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
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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
*