THE WALL FINITENESS OBSTRUCTION FOR A FIBRATION

ERIK KJÆR PEDERSEN AND LAWRENCE R. TAYLOR

1. We wish to study the Wall finiteness obstruction for the total space of a fibration F → E → B. Such a study was first done by V. J. Lal [5], who neglected the action of π_{1}(B) on the fibre. This was noticed by D. R. Anderson [2], who produced formulae for the finiteness obstruction of E in case the fibration is a flat bundle [2, 1]. Our main theorem gives a partial calculation of the general case: If B and E are finitely dominated and H_{∗}(F) is a finitely generated Z-module, we compute the image of the Wall obstruction of E in K_{0}(Zπ_{1}(B)). In case π_{1}(E) → π_{1}(B) is monic, we may refine this to compute the actual obstruction for E in terms of information on F and B. In this case the assumption that E is finitely dominated may be replaced by F being the homotopy type of a finite complex.

We begin by establishing some terminology. Let X be a path connected space. We let S_{∗}(X) denote the singular chain complex of the universal cover of X. There is an action of π_{1}(X) on S_{∗}(X) making it a free Zπ_{1}X module. Given a ring Λ and a ring homomorphism Zπ_{1}X → Λ, we say that X is Λ-dominated iff S_{∗}(X) ⊗_{Zπ_{1}X} Λ is chain homotopy equivalent to a complex P_{∗} of Λ-modules, where each P_{i} is a finitely generated projective Λ-module and all P_{i} are zero except for finitely many.

If X is Λ-dominated, we can form σ(X; Λ) = ∑_{i=−∞}^{∞}(−1)^{i}[P_{i}] in K_{0}(Λ). Although many choices were necessary to form σ(X, Λ), it is easy to check that σ(X; Λ) is independent of all these choices. Also one sees that σ(X; Λ) is a homotopy invariant and natural with respect to homomorphisms Λ → Λ_{1}.

We say that X is finitely dominated if X is Zπ_{1}X-dominated (where Zπ_{1}X → Zπ_{1}X is the identity) and π_{1}X is finitely presented. In this case we write σ(X; Zπ_{1}X) = σ(X).

Wall [7, 8] has proved that a CW complex is dominated by a finite CW complex iff it is finitely dominated and that it has the homotopy type of a finite CW complex iff σ(X) ∈ K_{0}(Zπ_{1}X) vanishes in K̃_{0}(Zπ_{1}X). We call the image of σ(X) in K̃_{0}(Zπ_{1}X) the Wall finiteness obstruction for X.

We say that X is homologically finite if X is Z-dominated, where Zπ_{1}X → Z is the natural map. It is easy to see that this is the case iff ∑_{i=0}^{∞} H_{i}(X; Z) is a finitely generated abelian group.

To describe the action of the fundamental group on the homology of the fibre we need a functor G(π), π a group, and a pairing G(π) × K_{0}(Zπ_{1}X) → K_{0}(Zπ). We let G(π) be the Grothendieck group of the abelian category of Zπ-modules which are finitely generated abelian groups. We show that G(π) is a ring, and we define a pairing G(π) × K_{0}(Zπ) → K_{0}(Zπ) making K_{0}Zπ) into a module over G(π). We actually show that G(π) is isomorphic to Swan’s Grothendieck group of integral representations of π [6].

Partially supported by the Danish Research Council.

Partially supported by a N. S. F. grant.

Given a Serre fibration F → E ⟶^{p} B with B connected and F homologically finite, we can define χ(p) ∈ G(π) as ∑_{i=0}^{∞}(−1)^{i}[H_{i}(F; Z)]. χ(p) can be seen to be independent of a choice of basepoint for B. We have

Theorem 1.1. Let F → E → B be a Serre fibration with E and B connected. Suppose B is finitely dominated and F homologically finite. Then E is Zπ_{1}B-dominated (using the homomorphism Zπ_{1}E → Zπ_{1}B induced by p) and

σ(E; Zπ_{1}B) = χ(p) · σ(B).

