DETECTING ELEMENTS AND LUSTERNIK-SCHNIRELMANN CATEGORY OF 3-MANIFOLDS JOHN OPREA AND YULI RUDYAK Abstract.In this paper, we give a new simplified calculation of the Lust* *ernik- Schnirelmann category of closed 3-manifolds. We also describe when 3-man* *ifolds have detecting elements and prove that 3-manifolds satisfy the equality * *of the Ganea conjecture. 1.Introduction The Lusternik-Schnirelmann category of a space X, denoted cat(X), is defined to be the minimal integer k such that there exists an open covering {A0, . .,.A* *k} of X with each Ai contractible to a point in X. Category, while easy to define,* * is notoriously difficult to compute in general. In particular, except for K(ß, 1)'* *s, it cannot be expected that the category of a space is determined by its fundamental group. In [GoGo ], however, the following interesting result was proved. 1.1. Theorem. Let M3 be a closed 3-dimensional manifold. Then 8 ><1 if ß1(M) = {1} cat(M) = >2 if ß1(M) is free. :3 otherwise In this paper, we will give a somewhat simplified proof of this theorem using* * the relatively new approximating invariant for category, category weight. Throughou* *t, we use only basic results about 3-manifolds found, for instance, in [H ]. But w* *e shall also do more. We will prove that most 3-manifolds possess a detecting element; that is, an element whose category weight is equal to the category of M (see [R* *3 ]). It is known that a detectable space (i.e., a space possessing detecting element* *s) has some special properties which allow solutions of certain well-known problems ([R3 ]). For example, from the existence of detecting elements, we prove that c* *losed 3-manifolds satisfy the Ganea conjecture. 1.2. Corollary. For every closed 3-manifold M, cat(M x Sn) = cat(M) + 1. This result is not obtainable from knowing the category alone, so the detecti* *ng element approach is a significant embellishment of Theorem 1.1. Another well- known problem is the relationship between degree 1 maps of manifolds and LS- category. For closed, 3-manifolds, we have 1.3. Corollary. Let f :M ! N be a degree 1 map of oriented 3-manifolds. Then catM catf = catN. ____________ Date: September 26, 2002. 1991 Mathematics Subject Classification. Primary 55M30; Secondary 57M99. 1 2 JOHN OPREA AND YULI RUDYAK We now turn to the fundamentals of 3-manifolds. 2.Preliminaries on 3-Manifolds 2.1. Definition. A 3-manifold M is irreducible if every embedded two-sphere S2 * *,! M bounds an embedded disk D3 ,! M. A 3-manifold M is prime if M = P # Q implies that either P = S3 or Q = S3. Here, "=" denotes diffeomorphism and # is the connected sum. The following two results clarify the relation between prime and irreducible * *man- ifolds. 2.2. Lemma. If M3 is irreducible, then it is prime. Proof.Suppose M is irreducible. In order to split M as M = P # Q, there must be an embedded S2 which separates M into two components (i.e. P - D3 and Q - D3). But any such S2 bounds an embedded disk D3 by irreducibility, so M can only split as M = M0# S3 (since S3 - D3 is a disk D3). This says that M is prime. 2.3. Lemma. If M is a prime 3-manifold and M is not irreducible, then M is the total space of a 2-sphere bundle over S1. Proof.See [H , Lemma 3.13] The fundamental structural result about 3-manifolds is the following 2.4. Theorem (Prime Decomposition). A 3-manifold M may be written as M = M1# M2# . .#.Mk, where each Mj is prime. Furthermore, such a prime decomposition is unique up to re-arrangement of summands. Proof.See [H , Theorems 3.15 and 3.21] The Sphere theorem says that, for an orientable 3-manifold M, ß2(M) 6= 0 implies that some element of ß2(M) is represented by an embedding S2 ,! M. We will require the following generalization. 