math.GT — Geometric Topology

Integral tail signatures and the Hilbert--Smith conjecture

Contributed by Alexander Dabrowski

The obstruction to an effective $p$-adic manifold action is an integer signature tail. On asymptotically controlled rational modules over one fixed free $C_p$-space, finite-cover Mayer--Vietoris makes the difference between $p$ copies and quotient-permutation tensoring nilpotent. Every degree-four class therefore has eventually $p$-divisible signatures. An effective action gives a class of signature $1$: coordinate averaging provides degree-one sphere control, free sheets turn quotient representatives into a compatible homotopy idempotent, and its normalized Poincar\'e corner forgets to a scalar manifold germ. Successive localization boundaries recover one copy of $\CP^2$, up to sign. The resulting contradiction proves the Hilbert--Smith conjecture.