## The signature of a fibre bundle is multiplicative mod 4.(English)Zbl 1136.55013

The purpose of this paper is to study the signature of a fibre bundle. The structure of this note is: Signed $$K$$-theory, Round $$L$$-theory, Absolute torsion and signatures, Absolute torsion structures on polyhedra (and on manifolds), Fibre transport on transfer, Absolute torsion, Structures of fibre bundles, Fibrations of PD space, Filtered chain complexes.
In the last part of the paper, the main result is an invariance theorem identifying the torsion of a contractible filtered chain complex with the torsion of the contractible chain complex of filtration quotients. All the fundamental concepts are clearly defined. The authors give a formula for the absolute torsion of the total space $$E$$ ($$E$$ is a PL fibred bundle) using absolute torsion of the base and fibre. Then the signature of $$E$$ is congruent mod 4 to the product of the signatures of the fibre and base. The main results are the following:
Theorem 1: Let $$F@>q>> E@> p>> B$$ be a PL fibre bundle of closed, connected, compatibly oriented PL manifolds. Then $$\text{sign}(E)\equiv \text{sign}(F)\cdot \text{sign}(B)\text{\,mod\,}4$$.
Theorem 2: Let $$F@>q>> E@>p>> B$$ be a PL fibre bundle of closed, connected, compatibly oriented PL manifolds. Then, if $$n= \dim E$$, $\tau^{\text{New}}(E)= p^!(\tau^{\text{New}}(B))+ \chi(B)q_*(\tau^{\text{New}}(F))\in \widehat H^n(\mathbb{Z}/2; K_1(\mathbb{Z}\pi_1(E)).$
In this statement the maps $$p^!$$ and $$q_*$$ are the transfer and push-forward maps associated to the fibre bundle. The exposition is generally easy to read. Some eamples are examined.

### MSC:

 55R25 Sphere bundles and vector bundles in algebraic topology
Full Text:

### References:

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