2.5. Theorem (The Projective Plane Theorem). Let M be a 3-manifold with ß2(M) 6= 0. Then there exists a map g :S2 ! M with the following properties. (1) The map g is not null-homotopic. (2) The map g :S2 ! g(S2) is a covering map. (3) g(S2) is a 2-sided submanifold (2-sphere or projective plane) in M. Proof.See [H , Theorem 4.12]. With these preliminaries, we can prove the folowing important characterization. 2.6. Proposition. Let M be a closed 3-manifold. Then, (1) If ß = ß1(M) is infinite and ß2(M) = 0, then M = K(ß, 1). (2) If ß1(M) is finite, then the universal covering of M is a homotopy 3-sph* *ere and M is orientable. DETECTING ELEMENTS AND LUSTERNIK-SCHNIRELMANN CATEGORY OF 3-MANIFOLDS3 Proof.For (1), assume that ß1(M) is infinite. Let p: fM ! M be the universal covering of M. Since ß2(M) = 0, we conclude that H2(fM) = 0. Since ß1(M) is infinite, we conclude that fM is not compact, and therefore H3(fM) = 0. Hence, * *fM is acyclic. Moreover, fM is simply-connected, and, by the Whitehead theorem, it* * is therefore contractible. Hence, M = K(ß1(M), 1). For (2), assume that ß1(M) is finite. Then the universal cover fM of M is a closed simply connected manifold. So, by Poincar'e duality, H2(fM) = 0, and hen* *ce, by the Hurewicz theorem, ß2(fM) = 0. Thus, ß2(M) = 0. Furthermore, H3(fM) = Z = ß3(fM), again by the Hurewicz theorem. Therefore, the generator of ß3(fM) = Z provides a degree 1 map S3 ! fM(i.e. an isomorphism on H3). Since fM and S3 are simply connected, the Whitehead theorem implies that fM ' S3. To see that M is orientable, we simply note that each g 2 ß1(M), thought of a* *s a covering transformation on the orientable manifold fM, acts to preserve orienta* *tion. This is seen by supposing the opposite; namely, that g reverses orientation. No* *w, because fM ' S3, homotopy classes of maps fM ! fM are classified by degree. Since g is a homeomorphism which reverses orientation, its degree is -1. But th* *en the Lefschetz number of g is L(g) = 2, implying the existence of a fixed point and contradicting the fact that g is a covering transformation. Hence, all cove* *ring transformations preserve orientation, so M = fM=ß1(M) is orientable. These are the only ingredients from 3-manifold theory that we shall need. In the next section, we introduce the main technical tool, the approximating invariant category weight. 3.Category Weight and Detecting Elements 3.1. Definition ([BG , Fe, F]). Let f :X ! Y be a map of finite CW -spaces. The Lusternik-Schnirelmann category of f, denoted cat(f), is defined to be the minimal integer k such that there exists an open covering {A0, . .,.Ak} of X wi* *th the property that each of the restrictions f|Ai:Ai ! Y , i = 0, 1, . .,.k is nu* *ll- homotopic. Clearly, cat(X) = cat(1X ) and. Also, it is easy to see that cat(f) cat(X) si* *nce f is null-homotopic on any subset which is contractible in X. 3.2. Definition. The category weight of a non-zero cohomology class u 2 H*(X; R) (for some, possibly local, coefficient ring R) is defined by wgt(u) k if and only ifOE*(u) = 0 for anyOE: A ! X withcat(OE) < k. 3.3. Remarks. 1. The idea of category weight was suggested by Fadell and Hussei* *ni (see [FH ]). In fact, they considered an invariant similar to our wgt (denoted * *in [FH ] by cwgt), but where the defining maps OE: A ! X were required to be inclusions instead of general maps. Because of this, cwgt was not a homotopy invariant, and this made it a delicate quantity in homotopy calculations. Rudyak in [R2 , R3 ] and Strom in [S] suggested the homotopy invariant version of category weight as defined in Definition 3.2. Rudyak called it strict category weight (using the n* *otation swgt(u)) and Strom called it essential category weight (using the notation E(u)* *). 