ∞ -operad, a diagonal is already known. As the action ofthe sequence operad on the normalized cochain is transparent, the tensor product provides a simplemodel of product and smash of spaces, for example. Notation and Terminology (1) We fix a base ring k . All complexes are defined over k and considered as cohomologically graded, i.e., differentials raise degree. For usually homologicallygraded complexes such as operads, we implicitly regard them as cohomologically graded by negatingdegree. For an element x of a complex, | x | denotes its degree and we put || x || = | x | − k ≥
1, ¯ k denotes the ordered set { , . . . , k } with the usual order.Let S be a finite set. | S | denotes the cardinality of S . || S || denotes | S | − S is non-empty, 0otherwise.(2) For k ≥
0, Σ k denotes the k -th symmetric group. As usual, an operad O = {O ( k ) } k ≥ is asequence of Σ k -modules O ( k ) equipped with composition multiplications which satisfy associativityand equivariance (see [3]). We denote by ¯ O the sequence obtained from O by replacing O (0) withthe zero module. ¯ O has natural operad structure induced from O .The sequence operad is defined in [5]. The surjection operad in [8] is the same thing except forsign difference. By definition, its k -th module is the free graded abelian group generated by non-degenerate sequences f : ¯ m → ¯ k , whose homological degree is m − k . As in [5], a non-degeneratesequence f is presented as ( f (1) f (2) . . . f ( m )). For example, (12) denotes the identity on ¯2. Wedenote by E the sequence operad tensored with k . We entirely follow the sign rules of [5]. Fordifferential d ( ∂ in [5]), we write that df = m X q =1 ( − τ ′ f ( q ) d q f, where d q f = f ¯ m −{ q } and τ ′ f ( q ) = τ f ( q ) − f ( q ) in the notation of [5]. To simplify notations we put f · g := (12)( f, g ) for f, g ∈ E . We denote by ⋄ the action of the symmetric groups, like f ⋄ σ for f ∈ E ( k ) and σ ∈ Σ k . We omit · and ⋄ if it does not cause confusion.We use chain maps r a : E ( k ) → E ( k − ι a : E ( k − → E ( k ) and a chain homotopy s a : E ( k ) →E ( k ) for a ∈ ¯ k . For a non-degenerate sequence f : ¯ m → ¯ k , r a ( f ) = 0 if | f − ( a ) | ≥