We obtain a number of corollaries: Let π denote the image of π_{1}E in π_{1}B. Since H_{∗}(F) is homologically finite, π has finite index in π_{1}B, and hence the restriction map Res : K_{0}(Zπ_{1}B) → K_{0}(Zπ) is defined. π acts on F by taking each component to itself. Define χ_{0}(p) ∈ G(π) to be ∑_{i=0}^{∞}(−1)^{i}[H_{i}(F_{0}, Z)], where F_{0} denotes one component of F. We then have

Corollary 1.2. Hypothesis as above. Then E is Zπ-dominated and

σ(E; π) = χ(p) · Res σ(B).

Remarks. If π_{1}E → π_{1}B is injective, then E is finitely dominated and the corollary computes the Wall finiteness obstruction of E. If π_{1}B acts trivially on H_{∗}(F, Z) then χ(p) becomes the classical Euler characteristic of F and · denotes the usual Z-module structure on K_{0}.

As an example. let S^{k} → E_{k} → B be a spherical fibration with ω : π_{1}B → Z_{2} denoting the first Stiefel-Whitney class. Let Θ denote the kernel of ω and let F_{∗} : K_{0}(Zπ_{1}B) → K_{0}(Zπ_{1}B) be the result of restricting to Θ and then inducting up to Zπ_{1}B. Then

Corollary 1.3. Assume B is finitely dominated. Then:

(1) If ω is trivial, then

σ(E_{k}) = \begin{cases}0 & if k is odd, \\ 2σ(B) & if k is even. \end{cases}

(2) If ω is onto, then

σ(E_{k}) = \begin{cases}F_{∗}σ(B) & if k is even, \\ 2σ(B) − F_{∗}σ(B) & if k is odd. \end{cases}

(In case k = 0 or 1, σ(E_{k}) should be replaced by σ(E_{k}, Zπ_{1}B).)

Proof. Case 1 follows immediately from the remarks above. To see case 2, notice that the action of π_{1}B on the homology of the fibre factors through Z_{2} and χ(p) is the image of 1 + (−1)^{k}g under the natural map G(Z_{2}) → G(π_{1}B), where g represents Z with nontrivial action of Z_{2}. We denote this by Z^{t}. If P is a projective Z(π_{1}B)-module representing σ(B), we need to compute P ⊗ Z^{t}, where the action of π_{1}B on Z is through ω. However, in K_{0}(Z(π_{1}B)) we have

[P ⊗ Z^{t}] = [P ⊗ (Z^{t} + Z)] − [P]
= [P ⊗ (Z[Z_{2}])] − [P]
= F_{∗}[P] − [P],

and the result follows. □

This corollary gives us a general method for constructing examples of the type considered by Anderson [2].

Corollary 1.4. Let π be a finitely presented group, ω : π → Z_{2} a nontrivial map and Θ the kernel of ω. Assume there is an element α ∈ K_{0}(Zπ) such that α is in the kernel of the restriction map K_{0}(Zπ) → K_{0}(ZΘ) and such that 2α ≠ 0. By Wall [7] we may find a complex B with π_{1}B = π and σ(B) = α. Given k, it is easy to find a vector bundle of dimension k + 1 with first Stiefel-Whitney class given by ω. Let E_{k} denote the corresponding sphere-bundle; then

σ(E_{k}) \begin{cases}= 0, & k even, \\ ≠ 0, & k odd. \end{cases}

As an example we may take π = Z_{6} + Z_{3}. Since by Fröhlich [4], K̃_{0}(Z[Z_{3} ⊕ Z_{3}]) has order 3 while K̃_{0}(Z(Z_{6} ⊕ Z_{3})) has order 81, any element α in ker(Res) will satisfy the conditions of the corollary.

