Title:

On the enumeration of crosssections and unstable vector bundles

This thesis is an attempt to generalise the methods of two papers [9], [11] by James and Thomas on enumeration problems. The first five sections consider the problem of enumerating the homotopy classes of crosssections of a fibration F c X p B. In §l, we define certain groups no*(p) by stabilizing (in a suitable sense) the homotopy groups of the space of crosssections of p. We obtain generalised 'difference classes' and 'obstruction classes' as elements of noo (p) and no 1 (p) respectively, and hence relate these to the enumeration problem. §2 develops the theory of modified Postnikov towers (MPT's) for fibrations in which the fundamental group of the base acts nontrivially. For this we need techniques for handling kinvariants with local coefficients, but the theory in other respects parallels that of Mahowald [13] and Thomas [23]. An existence theorem is proved in 2.10. §3 introduces the notion of 'stable modified Postnikov tower', which is a crude device to facilitate the setting up of the spectral sequence in the following section. A stable MPT for a fibration kills the stable homotopy groups of the fibre, just as an MPT kills the unstable ones. The existence of stable towers follows from the possibility of 'delooping' MPT's in the stable range (3.3). In §4 we set up a spectral sequence (4.19) which, in the simplest case, has the form E~,t ~ Ht(B;rrEsF) => rr: (p) where F is the s'th stable homotopy group of F. This is our main tool for calculating rr: (p) , and for enumerating crosssections. An essentially similar approach to the enumeration problem has been used (independently) by J.F. lTcClendon [12]. Our spectral sequence appears to be a formalisation of the last section of [12]. In §5 we compute MPT's for the fibrations BOn > BO. These extend the calculations of Mahowald [13] for BSOn > BSO to the nonorientable case and to the next two homotopy groups of the fibre. Instead of giving the defining relations of the kinvariants, we display the corresponding differentials in our spectral sequence: these contain equivalent information. We apply the results of the computations to determine the number of regular homotopy classes of immersions of real projective nspace Pn in Euclidean space in dimensions near the stable range. §6 considers the problem of determining the number of real kplane bundles over a complex X which are stably equivalent to a given bundle. The methods here are developments of those of James and Thomas [9]. We generalise one of the theorems of [9] to nonorientable bundles, and enumerate (nl) and (n2) dimensional normal bundles to Pn in certain cases.
