S2- and P 2-category of manifolds J. C. G´omez-Larra~naga∗ F. Gonz´alez-Acu~nay Wolfgang Heilz Abstract n 2 2 A closed topological n-manifold M is of S - (resp. P )-category 2 if it can be covered by two open subsets W1,W2 such that the inclusions n 2 2 Wi ! M factor homotopically through maps Wi ! S (resp. P ). 2 2 We characterize all closed n-manifolds of S -category 2 and of P - category 2. 1 2 1 Introduction While studying the minimal number of critical points of a closed smooth n- manifold M n, denoted by crit(M n), Lusternik and Schnirelmann introduced what is now called the Lusternik-Schnirelmann category of M n, denoted by cat(M n), which is defined to be the the smallest number of sets, open and contractible in M n that are needed to cover M n. They showed that cat(M n) is a homotopy type invariant with values between 2 and n+1 and furthermore that cat(M n) ≤ crit(M n). This invariant has been widely studied, many references can be found in [CLOT]. ∗Centro de Investigaci´onen Matem´aticas,A.P. 402, Guanajuato 36000, Gto. M´exico and Instituto de Matem´aticas,UNAM, Ciudad Universitaria, 04510 M´exico,D.F. M´exico. [email protected] yInstituto de Matem´aticas,UNAM, 62210 Cuernavaca, Morelos, M´exicoand Cen- tro de Investigaci´on en Matem´aticas, A.P. 402, Guanajuato 36000, Gto. M´exico. fi[email protected] zDepartment of Mathematics, Florida State University, Tallahasee, FL 32306, USA. [email protected] 1AMS classification numbers: 57N10, 57N13, 57N15, 57M30 2Key words and phrases: Lusternik-Schnirelmann category, coverings of n- manifolds with open S2-contractible or P2-contractible subsets 1 In 1968 Clapp and Puppe [CP] generalized this invariant as follows: Let A be a class of topological spaces. For a space A 2 A a subset B in M n is A-contractible if there are maps f : B −! A and α : A −! M n such that n n the inclusion map i : B −! M is homotopic to α · f. Then catA M is the smallest number m such that M n can be covered by m open sets, each A-contractible in M n, for some A 2 A. If A = fAg consists only of one n n space A write catA M instead of catfAg M . Clapp and Puppe also pointed n out relations between catA M and the set of critical points of smooth func- tions of M n to R. For n = 3 Khimshiashvili and Siersma [KhS] obtained a 3 relation between catS1 (M ) and the set of critical circles of smooth functions M 3 ! R. In [GGH],[GGH1],[GGH2] we obtained a complete classification n of the closed (topological) n-manifolds with catS1 (M ) = 2. n Motivated by the work of Gromov [G] (see also [I]) we define catameM to be the smallest number of open and amenable sets needed to cover M n; here a set A ⊂ M is amenable if for each path-component Ak of A the im- n age of the inclusion induced homomorphism im(ι∗ : π(Ak) ! π(M )) is an amenable group. Gromov has shown [G] that if M n is a closed n-manifold n with positive simplicial volume then catame(M ) = n + 1 . Hence, by Perel- 3 man (see [MT]), if catameM ≤ 3 then M is a graph manifold. If A is the class of connected CW-complexes with amenable fundamental groups then n n n catame M ≤ catA M ≤ catK M ≤ n + 1 for any K in A. Examples of such K are P (a point), S1, S2, P2, S1×~ S1 (an S1-bundle over S1). n n 2 2 In the present paper we consider the cases of catS (M ) and catP (M ). The main results are Theorem 1 which gives a classification of (topolog- n ical) n-manifolds with catS2 (M ) = 2, and Theorem 2 which exhibits a complete list of the fundamental groups of all (topological) n-manifolds with n 2 catP (M ) = 2. In particular, for n = 3 we obtain in Corollary 2 a complete 3 2 list of all 3-manifolds of catP (M ) = 2. The paper is organized as follows: In section 2 we point out that if n n catK (M ) = 2 for a CW-complex K then M can be covered by two compact K-contractible submanifolds that meet only along their boundaries and we show how to pull back K-contractible subsets of M to covering spaces of M. In section 3 we associate to a decomposition of M into two K-contractible submanifolds (where K = S2 or P2) a graph of groups and compute the fundamental group of this graph of groups. This, together with information 2 about the homology of M n developed in section 4 is used to prove Theorem 1 in section 5 and Theorem 2 in section 6. 