2. In this section we define algebraic functors G(π) and K′_{0}(Zπ) and relate these to well-known algebraic objects, namely G_{Z}(π), Swan’s Grothendieck group of integral representations, and K_{0}(Zπ).

We define G(π) to be the Grothendieck construction on Zπ-modules that are finitely generated abelian groups, i. e., as generators we take isomorphism classes of Zπ-modules that are finitely generated abelian groups, and if 0 → A → B → C → 0 is an exact sequence of such modules we have the relation [A] + [C] = [B]. Similarly we define K′_{0}(Zπ) to be the Grothendieck group of isomorphism classes of finitely generated Zπ-modules of homological dimension ≤ 1.

Lemma 2.1. Let T be a Zπ-module which is finite. Then there is an exact sequence of Zπ-modules 0 → F_{2} → F_{1} → T → 0 where F_{1} and F_{2} are finitely generated free abelian groups.

Proof. Since T is a finite group, Aut(T) is also finite. Hence π acts on T through a quotient group π′ which is finite. We can find a free Zπ′-module F_{1} and an epimorphism F_{1} → T. Let F_{2} be the kernel. Then F_{1} and F_{2} are finitely generated free abelian groups, and 0 → F_{2} → F_{1} → T → 0 is an exact sequence of Zπ′- and hence Zπ-modules. □

Lemma 2.2. The natural maps G_{Z}(π) → G(π) and K_{0}(Zπ) → K′_{0}(Zπ) are isomorphisms.

Proof. G_{Z}(π) is the Grothendieck group of Zπ-modules which are finitely generated free abelian groups. Let A represent an element of G(π). Since A is an abelian group, we have a short exact sequence 0 → T → A → F → 0 where T is torsion and F is free. This will be an exact sequence of Zπ-modules. By Lemma 2.1 there is an exact sequence 0 → F_{2} → F_{1} → T → 0, where F_{1} and F_{2} represent elements of G_{Z}(π). We thus have [A] = [F] + [F_{1}] − [F_{2}] in G(π) and we may define a map Σ : G(π) → G_{Z}(π) by

Σ[A] = [F] + [F_{1}] − [F_{2}].

We need only see that Σ is well defined, because by the above remarks, if I : G_{Z}(π) → G(π) denotes the natural map, it will be clear that IΣ = id and ΣI = id. As for well-definedness, it is enough to show that Σ[T] is well defined and to show that if 0 → A_{1} → A_{2} → A_{3} → 0 is exact, then

Σ[A_{2}] = Σ[A_{1}] + Σ[A_{2}]

Suppose we have 0 → H_{2} → H_{1} → T → 0, an exact sequence of Zπ-modules with H_{1} and H_{2} finitely generated free abelian groups. Let K be the pullback

\xymatrix{K \ar[r] \ar[d] & F_{1} \ar[d] \\ H_{1} \ar[r] & T}

Then 0 → H_{2} → K → F_{1} → 0 and 0 → F_{2} → K → H_{1} → 0 are exact, and K is a finitely generated free abelian group. Hence in G_{Z}(π), [H_{1}] − [H_{2}] = [F_{1}] − [F_{2}], so Σ[T] is well defined.

We prove our equation for 0 → A_{1} → A_{2} → A_{3} → 0 in several steps. First suppose that A_{1} and A_{2} are free abelian groups. Consider

[figure: 0; 0; 0; A_{1}; K; T; 0; 0; A_{1}; A_{2}; A_{3}; 0; F; F; 0; 0; 0]

where K is the kernel of A_{2} → F. Then K is a finitely generated free abelian group, so in G_{Z}(π) we have [A_{2}] = [K] + [F], so Σ[A_{3}] = [F] + Σ[T] = [F] + [K] − [A_{1}] = [A_{2}] − [A_{1}], since 0 → A_{1} → K → T → 0 is exact.