4 JOHN OPREA AND YULI RUDYAK At the Mt. Holyoke conference for which these proceedings are a record, both creators agreed to adopt the notation wgt and call it simply category weight. 2. In fact, one can define category weight for u 2 F *(X) where F is a suitab* *le functor on the category of topological spaces (e.g. F (X) = [X, Y ] or F is an arbitrary cohomology theory), see [R2 , R3, S]. However, Definition 3.2 is enou* *gh for our goals here. 3. There is an alternative definition of category weight which is actually mo* *re useful than the one given in Definition 3.2. Recall that the Ganea fibration pj* *:Gj(X) ! X is defined inductively starting with the path fibration p0: P X = G0(X) ! X having fibre X. Then given the fibration pi:Gi(X) ! X with fibre Fi = *(i+1) X, the fibration pi+1 is constructed by taking the cofibre Z of the incl* *u- sion Fi ! Gi(X) and extending pi to a map Z ! X (which is possible since the composition Fi! Gi(X) pi!X is null-homotopic. Finally, convert the map Z ! X to a fibration pi+1:Gi+1(X) ' Z ! X. Then it is known that cat(X) = k if and only if k is the least integer such that pk: Gk(X) ! X has a section, [G , Sv].* * It can also be shown that, for a cohomology class u 2 H*(X; R), wgt(u) = k if and only if k is the greatest integer such that p*k-1(u) = 0, [R3 , S]. We shall us* *e this below in giving a proof of Proposition 3.4 (4). 3.4. Proposition ([R3 , S]). Category weight has the following properties. (1) 1 wgt(u) cat(X), for all u 2 eH*(X; R), u 6= 0. (2) For every f :Y ! X and u 2 H*(X; R) with f*(u) 6= 0 we have cat(f) wgt(u) and wgt(f*(u)) wgt(u). (3) wgt(u [ v) wgt(u) + wgt(v). (4) For every u 2 Hs(K(ß, 1); R), u 6= 0, we have wgt(u) s. Proof.We will only prove (4) since the other results are proven in the referenc* *es cited. If X = K(ß, 1), then X has the homotopy type of a discrete set of points and, consequently, F1 = X * X is, up to homotopy, a wedge of circles. Also, G0(X) = P X ' *, so the cofibre of X ! G0(X) has the type of a wedge of circles. Then G1(X) has the homotopy type of a 1-dimensional space. Similarly, it is easy to see that Gk(X) has the homotopy type of a k-dimensional space. If u 2 Hs(K(ß, 1); R), then p*s-1(u) = 0 since Gs-1(X) is s-dimensional. By the equivalent definition of wgt given in Remark 3.3 (3), we see that wgt(u) s. 3.5. Definition. We say that u 2 H*(X; R) is a detecting element for X if wgt(u* *) = cat(X). We say that a space X is detectable if it possesses a detecting element. Recall that the cup-length of a space X with respect to a ring R is defined as fi clR(X) = max{k fiu1 [ . .[.uk 6= 0 for someu 2 eH*(X; R)}. 3.6. Lemma. If cat(X) = clR(X) for some ring R then the space X is detectable. Proof.It is well known that cat(X) clR(X) for every R. Now, let cat(X) = k and suppose that there are u1, . .,.uk 2 eH*(X; R) with u1[. .[.uk 6= 0. Then, usin* *g the first and third properties of Proposition 3.4, we conclude that wgt(u1[. .[.uk)* * = k. Thus, u1 [ . .[.uk is a detecting element for X. DETECTING ELEMENTS AND LUSTERNIK-SCHNIRELMANN CATEGORY OF 3-MANIFOLDS5 4.Basic Special Cases First, recall that cat(X) dim(X) for every connected CW -space X. In par- ticular, cat(M) 3 for every (connected) 3-manifold M. We also notice that, by Lemma 3.6, a space X is detectable whenever cat(X) = clR(X) for some R. Here is a first step in understanding the category of 3-manifolds. 