Celebratio Mathematica

incollection L. L. Larmore and E. Thomas: “Group extensions and principal fibrations” in Proceedings of the advanced study institute on algebraic topology (Aarhus, Denmark, 10–13 August 1970), vol. 2. Various publications series 13. Aarhus Universitet Matematisk Enstitut, 1971. Also published as Math. Scand. 30 (1972).

L. L. Larmore and E. Thomas: “Group extensions and principal fibrations,” Math. Scand. 30 (1972), pp. 227–248. Also appeared in Proceedings of the advanced study institute on algebraic topology (1970). MR 0328935 Zbl 0254.55009 article

We consider in this paper the following extension problem. Suppose we have a principal fibration \[ \Omega C \stackrel{i}{\longrightarrow} E \stackrel{\pi}{\longrightarrow}B, \] with classifying map \( \theta:B\to C \). One then has an extended sequence of fibrations \[ \cdots \to \Omega^{m+1}C \stackrel{{}^{m}i}{\longrightarrow} \Omega^m E\stackrel{{}^{m}\pi}{\longrightarrow} \Omega^m B \to \cdots \to B \stackrel{\theta}{\longrightarrow} C. \] (For any map \( f \) we write \( ^mf = \Omega^m f \), \( m\geq 1 \).) Given a space \( X \) a standard problem in topology is to compute the group \( [X,\Omega^m E] \). We have the exact sequence \[ e: 0 \to (\operatorname{Coker}^{m+1}\theta_*)\to [X,\Omega^m E] \to (\operatorname{Ker}^m\theta_*) \to 0, \] and our problem is: Compute the extension \( e \).

L. L. Larmore and E. Thomas: “Mappings into loop spaces and central group extensions,” Math. Z. 128 : 4 (1972), pp. 277–296. MR 0317318 Zbl 0254.55010 article

L. L. Larmore and E. Thomas: “Group extensions and twisted cohomology theories,” Ill. J. Math. 17 : 3 (1973), pp. 397–410. MR 0336744 Zbl 0274.55022 article

In this paper we continue the study of group extensions initiated in [Larmore and Thomas 1972]. The specific problem discussed there was the computation of extensions in the exact sequence of groups obtained by mapping a space into a principal fibration sequence. Here we consider the same problem, but in a different category — the category of spaces “over and under” a fixed space (see [McClendon 1969; Becker 1968]). This means in particular that the solution to the extension problem is given in terms of “twisted” cohomology operations [McClendon 1969], whereas in [Larmore and Thomas 1972] only ordinary cohomology operations were needed.

In §1 we discuss the category we will use. In §2 we state our extension problem, and in §§3–4 we give a general solution. Finally in §§5–6 we give applications of our theory — in §5 we compute the (affine) group of immersions of an \( n \) manifold in \( R^{2n-1} \), while in §6 we compute the (affine) group of vector 1-fields on a manifold.

L. L. Larmore and E. Thomas: “On the fundamental group of a space of sections,” Math. Scand. 47 : 2 (1980), pp. 232–246. MR 612697 Zbl 0463.55010 article

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P. Emery Thomas

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