2 K-contractible subsets In this section we assume that M = M n is a closed connected n-manifold and K is a CW-complex. A subset W of M is K-contractible (in M) if there are maps f : W ! K and α : K ! M such that the inclusion ι : W ! M is homotopic to α · f. catK (M) is the smallest number m such that M can be covered by m open K-contractible subsets. Note that a subset of a K-contractible set is also K-contractible. It is easy to show that catK is a homotopy type invariant. In particular, if catK (M) = 2 then M is covered by two open sets W0, W1 and for i = 0; 1, there are maps fi and αi such that the diagram below is homotopy commutative: ι - Wi M @ α fi @R i K The following proposition allows us to replace the open sets Wi by com- pact submanifolds that meet only along their boundaries. Proposition 1. If catK M = 2 then M can be expressed as a union of two compact K-contractible n-submanifolds W0, W1 such that W0 \ W1 = @W0 = @W1. This was proved in [GGH] for K = S1 using topological transversality (see [KS] and [Q]). The same proof applies for any finite complex K. 3 ~ ~ Now suppose p : M ! M is a covering map. For α : K ! M let Kp be the pullback of M~ p ? α KM- ~ ~ ~ ~ ~ i.e. Kp = f (x; y) 2 K × M j α(x) = p(y) g and let q : Kp ! K,α ~ : Kp ! M be the maps induced by the projections q(x; y) = x,α ~(x; y) = y. Lemma 1. Let W,! M be K-contractible in M with ι ' α · f and let ~ ~ −1 ~ p : M ! M be a covering map. Then W := p (W ) is Kp-contractible in M~ . Proof. We have a diagram ~ι W~ - M~ @ f~ α~ @ @R ~ Kp p0 q p ? K f @ α@ ? @@R ? ι WM- where p0 is the restriction of p and ~ι is the inclusion. The homotopy ι ' α · f lifts to a homotopy ~ι ' h~ for some map h~ : W~ ! M~ such that (α·f)·p0 = p·h~. Now define f~ by f~(z) = (fp0(z); h~(z)) to get q · f~ = f · p0 and (~α · f~) = h~ ' ~ι. 4 3 Fundamental group n In this section we consider the structure of π1(M ) for a closed n-manifold n n n 2 2 M with catS (M ) = 2 or catP (M ) = 2 by using the theory of graphs of groups ([S]). 1 1 2 2 Since clearly catS (S ) = catP (S ) = 2 we assume from now on that n > 1. By Proposition 1 we may assume that n n • M = W0 [W1 such that F := W0 \W1 = @W0 = @W1. Here Wi = Wi are K-contractible n-submanifolds of M where K = S2 or K = P2. Consider the graph G of (M; F ) whose vertices (resp. edges) are in one- j to-one correspondence with the components Wi of Wi, i = 0; 1 (resp. with j k j the components Fjk = W0 \ W1 of F ). Vertices of G corresponding to W0 k j k and W1 are joined by the edges corresponding to the components of W0 \W1 . For the associated graph G of groups the group Gv associated to a vertex v j j corresponding to a component Wi of Wi is im(π1(Wi ) ! π1(M)) and the group Ge associated to an edge e corresponding to a component Fk of F is im(π1(Fk) ! π1(M)). In our case these groups are either Z2 or trivial. 0 For the vertices v,v of e the monomorphisms Ge ! Gv and Ge ! Gv0 are induced by inclusions. The fundamental group of M is isomorphic to the fundamental group πG of G (see for example [SW]). For the computation of πG we follow [S]: Pick an orientation of each edge of G. For each (oriented) edge e from a vertex v to a vertex v0 the corre- sponding element in πG is denoted by ge. The monomorphism Ge ! Gv (resp. Ge ! Gv0 ) sends a generator ae of Ge to a generator bv of Gv (resp. to a generator bv0 of Gv0 ). Let T be a maximal tree T in G. Then πG is generated by the ge for each (oriented) edge e in G − T and the generators −1 bv of Gv and defining relations are gebvge = bv0 for e 2 G − T and bv = bv0 for e 2 T . 5 From this presentation of πG it follows that if all vertex groups of G are trivial then πG ∼= F, for some free group F, hence n Lemma 2.