4.1. Proposition. If M is a 3-manifold with finite fundamental group of order d > 1, then cat(M) = 3, and every non-zero element of H3(M; Z=d) is a detecting element for M. Moreover, if d is even, then every non-zero element of H3(M; Z=2) is a detecting element for M as well. Proof.Since ß1(M) is finite, ß2(M) = 0 because, by Proposition 2.6, the univers* *al cover is a homotopy sphere. Hence. there is the Hopf exact sequence ß3(M) --h--! H3(M) --q--! H3(ß) ! 0 where h is the Hurewicz homomorphism (e.g. see [Br, Theorem II.5.2]. Since, by Proposition 2.6, the d-fold universal covering fM ! M is a d-sheeted covering, M is orientable and fM is a homotopy sphere, we conclude that h has the form ß3(M) = Z ! Z = H3(M), a 7! d . a. Hence, H3(ß) = Z=d. Also consider the induced homomorphism Hom (H3(ß); Z=d) ! Hom (H3(M); Z=d). It is certainly injective since H3(M) ! H3(ß) is surjective. However, it is also true that, for any OE 2 Hom (H3(M); Z=d), Im(h) = dZ Ker(* *OE), so there exists ~OE2 Hom (H3(ß); Z=d) with ~OE7! OE. Thus, we have an isomorphi* *sm ~= Hom (H3(ß); Z=d) ! Hom (H3(M); Z=d). Now consider the diagram * H3(ß; Z=d) --q--! H3(M; Z=d) ?? ? y ?y * Hom (H3(ß); Z=d)--q--! Hom (H3(M); Z=d). By Proposition 3.4, (4), a non-zero element of H3(ß; Z=d) has category weight at least 3. The right arrow is an isomorphism because H2(M) is free abelian since M is orientable. The bottom arrow is an isomorphism by the argument above. Finally, the left arrow is a surjection by the Universal Coefficient For* *mula. Therefore, the top arrow is a surjection as well. In particular, by Proposition* * 3.4 (2), every non-zero element of H3(M; Z=d) has category weight at least 3. But cat(M) dim(M) = 3, so cat(M) = 3, and every non-zero element of H3(M; Z=d) is a detecting element for M. 4.2. Remark. Using the approach as in Proposition 4.1, it is also possible to p* *rove the following result originally due to Krasnoselski [Kra] and, in fact, re-prov* *ed in [GoGo ]: For a free action of the finite group G on a homotopy sphere S ' S2n+1, cat(S=G) = 2n + 1 = dim(S=G). Here is another basic result which follows from the characterization of prime non-irreducible 3-manifolds. 6 JOHN OPREA AND YULI RUDYAK 4.3. Proposition. Let M be a prime 3-manifold which is not irreducible. Then cat(M) = 2 = clZ=2(M), and M is detectable. Proof.In view of Lemma 2.3, M is the total space of a 2-sphere bundle over S1. So, M is either S1 x S2 or the mapping torus of the map r :S2 ! S2, r(x) = -x where S2 is regarded as the set of unit vectors in R3. It is easy to see that,* * in both of the cases, M = (S1 _ S2) [ e3 where e3 is a 3-cell attached to the wedge S1 _ S2. Thus, because a wedge of spheres has category one and a mapping cone can increase category by at most one, we obtain cat(M) 2. Futhermore, because ß1(M) = Z, we conclude that H1(M; Z=2) = Z=2. So, because of Poincar'e duality (with Z=2-coefficients), we have clZ=2(M) 2. Thu* *s, clZ=2(M) = 2 = cat(M), and M is detectable. The next two results treat the case of infinite fundamental group, excluding * *the S2-bundles over S1. 4.4. Proposition. If M is a 3-manifold with ß1(M) infinite and ß2(M) = 0, then cat(M) = 3 and M is detectable. Proof.By Proposition 2.6, M = K(ß1(M), 1), so, by Proposition 3.4, every non- zero element of H3(M; R) has category weight 3. (Notice that, for example, H3(M; Z=2) 6= 0). Thus, because cat(M) dim(M) = 3, each of these elements is a detecting element. 