SMOOTHING H-SPACES

ERIK KJÆR PEDERSEN

Consider a partition of all primes Π = P_{1} ∪ ⋯ P_{k}, and let G_{i} be quasi-finite complexes such that (G_{i})_{P_{i}}, G_{i} localized at P_{i} are H-spaces, that are rationally equivalent as H-spaces. The homotopy pullback

\xymatrix{(G_{1})_{P_{1}} \ar[r] & (G_{1})_{0} \ar[d]^{≅} \\ (G_{2})_{P_{2}} \ar[r] & (G_{2})_{0} \ar[d]^{≅} \\  & ⋮ \ar[d]^{≅} \\ (G_{k})_{P_{k}} \ar[r] & (G_{k})_{0}}

is well known to be a quasifinite H-space [1]. We will say that X is obtained by Zabrodsky mixing of G_{1}, ⋯, G_{k} at P_{1}, ⋯, P_{k}. Note that X depends on the rational equivalences chosen.

We consider H-spaces obtained by mixing the following types of spaces:

At the prime 2: products of Lie groups and S^{7} and RP^{7}.

At odd primes: products of

(a) Lie groups, S^{7}, RP^{7} and

(b) any stably reducible quasi-finite P. D. space with no 3-dimensional generator of the rational exterior algebra cohomology and finitely presented fundamental group, and

(c) principal S^{3}-bundles with basespace a stably reducible quasi-finite P. D. space We prove the following theorems:

Theorem. Let X be as above, 1-connected; then X is of the homotopy type of a parallellizable differentiable manifold.

We notice that by choosing all G_{i} equal, but choosing different rational equivalences we get the following:

Corollary. Let Y be in the genus of a simply connected Lie group G (i. e. Y_{p} ≅ G_{p} for all primes p); then Y is homotopy equivalent to a parallellizable manifold.

In the nonsimply connected context we prove the following:

Theorem. Let X be a as above; then X is of the homotopy type of a finite complex, and in the genus of X is a parallellizable differentiable manifold.

Indication of proof. The case where X_{(2)} ≅ (RP^{3})^{k} × (S^{7})^{l} × (RP^{7})^{M} requires special arguments. In all other cases we proceed by constructing a fibration S^{1} → X → Y, where Y is a stably reducible P. D. space. The map p induces isomorphisms on fundamental groups, and the fibration is orientable. We then use [2] to compute the Wall finiteness obstruction for X. The formula in this case says σ(X) = χ((S^{1}) · σ(Y), χ the euler characteristic; hence σ(X) = 0, so X is homotopy equivalent to a finite CW complex.

We then consider X as a P. D. boundary of the corresponding D^{2}-fibration D^{2} → E → Y, and notice that the classifying map E → BG reduces to BO since Y is stably reducible and S^{1}-fibrations are equivalent to O(2)-bundles. This allows us to set up a surgery problem

(M.∂M) → (E, Y)

and we proceed to show the surgery problem ∂M → Y has obstruction 0. The reduction of E is trivial when restricted to Y; hence Y is of the homotopy type of a parallellizable differentiable manifold.

References

[1] P. Hilton, G. Mislin, and J. Roitberg, Localization of Nilpotent Groups ands Spaces, North Holland Mathematics studies, vol. 15, North - Holland Publishing co., Amsterdam - New York, 1975.

[2] E. K. Pedersen and L. Taylor, The Wall finiteness obstruction for a fibration, Amer. J. Math. 100 (1978), 887–896.

Odense University