The next case to consider is the case that A_{1}, A_{2} and A_{3} are all torsion groups. As in the proof of Lemma 2.1 we may find a finite group π′ so that the action of π on each A_{i} goes through a map π → π′ → Aut(A_{i}). Then we can produce a diagram of Zπ′-modules:

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & K_{1} \ar[r] \ar[d] & K_{2} \ar[d] \ar[r] & K_{3} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & F_{1} \ar[r] \ar[d] & F_{2} \ar[d] \ar[r] & F_{3} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & A_{1} \ar[r] \ar[d] & A_{2} \ar[d] \ar[r] & A_{3} \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

where F_{i} and K_{i} are finitely generated free abelian groups that are all Zπ′- and hence Zπmodules. It is now easy to check that our equation holds in G_{Z}(π).

Now suppose A_{1} is a torsion group, and consider

[figure: 0; 0; 0; 0; A_{1}; T_{2}; T_{3}; 0; 0; A_{1}; A_{2}; A_{3}; 0; F_{2}; F_{3}; 0; 0; 0]

Our equation holds for all three columns and two of the rows, so our equation holds for the last row.

In the general case, consider

\xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & T_{1} \ar[d] \ar[r] & T_{2} \ar[d] \ar[r] & S \ar[d] \ar[r] & 0 \\ 0 \ar[r] & A_{1} \ar[d] \ar[r] & A_{2} \ar[d] \ar[r] & A_{3} \ar[d] \ar[r] & 0 \\ 0 \ar[r] & F_{1} \ar[d] \ar[r] & F_{2} \ar[d] \ar[r] & R \ar[d] \ar[r] & 0 \\  & 0 & 0 & 0}

Here F_{1} and F_{2} are free abelian groups, and S is just the quotient of T_{2} by T_{1} and hence is torsion. Our equation holds for all columns and two of the three rows, so it holds for the third row.

For the proof that K_{0}(Zπ) → K′_{0}(Zπ) is an isomorphism we refer the reader to Bass [3, p. 407].

□

Now Swan defines pairings G_{Z}(π) × G_{Z}(π) → G_{Z}(π) and G_{Z}(π) × K_{0}(π) → K_{0}(Zπ) by tensor product over the integers, the π-action in each case being just g(a ⊗ b) = ga ⊗ gb.

For all of this, see Bass [3, p. 563–565]. The tensor product over Z clearly defines a pairing G(π) × G(π) to G(π), so that I becomes a map of rings.

Proposition 2.3. The tensor product over Z defines a pairing G(π) × K_{0}(Zπ) → K′_{0}(Zπ).

Proof. Let P be a projective Zπ-module and A a Zπ-modules which is a finitely generated abelian group. We need to show that A ⊗_{Z} P is a Zπ-module of homological dimension ≤ 1. However, P, being a projective Zπ-module, is a free abelian group (not necessarily finitely generated), so tensoring over Z with P is exact. Given the short exact sequence of Zπ-modules 0 → T → A → F → 0, where T is a torsion and F a free abelian group, we have that 0 → T ⊗_{Z} P → A ⊗_{Z} P → F ⊗_{Z} P → 0 is exact and F ⊗_{Z} P is projective. To show h.d.(A ⊗_{Z} P) ≤ 1 it thus suffices to show h.d.(T ⊗_{Z} P) ≤ 1. But use Lemma 2.1 to find an exact sequence 0 → F_{2} → F_{1} → T with F_{i} Zπ-modules that are free abelian groups. Then 0 → F_{2} ⊗_{Z} P → F_{1} ⊗_{Z} P → T ⊗_{Z} P → 0 is exact and F_{i} ⊗_{Z} P is projective, so h.d.(T ⊗_{Z} P) ≤ 1. □

3. In this section we prove our main algebraic result. It says that complexes with “nice” homology must be Λ-dominated and that we can compute the Wall obstruction from the homology. A complex is Λ-dominated iff is is chain homotopy equivalent to a complex of finitely generated projective Λ-modules with only finitely many nonzero terms. Of course, X is Λ-dominated iff S_{∗}(X) ⊗_{Zπ_{1}X} Λ is Λ-dominated. A Λ-dominated complex C_{∗} has an invariant σ(C_{∗}) ∈ K_{0}(Λ) defined the usual way as an alternating sum, σ(S_{∗}(X) ⊗_{Zπ_{1}X} Λ) = σ(X, Λ).

