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arXiv:math/0209119v6 [math.GT] 18 Feb 2005

THE FIBERED ISOMORPHISM

CONJECTURE FOR COMPLEX MANIFOLDS

S. K. Roushon

School of Mathematics

Tata Institute

Homi Bhabha Road

Mumbai 400 005, India.

email address:- roushon@math.tifr.res.in

Abstract. In this paper we show that the Fibered Isomorphism Conjecture of

Farrell and Jones corresponding to the stable topological pseudoisotopy functor is

true for the fundamental groups of a class of complex manifolds. A consequence

of this result is that the Whitehead group, reduced projective class groups and the

negative K-groups of the fundamental groups of these manifolds vanish whenever

the fundamental group is torsion free. We also prove the same results for a class

of real manifolds including a large class of 3-manifolds which has a finite sheeted

cover fibering over the circle.

Keywords and phrases: Complex projective variety, complex surfaces, Whitehead

group, fibered isomorphism conjecture, negative K-groups.

2000 Mathematics Subject Classification: Primary: 57N37, 19J10, 14J99. Sec-

ondary: 19D35.

Abbreviated title: Isomorphism conjecture for complex manifolds

February 18, 2005

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2 S. K. ROUSHON

0. Introduction

In this paper we consider proving the Farrell-Jones Fibered Isomorphism Con-

jecture (FIC) corresponding to the stable topological pseudoisotopy functor for

the fundamental groups of complex manifolds. This conjecture is already proved

for several classes of groups including fundamental groups of closed nonpositively

curved Riemannian manifolds ([6]), for cocompact discrete subgroup of virtually

connected Lie groups ([6]) and for the class of virtually strongly poly-free groups

([9]). A rich class of complex manifolds are smooth complex projective algebraic

varieties. Some results are known on the structure of the fundamental groups of a

large class of complex surfaces. We make use of these informations for the proofs

of our results. The method of the proofs also generalizes to consider some real

manifolds and to prove the FIC for the fundamental groups.

The main input to the proofs of Theorem 1.3 and 1.5 comes from theorem 7.1

of Farrell and Linnell in [7] where they proved that if the Fibered Isomorphism

Conjecture is true for all the groups in a directed system of groups then it is true

for the direct limit also. The proof of our Main Lemma uses this result to show

that the FIC is true for a large class of mapping torus of the fundamental group of

a closed orientable real 2-manifold and also for a certain class of mapping tori of

infinitely generated free groups. The key idea to prove the Main Lemma was that

except for two closed surfaces the covering space corresponding to the commutator

subgroup of the fundamental group of all other surfaces have one topological end.

Though the spaces involved in the theorems have finitely presented fundamental

groups, during the proof we encounter some infinitely generated groups and these

are the places where we use the Main Lemma crucially.

Throughout the paper whenever we encounter a 3-manifold which fiber over the

circle with fiber a surface of genus ≥ 2 we make the assumption that the monodromy

diffeomorphism is special (see Definition in Section 1). In [27] we have proved that

we can remove this assumption of being ‘special’ provided the FIC is true for A-

groups (see [27] for definition). Roughly speaking a torsion free A-group is a discrete

group which is isomorphic to the fundamental group of a complete nonpositively

curved Riemannian manifold whose metric is A-regular. In a recent paper ([16])

L.E. Jones proved that the assembly map in the statement of the FIC induces a

surjective homomorphism on the homotopy group level for any torsion free A-group.

He also stated some conjectured theorems ([[16], conjectured theorems 6.7 and 6.9])

which imply that the ‘torsion free’ assumption can be dropped from the statement

of the result. On the other hand the FIC states that the above homomorphism

should be an isomorphism.

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THE FIBERED ISOMORPHISM CONJECTURE FOR COMPLEX MANIFOLDS3

In Section 1 using Lefschetz hyperplane section theorem we deduce that if the

FIC is true for the fundamental group of smooth complex algebraic surfaces then it

is true for the fundamental group of any smooth complex algebraic variety. Also we

state our results in this section. Section 2 recalls the Farrell and Jones Fibered Iso-

morphism Conjecture and states the known results we need. In Section 3 we make

a brief trip to classification of complex surfaces and their topological properties.

Sections 4 and 5 contain the proofs of the results stated in Section 1 and we prove

the Main Lemma in Section 6. In Section 7 we prove the FIC for a class of virtually

fibered 3-manifolds. Section 8 contains examples of special diffeomorphisms.

1. Reduction to surface case and statements of results

Let X be a smooth complex projective algebraic variety of dimension n. By

definition X ⊂ CPmfor some m and is a complex submanifold of CPm. It is well-

known that not all complex manifold is a complex submanifold of some complex

projective space; otherwise it will become algebraic. In Section 3 we will mention

such examples in the case of surfaces.

There is a natural collection of complex submanifolds of X arising from taking

intersection of X with hyperplanes H in CPm. For a general hyperplane H the

intersection of H with X is a connected complex submanifold of X ([[2], chapter

I, corollary 20.3]). Let H0be such a hyperplane. Then the two manifolds X and

X∩H0shares similar homological and homotopical properties up to a certain degree.

This is the content of the Lefschetz hyperplane section theorem ([[2], chapter I,

theorem 20.4]).

Lefschetz hyperplane section theorem. For n ≥ 2 the inclusion map X∩H0⊂

X induces the following isomorphisms.

Hi(X ∩ H0,Z) → Hi(X,Z)

πi(X ∩ H0,Z) → πi(X,Z)

for 0 ≤ i ≤ n − 2.

