Spherical bundles adapted to a -fibration

Author:
J. P. E. Hodgson

Journal:
Trans. Amer. Math. Soc. **263** (1981), 355-361

MSC:
Primary 55R25; Secondary 57Q50

DOI:
https://doi.org/10.1090/S0002-9947-1981-0594413-8

MathSciNet review:
594413

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Abstract: A spherical fibration is said to be adapted to a -fibration if there is a fibration with fibre the quotient of a sphere by a free -action and such that the composition . In this paper it is shown that for spherical bundles in the PL, TOP or Homotopy categories that are adapted to - or -fibrations there is a procedure analogous to the splitting principle for vector bundles that enables one to define characteristic classes for these fibrations and to relate them to the usual characteristic classes. The methods are applied to show that a spherical fibration over a -connected base which is adapted to an -fibration admits a PL structure.

**[1]**E. Akin,*Transverse cellular maps of polyhedra*, Trans. Amer. Math. Soc.**169**(1972), 401-438. MR**0326745 (48:5088)****[2]**M. F. Atiyah,*-theory*, Benjamin, New York, 1967. MR**0224083 (36:7130)****[3]**A. Dold,*Relations between ordinary and extraordinary cohomology*, Proc. Aarhus Sympos., Aarhus Univ., Aarhus, 1962, pp. 2-9.**[4]**N. Levitt,*Poincaré duality cobordism*, Ann. of Math. (2)**96**(1972), 211-244. MR**0314059 (47:2611)****[5]**N. Levitt and J. Morgan,*Transversality structures and PL structures on spherical fibration*, Bull. Amer. Math. Soc.**78**(1972), 1064-1068. MR**0314050 (47:2602)****[6]**I. Madsen and R. J. Milgram,*The universal smooth surgery class*, Comment. Math. Helv.**50**(1975), 281-310. MR**0383404 (52:4285)****[7]**J. Milnor and J. Stasheff,*Characteristic classes*, Ann. of Math. Studies, no. 76, Princeton Univ. Press, Princeton, N. J., 1974. MR**0440554 (55:13428)****[8]**F. Quinn,*Surgery on Poincaré and normal spaces*, Bull. Amer. Math. Soc.**78**(1972), 262-267. MR**0296955 (45:6014)****[9]**D. Sullivan,*Geometric homotopy theory*. I.*Localization*, Periodicity and Galois Symmetry, M.I.T. Press, Cambridge, Mass., 1970.**[10]**-,*Geometric topology*, Lecture Notes, Princeton Univ., Princeton, N. J., 1967.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1981-0594413-8

Keywords:
Block bundle,
spherical fibration,
projectivisation,
characteristic classes,
transversality obstructions

Article copyright:
© Copyright 1981
American Mathematical Society