4.5. Proposition. If M is an irreducible 3-manifold with ß1(M) infinite and ß2(M) 6= 0, then cat(M) = 3 = clZ=2(M). In particular, M is detectable. Fur- thermore, M is non-orientable. Proof.Consider a map g :S2 ! M as in Theorem 2.5. Since M is irreducible, we conclude that g(S2) is a 2-sided projective plane in M. Let i: RP2 ! M be the corresponding embedding, and let [RP2] 2 H2(RP2; Z=2) denote the fundamental class modulo 2 of RP2. Let wk and __wkdenote the k-th Stiefel-Whitney class of M and RP2, respective* *ly. Since the 1-dimensional normal bundle of i is trivial, we conclude that i*wk = * *__wk. We can now compute the Kronecker products = = <__w2, [RP2]> = 1, and so i*[RP2] 6= 0 2 H2(M; Z=2). Now, since <__w21, [RP2]> = 1, we conclude that i*w21= __w216= 0, and so w216= 0. So, by Poincar'e duality, there exists * *x 2 H1(M; Z=2) with xw216= 0. Thus, clZ=2(M) = 3. We also need the following fact which, in a sense, is a converse of Lemma 2.3. 4.6. Corollary. If M is a closed 3-manifold with non-trivial free fundamental g* *roup, then M is not irreducible. Proof.Notice that ß2(M) 6= 0. Indeed, if ß2(M) = 0 then, by Proposition 2.6 and the hypothesis that ß1(M) is free, M = K(ß1(M), 1) = _S1. But this is wrong since a wedge of circles has vanishing homology above degree 1 for any coefficients. DETECTING ELEMENTS AND LUSTERNIK-SCHNIRELMANN CATEGORY OF 3-MANIFOLDS7 Now, if M is irreducible then, by Proposition 4.5, clZ=2(M) = 3. But this is impossible. Indeed, let f :M ! K(ß1(M), 1) = _S1 be a map which induces an isomorphism of fundamental groups. Then f* :H1(K(ß1(M), 1); Z=2) ! H1(M; Z=2) is an isomorphism. Thus, x[y = 0 for all x, y 2 H1(M; Z=2), and so clZ=2(M) < 3. This is a contradiction. 4.7. Remark. If ß1(M) = Z then M = P # where is a homotopy sphere and P is prime. So, ß1(P ) = Z. But P is not irreducible by Corollary 4.6, so, beca* *use of Lemma 2.3, ß2(P ) = Z. In other words, ß2(M) = Z whenever ß1(M) = Z. Actually, the following general fact holds: for every closed 3-manifold M, the * *group ß1(M) completely determines ß2(M), see e.g. [R1 ]. 5.Detectability of 3-Manifolds 5.1. Proposition. Let M3 be a closed 3-manifold with ß1(M) free and non-trivial. Then cat(M) = 2, and M is detectable. Proof.Write M = M1# . .#.Mk with each Mj prime. Because ß1(M) = ß1(M1)* . .*.ß1(Mk) is free, each ßj = ß1(Mj) must be free (where we agree that the tri* *vial group is free). If Mjis irreducible with ßj 6= {1}, then this contradicts Corol* *lary 4.6. Therefore, all such Mj are non-irreducible primes; that is, the Mj are the mani* *folds considered in Proposition 4.3. Because of Lemma 2.3, these are the total spaces of S2-bundles over S1. There are only two such manifolds: one orientable and one non-orientable, and we denote both of them by S1 / S2. Of course, the Mj with ßj = {1} are homotopy spheres j. The key point now is that, for M = P # Q with P = # k(S1 / S2) and Q = # j j, M - D3 deformation retracts onto the 2-skeleton _k(S1_S2). Because of Proposition 4.3, cat(S1 / S2) = 2. This handles the "trivial" case where the connected sum degenerates to a single summand. Now suppose M = # jMj = P # Q, where Mj is either a homotopy sphere or S1 / S2 and P = # jtMjt, Q = # jsMjsarbitrarily split M. If we remove a disk from a 3-manifold N, then the inclusion S2 ,! N - D3 is the inclusion of a subcomplex; so therefore a cofibration. Thus, the pushout diagram S2_________//P - D3 | | | | fflffl| fflffl| Q - D3 _____//P # Q = M is a homotopy pushout as well. But then we may apply the standard estimate for the category of a double mapping cylinder (see [Har]) to obtain cat(M) cat(S2) + max{cat(P - D3), cat(Q - D3)} = 1 + max{cat(_jt(S1 _ S2)), cat(_js(S1 _ S2))} = 1 + 1 = 2. Of course, cup-length then shows that cat(M) = 2 and this completes the proof. 