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This thesis is concerned with the residues modulo 4 and 8 of the signature ofa 4k-dimensional oriented geometric Poincare complex. The Z_8-valuedBrown-Kervaire invariant of Z_4-valued quadratic forms is used to prove that ifthe signature is divisible by 4, the divisibility by 8 is detected by the Arfinvariant of a Z_2-valued quadratic form. This result is applied to a fibrationof geometric Poincare complexes with the total space 4k-dimensional. It isknown that the signature is multiplicative modulo 4, and that there areexamples of non-multiplicativity modulo 8. We identify the obstruction tomultiplicativity modulo 8 with the Arf invariant of the Z_2-valued quadraticform defined on an appropriate Z_2-cohomology vector space by dividing theZ_4-valued Pontryagin square by 2. The obstruction is then shown to vanishunder certain assumptions.

The signature modulo 8 of fibre bundles