The signature modulo 8 of fibre bundles

Abstract

This thesis is concerned with the residues modulo 4 and 8 of the signature of a 4k-dimensional oriented geometric Poincare complex. The Z_8-valued Brown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that if the signature is divisible by 4, the divisibility by 8 is detected by the Arf invariant of a Z_2-valued quadratic form. This result is applied to a fibration of geometric Poincare complexes with the total space 4k-dimensional. It is known that the signature is multiplicative modulo 4, and that there are examples of non-multiplicativity modulo 8. We identify the obstruction to multiplicativity modulo 8 with the Arf invariant of the Z_2-valued quadratic form defined on an appropriate Z_2-cohomology vector space by dividing the Z_4-valued Pontryagin square by 2. The obstruction is then shown to vanish under certain assumptions.