The principal fibration sequence and the second cohomotopy set. (English) Zbl 1270.55012
Kirby, Rob (ed.) et al., Proceedings of the Freedman Fest. Based on the conference on low-dimensional manifolds and high-dimensional categories, Berkeley, CA, USA, June 6–10, 2011 and the Freedman symposium, Santa Barbara, CA, USA, April 15–17, 2011 dedicated to Mike Freedman on the occasion of his 60th birthday. Coventry: Geometry & Topology Publications. Geometry and Topology Monographs 18, 235-251 (2012).
Let \(p:E\to B\) be a principal fibre space classified by \(w:B\to C\). \([X,\Omega C]\) acts on \([X,E]\), and there is an exact sequence containing \([X,E]\). This has been studied by F. P. Peterson [Trans. Am. Math. Soc. 86, 197–211 (1957; Zbl 0212.55902)] and Y. Nomura [Nagoya Math. J. 17, 111–145 (1960; Zbl 0125.11403)], and in the case where \(B\) and \(C\) are \(H\)-spaces, J. W. Rutter [Topology 6, 379–403 (1967; Zbl 0152.21804)].
The present paper is a very interesting re-visit and expansion of these ideas. The author relates these ideas to the space of lifts Lift\(^f_{_*}(X\to LY)\), where \(f:X\to Y\) is a continuous map and \(LY\) is the free loop space on \(Y\), as well as the path groupoid thought of as a torsor. As applications, he studies the cohomotopy group \([X,S^2]\), where dim\((X)\leq 4\), extending the earlier work of L. Pontrjagin (when dim\(X=3\)). The seminal work of N. E. Steenrod [Ann. Math. (2) 48, 290–320 (1947; Zbl 0030.41602)] analyzes \([X,S^n]\) when dim\(X = n+1\).
Except in the lowest dimensions, this is an abelian group which sits in a short exact sequence. Using the ideas of L. L. Larmore and E. Thomas [Math. Scand. 30, 227–248 (1972; Zbl 0254.55009)], the author gives a complete analysis of this extension, which contains \([X,S^n]\). The reviewer would like to dream that there is a functorial description of the extension or even better a functor from \([X,S^n]\) to abelilan groups (since the determination of the extension, with \([X,S^n]\) in the middle, actually over-determines the abelian groups \([X,S^n]\)).
For the entire collection see [Zbl 1253.00022].
The present paper is a very interesting re-visit and expansion of these ideas. The author relates these ideas to the space of lifts Lift\(^f_{_*}(X\to LY)\), where \(f:X\to Y\) is a continuous map and \(LY\) is the free loop space on \(Y\), as well as the path groupoid thought of as a torsor. As applications, he studies the cohomotopy group \([X,S^2]\), where dim\((X)\leq 4\), extending the earlier work of L. Pontrjagin (when dim\(X=3\)). The seminal work of N. E. Steenrod [Ann. Math. (2) 48, 290–320 (1947; Zbl 0030.41602)] analyzes \([X,S^n]\) when dim\(X = n+1\).
Except in the lowest dimensions, this is an abelian group which sits in a short exact sequence. Using the ideas of L. L. Larmore and E. Thomas [Math. Scand. 30, 227–248 (1972; Zbl 0254.55009)], the author gives a complete analysis of this extension, which contains \([X,S^n]\). The reviewer would like to dream that there is a functorial description of the extension or even better a functor from \([X,S^n]\) to abelilan groups (since the determination of the extension, with \([X,S^n]\) in the middle, actually over-determines the abelian groups \([X,S^n]\)).
For the entire collection see [Zbl 1253.00022].
Reviewer: Donald W. Kahn (Minneapolis)
