K-theory for spherical space forms

On smooth manifolds with the homotopy type of a homology sphere

“…Hence, f * is a surjection on the infinity page by the naturality of the AHSS. It follows that f * : K(X) → K(S m ) is surjective (for the case when X is a spherical space form, this follows from [23], Theorem 1-(b)). Hence, we have the following remark: Remark 3.9.…”

“…By an argument as in [34] Theorem 7.26, the diagram commutes up to homotopy. It is well-known that ϕ induces a surjection on K (see for example [23]). Thereby, inner automorphisms of π induce identity on K(X) as well.…”

“…In general, if S is the set of primes between p and 2p which divide the order of π, then the image of Ψ is K(X) (S,q 1 ) . We refer to [23] for the computation of K(X) (for X = Σ/π as above) and the results given in [35] (and although indirectly, in [36]) for the computation of Aut(X). Of course we only need these computations when π 1 (X) is of odd order.…”