A simplicial model for the Hopf map Orin R. Sauvageot* orin.sauvageot@epfl.ch October 29, 2003 Abstract We give an explicit simplicial model for the Hopf map S3 ! S2. For this purpose, we construct a model of S3 as a principal twisted cartesian product K x''S2, where K is a simplicial model for S1 acting by left multiplication on itself, S2 is given the simplest simplicial model and the twisting map is j : S2 n ! (K)n-1. We construct a Kan complex for the simplicial model K of S1. The simplicial model for the Hopf map is then the projection K x''S2 ! S2. Mathematics Subject Classification : 55U10, 55Q40; 55R10 1 Introduction The motivation for finding a simplicial model for the Hopf map arose when trying to find a simple test to decide whether the stabilisation of a certain interesting model category M is different from that of the category of chain complexes of abelian groups. As detailed in [2, chapter 6], consider the following situation. Let M be a symmetric monoidal model category whose stabilisation exists and suppose there is a monoidal Quillen adjunction F : sS $ M : G between the category of simplicial sets and M. In the stable category of chain complexes, the Hopf map vanishes. Therefore, if we have a good simplicial model for the Hopf map that allows us to show that the ______________________________ *'Ecole Polytechnique F'ed'erale de Lausanne - Institute of Geometry, Algebr* *a and Topology - CH-1015 Lausanne, Switzerland 1 multiply suspended images under the functor F of this simplicial model never vanish, then the stabilisation of M is different from that of chain complexes. The main result of this paper is that a very good simplicial model of the Hopf map is the projection p : K x''S2 ! S2 of a principal twisted cartesian product of a simplicial model K of S1 with the simplest simplicial model for S2. The proof of this result shows that we are able to model simplicially any S1-bundle of base S2. This paper is structured in the following manner. Section 2 recalls the notions and results related to principal twisted cartesian products. In sec- tion 3 we construct a Kan model for S1, which is required to carry enough structure. Section 4 gives explicit computations of the Kan model as well as of the twisting map. Finally, we prove the main result in section 5. We are indebted to Kathryn Hess and Andrew Tonks for the idea of using principal twisted cartesian products. The idea of using the Hopf map as a test came out from a discussion with Stefan Schwede. This work has been carried out with the financial support of the Swiss National Science Foundation. 2 Principal twisted cartesian products This section is devoted to explaining the tools for building a simplicial model for S3 with enough structure to capture the Hopf map. Definition 2.1. Let F and B be two simplicial sets. Let H be a simplicial group acting on the left on F . Let i : B ! H be a map of graded sets of degree -1 such that in : Bn ! Hn-1 satisfies the following identities: @0i(b) = (i(@0b))-1i(@1b) @ii(b) = i(@i+1b) for i > 0 sii(b) = i(si+1b) for i 0 i(s0b) = idn forb 2 Bn. The map i is the twisting map. A twisted cartesian product of fibre F , base B and group H is a simplicial set denoted F x``B satisfying (F x``B)n = Fn x Bn with faces and degeneracies as follows : 1. @i(f, b) = (@if, @ib) for i > 0 2 2. @0(f, b)) = (i(b)@0f, @0b) 3. si(f, b) = (sif, sib) for i 0. Furthermore, if F = H acting on itself by left multiplication, then F x``B is a principal twisted cartesian product (PTCP). We will also use the terminology "twisted cartesian product" for the pro- jection p : F x``B ! B. The following proposition is a classical result whose proof can be found in [1, proposition 18.4]. Proposition 2.2. Let p : F x``B ! B be a twisted cartesian product with group H. If the fiber