## The sectional category of spherical fibrations

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- by Don Stanley
- Proc. Amer. Math. Soc.
**128**(2000), 3137-3143 - DOI: https://doi.org/10.1090/S0002-9939-00-05468-X
- Published electronically: April 28, 2000
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## Abstract:

We give homological conditions which determine sectional category, secat, for rational spherical fibrations. In the odd dimensional case the secat is the least power of the Euler class which is trivial. In the even dimensional case secat is one when a certain homology class in twice the dimension of the sphere is $-1$ times a square. Otherwise secat is two. We apply our results to construct a fibration $p$ such that $\mathrm {secat}(p)=2$ and genus$(p)=\infty$. We also observe that secat, unlike cat, can decrease in a field extension of $\mathbb {Q}$.## References

- M. G. Barratt, V. K. A. M. Gugenheim, and J. C. Moore,
*On semisimplicial fibre-bundles*, Amer. J. Math.**81**(1959), 639–657. MR**111028**, DOI 10.2307/2372920 - Jean-Paul Doeraene,
*L.S.-category in a model category*, J. Pure Appl. Algebra**84**(1993), no. 3, 215–261. MR**1201256**, DOI 10.1016/0022-4049(93)90001-A - Yves Félix and Stephen Halperin,
*Rational LS category and its applications*, Trans. Amer. Math. Soc.**273**(1982), no. 1, 1–38. MR**664027**, DOI 10.1090/S0002-9947-1982-0664027-0 - J.-B. Gatsinzi,
*LS-category of classifying spaces. II*, Bull. Belg. Math. Soc. Simon Stevin**3**(1996), no. 2, 243–248. MR**1389618** - S. Halperin,
*Lectures on minimal models*, Mém. Soc. Math. France (N.S.)**9-10**(1983), 261. MR**736299** - Kathryn P. Hess,
*A proof of Ganea’s conjecture for rational spaces*, Topology**30**(1991), no. 2, 205–214. MR**1098914**, DOI 10.1016/0040-9383(91)90006-P - I. M. James,
*On category, in the sense of Lusternik-Schnirelmann*, Topology**17**(1978), no. 4, 331–348. MR**516214**, DOI 10.1016/0040-9383(78)90002-2 - M. A. Krasnosel’skii,
*Topological methods in the theory of nonlinear integral equations*, A Pergamon Press Book, The Macmillan Company, New York, 1964. Translated by A. H. Armstrong; translation edited by J. Burlak. MR**0159197** - John Milnor,
*On spaces having the homotopy type of a $\textrm {CW}$-complex*, Trans. Amer. Math. Soc.**90**(1959), 272–280. MR**100267**, DOI 10.1090/S0002-9947-1959-0100267-4 - Dennis Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math.**47**(1977), 269–331 (1978). MR**646078** - A. S. Švarc,
*The genus of a fiber space*, Dokl. Akad. Nauk SSSR (N.S.)**119**(1958), 219–222 (Russian). MR**0102812** - Daniel Tanré,
*Homotopie rationnelle: modèles de Chen, Quillen, Sullivan*, Lecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983 (French). MR**764769**, DOI 10.1007/BFb0071482 - George W. Whitehead,
*Elements of homotopy theory*, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR**516508** - Tadasi Nakayama,
*On Frobeniusean algebras. I*, Ann. of Math. (2)**40**(1939), 611–633. MR**16**, DOI 10.2307/1968946

## Bibliographic Information

**Don Stanley**- Affiliation: II Mathematisches Institut, Freie Univerität Berlin, Arnimallee 3, D-14195 Berlin, Germany
- Address at time of publication: Max-Plank-Institut für Mathematik, Vivatsgasse 7, D-53111 Bonn, Germany
- MR Author ID: 648490
- Email: stanley@math.fu-berlin.de, stanley@mpim-bonn.mpg.de
- Received by editor(s): December 10, 1998
- Published electronically: April 28, 2000
- Additional Notes: This work was supported by DFG grant Sche 328/2-1
- Communicated by: Ralph L. Cohen
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**128**(2000), 3137-3143 - MSC (1991): Primary 55R25, 55P62; Secondary 55M30
- DOI: https://doi.org/10.1090/S0002-9939-00-05468-X
- MathSciNet review: 1691006

Dedicated: This paper is dedicated to my son Russell