Proposition 3.1. Let C_{∗} be a projective chain complex over Λ with C_{∗} = 0, ∗ < 0. Suppose that H_{∗}(C) is a finitely generated Λ-module of homological dimension ≤ 1 and H_{∗}(C) = 0 for ∗ sufficiently large. Then C_{∗} is Λ-dominated and

σ(C_{∗}) = Σ(−1)^{i}[H_{i}(C)] ∈ K′_{0}(Λ)

under the natural isomorphism K_{0}(Λ) ≅ K′_{0}(Λ).

Proof. Consider the two short exact sequences

0 → Z_{i} → C_{i} → B_{i} → 0,
0 → B_{i+1} → Z_{i} → H_{i} → 0.

C_{∗} projective just means that each C_{i} is projective (perhaps not finitely generated). By induction we can prove that each B_{i} and Z_{i} are projective, so C_{i} ≅ Z_{i} ⊕ B_{i}.

Since H_{i}(C) is finitely generated and of homological dimension ≤ 1, we can find finitely generated projective modules Q_{i+1} and P_{i} with 0 → Q_{i+1} → P_{i} → H_{i} → 0 exact. If H_{i} = 0, choose P_{i} = Q_{i+1} = 0. Let D_{i} = P_{i} ⊕ Q_{i+1}. It is easy to find a map D_{∗} → C_{∗} inducing an isomorphism in homology, and hence a chain homotopy equivalence.

Obviously C_{∗} is Λ-dominated. To show σ(C_{∗}) = Σ(−1)^{i}[H_{i}(C)] is now a standard argument. □

To get the most out of proposition 3.1 we must observe

Proposition 3.2. Let 0 → A_{∗} → B_{∗} → C_{∗} → 0 be a short exact sequence of projective chain complexes over Λ. If any two of these complexes are Λ-dominated, then so is the third, and σ(B_{∗}) = σ(A_{∗}) + σ(C_{∗}).

Proof. If A_{∗} and C_{∗} are Λ-dominated, we can splice together the complexes for A_{∗} and C_{∗} to get one for B_{∗}. The argument is elementary, and the equation follows.

If A_{∗} and B_{∗} are Λ-dominated, C_{∗} is equivalent to the algebraic mapping cone of A_{∗} → B_{∗}. Let D_{∗} be the mapping cone. Then 0 → B_{∗} → D_{∗} → A_{∗−1} → 0 is exact, so D_{∗} is Λdominated, and hence so is C_{∗}. Again the equation follows. The other case is similar. □

4.

Proof of Main Theorem. We get our principal geometric insight from a filtration that we put on the base space of our fibration. We then look at the induced filtration on the total space and compute using the results of Sections and .

Without loss of generality we may replace B by a homotopy equivalent CW-complex, and consider the fibration induced over this complex. Wall [7] shows, that since B is finitely dominated, B may be chosen so that there is a finite subcomplex K ⊂ B with H_{∗}(B̃, K̃; Z) = 0 except for one dimension, and there it is a finitely generated projective Zπ_{1}B = Λ-module. Here B̃ and K̃ denote the universal covers of B and K respectively. We let Ẽ denote the pullback of

\xymatrix{ & B̃ \ar[d] \\ E \ar[r] & B}

and p̃: Ẽ → B̃ and π : B̃ → B be the obvious maps.

The filtration on B is defined as follows: If K has dimension n − 1, define B_{n} = B; B_{n−1} = K; B_{n−2} = (n − 2)-skeleton of K; …; B_{0} = 0-skeleton of K; B_{−1} = empty set.

