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Robinson, Christopher Alan
Languages: English
Types: Doctoral thesis
Subjects: QA
This thesis is an attempt to generalise the methods of two papers [9], [11] by James and Thomas on enumeration problems. \ud \ud The first five sections consider the problem of enumerating the homotopy classes of cross-sections 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 cross-sections of p. We obtain generalised 'difference classes' and 'obstruction classes' as elements of noo (p) and no -1 (p) respectively, and hence relate\ud these to the enumeration problem.\ud \ud §2 develops the theory of modified Postnikov towers (MPT's) for fibrations in which the fundamental group of the base acts non-trivially. For this we need techniques for handling k-invariants 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\ud 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 'de-looping' 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 cross-sections. 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].\ud \ud In §5 we compute MPT's for the fibrations BOn --> BO. These extend the calculations of Mahowald [13] for BSOn --> BSO to the non-orientable case and to the next two homotopy groups of the fibre. Instead of giving the defining relations of the k-invariants, 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 n-space Pn in Euclidean space in dimensions near the stable range.\ud \ud §6 considers the problem of determining the number of real k-plane 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\ud bundles, and enumerate (n-l)- and (n-2)- dimensional normal bundles to Pn in certain cases.
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    • 2.3 PropositiO.,!l. [(x,A),(KB(G,n),B)}--- ~(X,Aif*9). E -+ ~(G,n)
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