Let Gn be the class of fundamental groups of all smooth complex projective

algebraic varieties of dimension n. Then successively applying Lefschetz hyperplane

section theorem we get the following Lemma.

Lemma 1.1. ∪∞

n=2Gn⊂ G2.

Thus from Lemma 1.1 we see that if we want to prove the FIC for the fundamental

group of smooth projective algebraic varieties then it is enough to consider the

smooth projective algebraic surfaces.

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4 S. K. ROUSHON

The rest of this section contains the statements of the results. Before that we

recall some standard definition from algebraic geometry. By a complex surface we

mean a closed complex 2-manifold. By an algebraic complex surface we mean it is a

complex surface and is defined by finitely many homogeneous polynomials in n+1

variable in the complex projective space CPn. By a curve we will mean a complex

projective algebraic variety of dimension 1. κ(X) denotes the Kodaira dimension

of X. For definition of κ(X) see [[2], p. 23] or [[10], definition 1.6]. When X is a

complex surface κ(X) ∈ {−∞,0,1,2}.

Theorem 1.2. Let X be a complex surface of one of the following types.

(1) X is algebraic and κ(X) = −∞

(2) X is a Hopf surface

(3) κ(X) = 0

(4) X is an Inoue surface

(5) X is an elliptic surface

Then the Fibered Isomorphism Conjecture is true for π1(X).

The theorem below deals with some more complex manifolds.

theorem we need to make some definition. At first recall that it follows from a

result of Hillman [[12], theorem 7] that if a complex surface fibers over the circle

then the fiber is a Seifert fibered space (see Theorem 4.1) . Hence if X is such a

surface then π1(X) ≃ π1(S) ⋊ Z where S is a Seifert fibered space. Assume that

the monodromy diffeomorphism is a fiber preserving diffeomorphism of the Seifert

fibered space S. If π1(S) is infinite then there is an infinite cyclic normal subgroup

of π1(S) with quotient πorb

1 (B) where B is the base orbifold of S ([[11], chapter 12]).

Again if πorb

1 (B) is infinite then one can find a finite index characteristic subgroup

K of πorb

1 (B) which is isomorphic to a closed surface group (see Section 4). Note

that in this situation the monodromy diffeomorphism of the fiber bundle X → S1

induces an automorphism of πorb

1 (B). Since K is characteristic we have an exact

sequence

1 → K → πorb

To state the

1 (B) ⋊ Z → (πorb

1 (B)/K) ⋊ Z → 1

Let l be an element of (πorb

subgroup of finite index. Since K is a closed surface group the action of l (by

conjugation by a lift of l) on K is induced, up to conjugation, by a diffeomorphism

fl of a closed surface F so that π1(F) is isomorphic to K. Let us call fl a base

diffeomorphism associated to the infinite cyclic normal subgroup generated by l.

1 (B)/G)⋊Z which generates an infinite cyclic normal

Theorem 1.3. Let X be a complex surface which is the total space of a fiber bundle

over the circle S1. Under the above notations when F is not the 2-torus assume

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THE FIBERED ISOMORPHISM CONJECTURE FOR COMPLEX MANIFOLDS5

that there is a base diffeomorphism flwhich is special (see Definition below). Then

the FIC is true for π1(X).

In Section 4 we will also prove that the FIC is true for a class of complex surfaces

of Kodaira dimension 2. A large class of examples of surfaces of Kodaira dimension

2 are ramified 2-sheeted covering of CP2ramified along a curve. We will prove the

FIC for a class of such surfaces and will give an example to show that, given the

methods available, this is the best possible result we can prove.

Complex surfaces with Kodaira dimension 1 are elliptic surfaces (Theorem 3.3.1).

There is a notion of elliptic surfaces in the smooth category called C∞-elliptic

surface. These are smooth real 4-manifold which are locally modelled on complex

elliptic surfaces. We recall the definition of C∞-elliptic surface in Section 3. The

fundamental groups of these 4-manifolds have close properties with that of complex

elliptic surfaces. We prove in Corollary 1.4 that the FIC is true for a class of these

manifolds also.

Corollary 1.4. The FIC is true for the fundamental group of a C∞-elliptic surface

X if X has no singular fiber and with cyclic monodromy.

There is another natural collection of smooth 4-manifolds which are fiber bundles

over real 2-manifolds with real 2-manifolds as fiber. In Theorem 1.5 below we prove

that the FIC is true for the fundamental group of a large class of manifolds from

this collection. A large class of complex surfaces belong to this collection where

both the fiber and base are 2-manifolds of genus ≥ 2. In this particular case the

fiber bundle projection is called Kodaira fibration (see [[2], chapter V, section 14]

for details). To state our next results we need the following definition.

Definition. Let F be a closed orientable surface and f is an orientation preserving

diffeomorphism of F. Let˜F be the covering of F corresponding to the commutator

subgroup of π1(F) and let˜f be the lift of f to˜F →˜F. Let p : M˜ f→ Mf be the

covering projection from the mapping torus of˜f to that of f.

We say f is a special diffeomorphism if one of the following holds.

(1) the mapping torus of f supports a nonpositively curved Riemannian metric.

(2) some power of f is isotopic to identity.

(3) the fundamental group of any component of p−1(S) is not free, where S

varies over all Seifert fibered pieces in the Jaco-Shalen and Johannson (JSJ)

decomposition of the mapping torus M(f).

We now recall JSJ-decomposition of a 3-manifold briefly. A 3-manifold is called

irreducible if any embedded 2-sphere in the manifold bounds an embedded 3-disc.