Notice that H_{∗}(B̃_{r}, B̃_{r−1}, Z) is always 0 except in one dimension and that there is a finitely generated projective Λ-module. Here B̃_{r} = π^{−1}(B_{r}).

Consider the Serre spectral sequence of (E_{r}, E_{r−1}) where Ẽ_{r} = P̃^{−1}(B̃_{r}). The two-term E_{p,q}^{2} is just H_{p}(B̃_{r}, B̃_{r−1}, H_{q}(F, Z)), but H_{p}(B̃_{r}, B̃_{r−1}, Z) is 0 or free as an abelian group, so E_{p,q}^{2} = H_{p}(B̃_{r}, B̃_{r−1}) ⊗ H_{q}(F, Z). There is an action of π_{1}(B) on the Serre spectral sequence, and the action of E^{2} is just the diagonal action on the tensor product, where π_{1}(B) acts on H_{q}(F, Z) as usual. Since H_{p}(B̃_{r}, B̃_{r−1}) = 0 except for one value of p, the Serre sequence collapses and we have

H_{r+q}(Ẽ_{r}, Ẽ_{r−1}) ≅ H_{r}(B̃_{r}, B̃_{r−1}) ⊗ H_{q}(F).

Proposition 2.3 shows that H_{r} + q(Ẽ_{r}, Ẽ_{r−1}) is a finitely generated Λ-module of homological dimension ≤ 1. Proposition 3.1 shows that the chain complex for the pair (Ẽ_{r}, Ẽ_{r−1}) is Λ-dominated and σ(Ẽ_{r}, Ẽ_{r−1}) = χ(p) · (−1)^{r}[H_{r}(B̃_{r}, B̃_{r−1})].

An easy induction argument using proposition 3.2 shows that each Ẽ_{r} is Λ-dominated and that σ(Ẽ_{r}) = χ(p) · σ(B_{r}). Since it is a standard fact that the singular chains of Ẽ is isomorphic to S_{∗}(E) ⊗_{Zπ_{1}E} Zπ_{1}B, we have proved out main theorem:

σ(E, Λ) = χ(p) · σ(B).
□

Proof of Corollary. Let \overline{B} be the cover of B corresponding to the subgroup π ⊂ π_{1}B, where π is the image of π in π_{1}B. Then we have a fibration F_{0} → E → \overline{B}. Since B is finitely dominated and \overline{B} is a finite cover, \overline{B} is finitely dominated. Applying the theorem yield σ(E) = χ(p) · σ(\overline{B}) and σ(\overline{B}) = Res σ(B), as is easy to see directly from the definitions. □

We finally notice that if π_{1}(B) acts trivially on H_{∗}(F), multiplication with χ(p) becomes the usual multiplication with χ(F), since Z with trivial action is the unit of G(π_{1}B) and K_{0}(Zπ_{1}B) is a unital module.

References

[1] D. R. Anderson, Generalized product theorems for torsion invariants with applications to flat bundles, Bull. Amer. Math. Soc. (N.S.) 78 (1972), 465–469.

[2] , The obstruction to the finiteness of the total space of a flat bundle, Amer. J. Math. 95 (1973), 281–293.

[3] H. Bass, Algebraic K-theory, Benjamin, 1968.

[4] A. Fröhlich, On the classgroup of integral grouprings of finite abelian groups, Mathematica 16 (1969), 143–152.

[5] V. J. Lal, The wall obstruction of a fibration, Invent. Math. 6 (1968), 67–77.

[6] R. Swan, Induced representations and projective modules, Ann. of Math. (2) 71 (1960), 267–291.

[7] C. T. C. Wall, Finiteness conditions for CW -complexes, Ann. of Math. (2) 81 (1965), 56–69.

[8] , Finiteness conditions for CW -complexes II, Proc. Roy. Soc. London Ser. A 295 (1966), 129–139.

Odense, Denmark

Notre Dame University, USA