## Algebraic & Geometric Topology

### The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant

Carmen Rovi

#### Abstract

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle $F → E → B$ is multiplicative if the fundamental group $π 1 ( B )$ acts trivially on $H ∗ ( F ; ℝ )$, with $σ ( E ) = σ ( F ) σ ( B )$. Hambleton, Korzeniewski and Ranicki proved that in any case the signature is multiplicative modulo $4$, that is, $σ ( E ) = σ ( F ) σ ( B ) mod 4$. We present two results concerning the multiplicativity modulo $8$: firstly we identify $1 4 ( σ ( E ) − σ ( F ) σ ( B ) ) mod 2$ with a $ℤ 2$–valued Arf–Kervaire invariant of a Pontryagin squaring operation. Furthermore, we prove that if $F$ is $2 m$–dimensional and the action of $π 1 ( B )$ is trivial on $H m ( F , ℤ ) ∕ torsion ⊗ ℤ 4$, this Arf–Kervaire invariant takes value $0$ and hence the signature is multiplicative modulo $8$, that is, $σ ( E ) = σ ( F ) σ ( B ) mod 8$.

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 3 (2018), 1281-1322.

Dates
Revised: 13 August 2017
Accepted: 27 August 2017
First available in Project Euclid: 26 April 2018

https://projecteuclid.org/euclid.agt/1524708093

Digital Object Identifier
doi:10.2140/agt.2018.18.1281

Mathematical Reviews number (MathSciNet)
MR3784006

Zentralblatt MATH identifier
06866400

Subjects
Primary: 55R10: Fiber bundles 55R12: Transfer

#### Citation

Rovi, Carmen. The nonmultiplicativity of the signature modulo $8$ of a fibre bundle is an Arf–Kervaire invariant. Algebr. Geom. Topol. 18 (2018), no. 3, 1281--1322. doi:10.2140/agt.2018.18.1281. https://projecteuclid.org/euclid.agt/1524708093

#### References

• M F Atiyah, The signature of fibre-bundles, from “Global analysis: papers in honor of K Kodaira” (D C Spencer, S Iyanaga, editors), Univ. Tokyo Press (1969) 73–84
• M Banagl, A Ranicki, Generalized Arf invariants in algebraic $L$–theory, Adv. Math. 199 (2006) 542–668
• G E Bredon, Topology and geometry, Graduate Texts in Mathematics 139, Springer (1993)
• E H Brown, Jr, Generalizations of the Kervaire invariant, Ann. of Math. 95 (1972) 368–383
• R Brown, Elements of modern topology, McGraw-Hill, New York (1968)
• S S Chern, F Hirzebruch, J-P Serre, On the index of a fibered manifold, Proc. Amer. Math. Soc. 8 (1957) 587–596
• F Deloup, G Massuyeau, Quadratic functions on torsion groups, J. Pure Appl. Algebra 198 (2005) 105–121
• I Hambleton, A Korzeniewski, A Ranicki, The signature of a fibre bundle is multiplicative mod $4$, Geom. Topol. 11 (2007) 251–314
• A Hatcher, Algebraic topology, Cambridge Univ. Press (2002)
• A Korzeniewski, On the signature of fibre bundles and absolute Whitehead torsion, PhD thesis, University of Edinburgh (2005) \setbox0\makeatletter\@url http://www.maths.ed.ac.uk/~aar/papers/korzen.pdf {\unhbox0
• W Lück, The transfer maps induced in the algebraic $K_0$– and $K_1$–groups by a fibration, I, Math. Scand. 59 (1986) 93–121
• W Lück, A Ranicki, Surgery transfer, from “Algebraic topology and transformation groups” (T tom Dieck, editor), Lecture Notes in Math. 1361, Springer (1988) 167–246
• W Lück, A Ranicki, Surgery obstructions of fibre bundles, J. Pure Appl. Algebra 81 (1992) 139–189
• W Meyer, Die Signatur von Flächenbündeln, Math. Ann. 201 (1973) 239–264
• J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
• S Morita, On the Pontrjagin square and the signature, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 18 (1971) 405–414
• R E Mosher, M C Tangora, Cohomology operations and applications in homotopy theory, Harper & Row, New York (1968)
• L Pontrjagin, Characteristic cycles on manifolds, C. R. (Doklady) Acad. Sci. URSS 35 (1942) 34–37 In Russian
• A Ranicki, The algebraic theory of surgery, I: Foundations, Proc. London Math. Soc. 40 (1980) 87–192
• A A Ranicki, Algebraic $L$–theory and topological manifolds, Cambridge Tracts in Mathematics 102, Cambridge Univ. Press (1992)
• A Ranicki, L R Taylor, The mod $8$ signature of normal complexes, in preparation
• C Rovi, The signature modulo $8$ of fibre bundles, preprint (2015)
• R Schön, Fibrations over a CWh–base, Proc. Amer. Math. Soc. 62 (1976) 165–166
• E H Spanier, Algebraic topology, McGraw-Hill, New York (1966)
• J Stasheff, A classification theorem for fibre spaces, Topology 2 (1963) 239–246
• L R Taylor, Gauss sums in algebra and topology, preprint (2006) \setbox0\makeatletter\@url http://www.maths.ed.ac.uk/~aar/papers/taylorg.pdf {\unhbox0
• J H C Whitehead, On simply connected, $4$–dimensional polyhedra, Comment. Math. Helv. 22 (1949) 48–92
• J H C Whitehead, A certain exact sequence, Ann. of Math. 52 (1950) 51–110