Existence of finitely dominated CW -complexes with G_{1}(X) = π_{1}(X) and non-vanishing finiteness obstruction by Wolfgang Lück and Andreas Müller

Abstract We show for a finite abelian group G and any element in the image of the Swan homomorphism sw : ℤ/|G|^{∗} ⟶ K̃_{0}(ℤG) that it can be realized as the finiteness obstruction of a finitely dominated connected CW -complex X with fundamental group π_{1}(X) = G such that π_{1}(X) is equal to the subgroup G_{1}(X) defined by Gottlieb. This is motivated by the observation that any H-space X satisfies π_{1}(X) = G_{1}(X) and still the problem is open whether any finitely dominated H-space is up to homotopy finite.

key words: finiteness obstruction, Swan homomorphisms, Gottlieb’s subgroup of the fundamental group. AMS-classification number: 57Q12

The purpose of this note is to prove

Theorem 1 Let G be a finite abelian group and η be any element in the image of the Swan homomorphism sw : ℤ/|G|^{∗} ⟶ K̃_{0}(ℤG). Then there is a finitely dominated connected CW -complex X with the following properties:

1. G = π_{1}(X);

2. Gottlieb’s subgroup G_{1}(X) ⊂ π_{1}(X) is equal to π_{1}(X);

3. Wall’s finiteness obstruction õ(X) is η.

Recall that a space X satisfies π_{1}(X) = G_{1}(X) if and only if π_{1}(X) is abelian and for each w ∈ π_{1}(X) the associated deck transformation on the universal covering l_{w} : X̃ ⟶ X̃, which is a π_{1}(X)-equivariant map, is π_{1}(X)- homotopic to the identity [2]. The finiteness obstruction of a finitely dominated CW -complex was introduced by Wall [10]. A survey about the finiteness obstruction and nilpotent and simple spaces is given in [7]. The Swan homomorphism sw : ℤ/|G|^{∗} ⟶ K̃_{0}(ℤG) sends \overline{r} ∈ ℤ/|G|^{∗} to the class [ℤ/r] of the ℤG-module given by the cyclic group ℤ/r with the trivial G-action for any representative r ∈ ℤ of \overline{r}. This ℤG-module has a finite projective ℤGresolution P_{∗} and for any such P_{∗} the class [ℤ/r] is given by ∑_{p≥0}(−1)^{p} · [P_{n}] in K̃_{0}(ℤG) [8]. Computations of the image of the Swan homomorphism can be found for instance in [9].

The motivation for the study of the possible finiteness obstructions of finitely dominated CW -complexes X with G_{1}(X) = π_{1}(X) comes from the to the author’s knowledge still unsettled question whether a finitely dominated H-space is always up to homotopy finite. The point is that any H-space X satisfies G_{1}(X) = π_{1}(X). Mislin has shown that a finitely dominated H-space is finite up to homotopy if its fundamental group is of square free order [5, Theorem II on page 375]. We mention that a nilpotent space is finitely dominated if and only if H_{i}(X; ℤ) is finitely generated for all i and zero for sufficiently large i and that each H-space is nilpotent [4, Theorem A]. A space with G_{1}(X) = π_{1}(X) is nilpotent, but the converse is not true.

If π_{1}(X) is infinite, any finitely dominated CW -complex X with G_{1}(X) = π_{1}(X) is homotopy equivalent to a space of the form Z × S^{1} and hence its finiteness obstruction vanishes by the product formula [3, Prop. 4.3 on page 153]. So it suffices to consider finitely dominated CW -complexes X with G_{1}(X) = π_{1}(X) such that π_{1}(X) is finite.

Theorem 1 has been proven in [6, Theorem 2.4, page 203] if one substitutes the second condition in Theorem 1 by requiring that X is simple. So Mislin gives an example for X where for each element g ∈ G = π_{1}(X) the covering translation l_{g} : X̃ ⟶ X̃ is homotopic to the identity where we require that it is G-equivariantly homotopic to the identity.

Now we prove Theorem 1. The construction of the space X is a variation of the one in [6, Theorem 2.2, page 201]. Fix an integer n ≥ 3. We claim that there is a connected finite CW -complex A such that G = π_{1}(A) = G_{1}(A) and Ã is n + 2-connected. Since G is a product of cyclic groups, it suffices to treat the case where G is cyclic. Let G operate on ℂ^{n+1} by multiplication with a primitive |G|-th root of unity. It induces a free G-action on the unit sphere S^{2n+1}. Take A = S^{2n+1}/G. For the given η in the image of the Swan homomorphism we can choose an odd natural number r such that sw(\overline{r}) = η. Precisely as in [6, Theorem 2.2, page 201] we can attach n and n + 1-cells to A to obtain a connected finitely dominated CW -complex X with η as finiteness obstruction such that H_{k}(X̃, Ã) is zero for k ≠ n and, for k = n, is ℤG-isomorphic to ℤ/r with the trivial G-action. It remains to prove for g ∈ G that l_{g} : X̃ ⟶ X̃ is G-homotopic to the identity. Notice that there is already a G-homotopy h : Ã × I ⟶ Ã between l_{g} : Ã ⟶ Ã and the identity. Hence we have to extend (l_{g} ∐ id) ∪ h : X̃ × ∂I ∪ Ã ⟶ X̃ to a G-map H : X̃ × I ⟶ X̃.