8 JOHN OPREA AND YULI RUDYAK 5.2. Theorem. Let M be a 3-manifold whose fundamental group is non-trivial and not a free group. Then cat(M) = 3. Further, M is detectable unless it is non- orientable of the form P # Q, where P is non-orientable and Q is prime with odd torsion. Also, in the last case, the orientable double cover of M has category * *3. Proof.The case of finite ß1 is considered in Proposition 4.1. So, we assume that ß1(M) is infinite. We represent M as a connected sum M = N # P , where P is prime and ß1(P ) 6= {1}. Furthermore, we can always assume that ß1(P ) 6= Z, and therefore P is irreducible in view of Corollary 4.6. Now, because of the result* *s of x4, P posseses a detecting element u 2 H3(P ; R) for suitable R. Now suppose that M is orientable. Then there is a map f :M ! P of degree 1. (In greater detail, M = (N \ D [ (P \ D) where D is a 3-disk, and f :M ! P maps N \ D to the disk D in P and is the identity on P \ D.) Then f* :H3(P ; R) ! H3(M : R) is an isomorphism for every coefficient ring (group) R. Now, for the detecting element u above, f*(u) 6= 0, and, therefore, wgt(f*(u)) = 3. Thus, f** *(u) is a detecting element for M. ___ Now, if M is not orientable, then let M ! M be its orientable double cover * *___ (which also_is a closed 3-manifold). If ß1(M) has odd torsion,_then so does ß1(* *M_)._ Because M is orientable, the_argument_above says that cat(M ) = 3. But because M covers M, we know that cat(M ) cat(M). Therefore, cat(M) = 3. If, on the other hand, there is a prime component of M with non-free fundamental group having no odd torsion, then this component has a detecting element in 3-dimensional Z=* *2- cohomology. Therefore, M has a detecting element in Z=2-cohomology as well and cat(M) = 3. Now, if ß1(M) has odd torsion, then this occurs in individual prime component* *s. So, M may not have a detecting element only if we can write M = P # Q, where P is non-orientable and Q is a prime manifold having odd torsion. For completeness, note that cat( ) = 1 for every simply connected 3-manifold (= homotopy sphere) , and, therefore, every non-zero element u 2 H3( ) is a de- tecting element. Therefore, we now have proved Theorem 1.1 and augmented it by showing that most closed 3-manifolds possess detecting elements. The significan* *ce of this will be apparent in x6. 5.3. Remark. In fact, if we allow local coefficients, then all 3-manifolds with* * non- trivial and non-free fundamental groups have detecting elements. More specifica* *lly, by [Ber], cat(X) = n = dim(X) if and only if a certain element u 2 H1(X; I(ß)) * *has un 6= 0 in Hn(X; I(ß) . . .I(ß)). Here, ß = ß1(X) and I(ß) is the augmentation ideal in the group ring Zß. Since un is a cup product (with local coefficients)* *, it is a detecting element. 6.Two Applications A prime motivating problem in the study of Lusternik-Schnirelmann category has been the the Ganea conjecture; cat(X x Sn) = cat(X) + 1. We now know that the conjecture is not true in general, so it is even more interesting to unders* *tand when it is valid. For 3-manifolds, we have the following. 6.1. Corollary. For every closed 3-manifold M, cat(M x Sn) = cat(M) + 1. DETECTING ELEMENTS AND LUSTERNIK-SCHNIRELMANN CATEGORY OF 3-MANIFOLDS9 That is, the Ganea conjecture holds for M. Proof.First, suppose that M is detectable. Then the equality follows from the general result [R3 , Corollary 2.3], but the argument in this case is easy. Let* * u 2 H*(M; R) have wgt(u) = cat(M) and let v 2 Hn(Sn; R) be non-trivial, where, by the results above, we can always take R = Z or R = Z=d. Let ~u= p*M(u) and ~v= p*Sn(v), where pM :M x Sn ! M and pSn :M x Sn ! Sn are the respective projections. Clearly, ~u6= 0 and ~v6= 0 since the compositions pM