F is a Kan complex, then : 1. the projection p is a Kan fibration, and 2. if F = H, p is a principal fibration. Remark 2.3. Let S : TOP ! sS be the singular functor from the category TOP of topological spaces to the category sS of simplicial sets. A map f is a (Serre) fibration if and only if S(f) is a Kan fibration. Thus, a principal fibration (or fibre bundle) in TOP passes via the functor S to a principal fibration in sS. Since the Hopf map S3 ! S2 is a fibration in TOP , the corresponding simplicial model has to be a Kan fibration. As a consequence, if we want to model S3 as a PTCP K x''S2 ! S2, this has to be a Kan fibration, which it is when K is a Kan complex, by proposition 2.2. We construct such a PTCP in the following sections. 3 The simplicial model for S1 In short, to build a Kan model of S1 we let Z(2) denote a chain complex concentrated in degree two, and we apply a functor to obtain a simplicial abelian group Z(2). By applying the loop group functor G, the model of S1 is given by G Z(2). The latter is always a Kan complex, since every simplicial group is a Kan complex. More precisely we give the following definitions. Definition 3.1. Let sAb be the category of simplicial abelian groups and let CC be the category of chain complexes of abelian groups. We define the functor : CC ! sAb as follows. For any (X, @) 2 CC , the simplicial abelian group (X) is given by : 3 1. n-1MX n(X) = Xn oejk. .o.ej1Xr (1) r=0k=n-r where oejk. .o.ej1Xr is the abelianPgroup whose elements are the symbols oejk. .o.ej1x with x 2 Xr. The sum k=n-r is taken over all sequences of indices {ji} such that 0 j1 < j2 < . .<.jk < n. The addition of symbols is defined by oejk. .o.ej1x + oejk. .o.ej1y = oejk. .o.ej1(x + y). Degeneracies and faces are given by : 2. si : n(X) ! n+1(X) is defined by (a) six = oeix for x 2 Xn (b) if k = n - r and x 2 Xr then sioejk. .o.ej1x = oehk+1. .o.eh1x when sisjk. .s.j1= shk+1. .s.h1and where shk+1. .s.h1is written in the canonical form1, i.e. hk+1 > hk > ... > h1. 3. @i : n(X) ! n-1(X) is defined by (a) @nx = @(x) and @ix = 0 if i < n and x 2 Xn. (b) if k = n - r and x 2 Xr then 8 >> oehk. .o.eh1@(x) >: 0 if respectively 8 >> shk. .s.h1@r >: shk. .s.h1@j j < r where the right hand side is written in the canonical form. ______________________________ 1Every composition of degeneracies and/or faces can be written in the canoni* *cal form with the aid of the simplicial identities. 4 We now define the functor G. Definition 3.2. Let sGr the category of simplicial groups and let K be a simplicial set. We define the functor G : sS ! sGr as follows. The group Gn(K) = G(K)n is the free group generated by the elements of Kn+1 modulo the relations s0x = idn for all x 2 Kn. If x 2 Kn+1, let i(x) be the class of x in Gn(K). Faces and degeneracies of G(K) are defined on generators by the relations : i(@0x)@0i(x) = i(@1x) (2) @ii(x) = i(@i+1x) ifi > 0 (3) sii(x) = i(si+1x) ifi 0. (4) By extension we have homomorphisms @i : Gn(K) ! Gn-1(K) and si : Gn(K) ! Gn+1(K). Clearly, G(K) is a simplicial group. Remark 3.3. The morphism i of definition 3.2 is clearly a twisting map. Hence, for every simplicial abelian group K we have a twisted cartesian prod- uct G(K) x``K, which is acyclic. The reader may refer to [1, pp. 118-123] for details. By [1, Remarks 23.7], Z(2) is a K(Z, 2), hence a simplicial model for BS1. G Z(2) is then a model for BS1, hence for S1. 4 Some computations This section is devoted to clarifying the previous construction by giving ex- plicit computations of G Z(2) and the map j. For this we will choose a simplicial model for S2 consisting in one non degenerate simplex in degree two and only degeneracies above. To compute nZ(2) we use formula (1). Since Z(2) is concentrated in degree two, we obtain M nZ(2) = oejn-2. .o.ej1Z. (5) 0 j1<...