We use the equivariant obstruction theory as developed in [1, II.3]. Because of the obstruction sequence [1, Theorem 3.10 on page 115] and [1, Theorem 3.17 on page 120] it suffices to show

1. The primary obstruction γ((l_{g} ∐ id) ∪ h) ∈ H_{G}^{n+1}((X̃, Ã) × (I, ∂I); π_{n}(X̃)) vanishes;

2. H_{G}^{k+1}((X̃, Ã) × (I, ∂I); π_{k}(X̃)) is trivial for k ≥ n + 1.

We will identify H_{G}^{k+1}((X̃, Ã) × (I, ∂I); π_{k}(X̃)) and H_{G}^{k}(X̃, Ã); π_{k}(X̃)) by the suspension isomorphism in the sequel. Recall that H_{G}^{n}(X̃, Ã; π_{n}(X̃)) is defined by the cohomology of the cochain complex hom_{ℤG}(C_{∗}(X̃, Ã), π_{n}(X̃)). Since Ã is n + 2-connected and X obtained from A by attaching cells of dimension greater or equal to n, we get an isomorphism

H_{G}^{n}(X̃, Ã; π_{n}(X̃)) ⟶^{≅} hom_{ℤG}(H_{n}(X̃, Ã), π_{n}(X̃))
⟶^{≅} hom_{ℤG}(H_{n}(X̃, Ã), π_{n}(X̃, Ã))
⟶^{≅} hom_{ℤG}(H_{n}(X̃, Ã), H_{n}(X̃, Ã)).

One easily checks that the primary obstruction γ((l_{g} ∐ id) ∪ h) is sent under this isomorphism to H_{n}(l_{g}) − id (cf. [1, 3.18 and 3.19 on page 121]). Since G acts trivially on H_{n}(X̃, Ã) this difference and hence the primary obstruction vanish.

Since X̃ is obtained from Ã by attaching cells in dimensions n and n + 1, it remains to prove that H_{G}^{n+1}(X̃, Ã; π_{n+1}(X̃)) vanishes. Since H_{k}(X̃, Ã) vanishes or is isomorphic to ℤ/r with odd r for all k ≥ 0, it suffices to prove that π_{n+1}(X̃) is a finite abelian 2-group. Denote by X̃_{n} the n-skeleton of the relative CW -complex (X̃, Ã). Consider the following part of the long exact sequence of a triple

… ⟶ π_{n+1}(X̃_{n}, Ã) ⟶ π_{n+1}(X̃, Ã) ⟶ π_{n+1}(X̃, X̃_{n}) ⟶^{Δ} π_{n}(X̃_{n}, Ã) ⟶ …

By the Hurewicz we can identify Δ with the n + 1-differential in the cellular ℤG-chain complex of (X̃, Ã). Since H_{n+1}(X̃, Ã) vanishes, Δ is injective. Hence it suffices to show that π_{n+1}(X̃_{n}, Ã) is a finite abelian 2-group because π_{n+1}(X̃) ≅ π_{n+1}(X̃, Ã) is a quotient of π_{n+1}(X̃_{n}, Ã). Since Ã is n + 2connected, π_{n+1}(X̃_{n}, Ã) is isomorphic to π_{n+1}(X̃_{n}/Ã). As X̃_{n}/Ã is a wedge of copies of S^{n}-s and n ≥ 3 we conclude from the Freudenthal Suspension Theorem, that π_{n+1}(X̃/Ã) is isomorphic to a direct sum of copies of the stable homotopy group π_{1}^{s} which is ℤ/2. This finishes the proof of Theorem 1.

References

[1] tom Dieck, T.: “Transformation groups”, Studies in Math. 8, de Gruyter (1987)

[2] Gottlieb, D.: “A certain subgroup of the fundamental group”, Amer. J. of Math. 87, 840-856 (1965)

[3] Lück, W.: “The transfer maps induced in the algebraic K_{0}- and K_{1}- groups by a fibration II”, J. of Pure and Applied Algebra 45, 143 - 169 (1987)

[4] Mislin, G.: “Finitely dominated nilpotent spaces”, Ann. of Math. 103, 547 - 556 (1976)

[5] Mislin, G.: “Groups with cyclic subgroups and finiteness conditions for certain complexes”, Comm. Math. Helv. 52, 373 - 391 (1977)

[6] Mislin, G.: “The geometric realization of Wall obstructions by nilpotent and simple spaces”, Math. Proc. Camb. Phil. Soc. 87, 199 - 206 (1980)

[7] Mislin, G.: “The geometric realization of Wall obstructions by nilpotent and simple spaces”, in “Handbook of algebra”, editor: I.M. James, 1259 - 1291, Elsevier (1995)

[8] Swan, R.G.: “Periodic resolutions for finite groups”, Ann. of Math. 72, 267 - 291 (1960)

[9] Taylor, M. J.: “The locally free class group of prime power order”, J. of Algebra 50, 463 - 487 (1978)

[10] Wall, C.T.C.: “Finiteness conditions for CW -complexes”, Ann. of Math. 81, 59 - 69 (1965)

Address

Wolfgang Lück Andreas Müller Fachbereich für Mathematik und Informatik Fachbereich Mathematik Westfälische Wilhelms-Universität Johannes Gutenberg-Universität Münster Einsteinstr. 62 Staudingerweg 9 48149 Münster 55099 Mainz Germany Germany email: lueck@math.uni-muenster.de internet: http://wwwmath.uni-muenster.de/math/u/lueck/ Fax: 0251 83837

Version of June 11, 1997