M _incl//_M x Sn___//_M pSn Sn _incl//_M x Sn___//_Sn are the respective identity maps. By Proposition 3.4 (2), wgt(~u) wgt(u) = cat(M) and wgt(~v) wgt(v) = 1. Then the Künneth theorem says that 0 6= ~u[ ~v2 H*(M x Sn; R) and (using Proposition 3.4 (3) and the product inequality cat(X x Y ) cat(X) + cat(Y ))) cat(M) + 1 cat(M x Sn) wgt(~u[ ~v) wgt(~u) + wgt(~v) cat(M) + 1. Hence, cat(M x Sn) = cat(M) + 1. Now, suppose_that_M is not detectable. Then,_by Theorem 5.2, the oriented double cover M of_M_is detectable,_and_cat(M ) = 3. Therefore, in view of what * *we said above, cat(M xSn) = 4. But M xSn covers M xSn, and so cat(M xSn) 4. On the other hand, cat(M x Sn) cat(M) + 1 = 4 for general reasons. Thus, cat(M x Sn) = 4. 6.2. Corollary. Let f :M ! N be a degree 1 map of oriented 3-manifolds. Then catM catf = catN. Proof.Let u 2 H3(N; A) be a detecting element for N. (Recall that orientable 3-manifolds always have detecting elements.) Since deg(f) = 1, we conclude that f*(u) 6= 0. So, cat(f) wgt(u) by Proposition 3.4 (2). Thus cat(M) cat(f) wgt(u) = cat(N). Of course, cat(f) = cat(N) holds since cat(f) cat(N) for general reasons. 6.3. Corollary. Let f :M ! N be a degree 1 map of oriented 3-manifolds. If ß1(M) is free, then ß1(N) is. Proof.By Corollary 6.2, cat(N) 2, and so ß1(N) is free by Theorem 5.2 References [Ber] I. Berstein, On the Lusternik-Schnirelmann category of Grassmannians, Pr* *oc. Camb. Phil. Soc. 79 (1976) 129-134. [BG] I. Berstein and T. Ganea, The category of a map and of a cohomology clas* *s. Fund. Math. 50 (1961/1962) 265-279. [Br] K. Brown, Cohomology of groups. Graduate Texts in Mathematics 87, Spring* *er-Verlag, New York 1994. [FH] E. Fadell and S. Husseini, Category weight and Steenrod operations. Bol.* * Soc. Mat. Mexicana (2) 37 (1992) no. 1-2, 151-161. 10 JOHN OPREA AND YULI RUDYAK [Fe] A. I. Fet, A connection between the topological properties and the numbe* *r of extremals on a manifold (Russian), Doklady AN SSSR, 88 (1953) 415-417. [F] R. Fox, On the Lusternik-Schnirelmann category, Ann. of Math. 42 (1941) * *333-370. [G] T. Ganea Lusternik-Schnirelmann category and strong category Illinois J.* * Math. 11 (1967), 417-427 [GoGo] J.C. Gomez-Larranaga and F. Gonzalez-Acuna, Lusternik-Schnirelmann categ* *ory of 3- manifolds, Topology 31 (1992) 791-800. [Har] K.A. Hardie, On the category of the double mapping cylinder, T^ohoku Mat* *h. J. 25 (1973) 355-358. [H] J Hempel, 3-Manifolds. Ann. of Math. Studies 86, Princeton Univ. Press, * *Princeton, New Jersey 1976. [Ja] I. James, On category in the sense of Lusternik-Schnirelmann, Topology 1* *7 (1978) 331- 348. [Kra] M.A. Krasnosielski, On special coverings of a finite dimensional sphere * *(in Russian), Dokl. Akad. Nauk SSSR 103 (1955) 961-964. [R1] Yu. B. Rudyak, On the fundamental group of a three-dimensional manifold,* * Soviet Math. Doklady 14 (1973) 814-818. [R2] Yu. B. Rudyak, Category weight: new ideas concerning Lusternik-Schnirelm* *ann cate- gory. Homotopy and geometry (Warsaw, 1997), 47-61, Banach Center Publ., * *45, Polish Acad. Sci., Warsaw, 1998. [R3] Yu. B. Rudyak, On category weight and its applications. Topology 38 (199* *9) no. 1, 37-55. [S] J. Strom, Category weight and essential category weight, Thesis, Univ. o* *f Wisconsin 1997. [Sv] A. ~Svarc, The genus of a fiber space. Amer. Math. Soc. Translations 55 * *(1966), 49-140. Department of Mathematics, Cleveland State University, Cleveland Ohio 44115 U.S.A. E-mail address: oprea@math.csuohio.edu Department of Mathematics, University of Florida, 358 Little Hall, PO Box 118* *105, Gainesville, FL 32611-8105 U.S.A. E-mail address: rudyak@math.ufl.edu