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Cochain operations and higher cohomology operations. (English) Zbl 1003.55005

At the minimum higher-order operations can be defined in terms the following data: a principal fibration \(F@>j>>E@>p>>B\) with principal action \(\mu:F\times E\to E\) and a map \(\varphi:E\to D\). Then for any space \(X\) we have \(Sp(X)\): the set of homotopy classes of maps \(X\to B\) such that \(\varepsilon \simeq p\circ\overline \varepsilon\) for some \(\overline\varepsilon :X\to E\) and \(T\varphi X=\text{im}(\varphi_\#: [X,E]\to [X,D])/R\) where \(R\) is the equivalence relation induced by \(\mu\). A higher-order operation is a natural transformation \(Sp()\to T\varphi()\). This generality does not embrace a useful theory. In practice, universal examples for operations are built inductively through towers of fibrations. The purpose of this paper is to describe unstable higher-order cohomology operations at the level of normalized simplicial cochains.

MSC:

55S20 Secondary and higher cohomology operations in algebraic topology
55U15 Chain complexes in algebraic topology
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References:

[1] F. Adams , On the non-existence of elements of Hopf invariant one , Ann. of Math. 72 , 20 - 104 ( 1960 ) MR 141119 | Zbl 0096.17404 · Zbl 0096.17404 · doi:10.2307/1970147
[2] S. Klaus , Brown-Kervaire invariants , Dissertation Univ. Mainz ( 1995 ), Shaker Verlag ( 1995 ), Aachen [3] S. Klaus , The Ochanine k-invariant is a Brown-Kervaire invariant , Topology 36 , 257 - 270 ( 1997 ) MR 1410474 | Zbl 0864.57025 · Zbl 0864.57025 · doi:10.1016/0040-9383(95)00063-1
[3] S. Klaus , On simplicial loops and H-spaces Preprint Mathematisches Forschungsinstitut Oberwolfach ( 1999 ), to appear in Topology and its Applications MR 1824167 | Zbl 0989.55005 · Zbl 0989.55005 · doi:10.1016/S0166-8641(00)00005-5
[4] S. Klaus , Towers and pyramids I , Preprint Mathematisches Forschungsinstitut Oberwolfach ( 1999 ), to appear in Forum Mathematicum MR 1858494 | Zbl 0982.55003 · Zbl 0982.55003 · doi:10.1515/form.2001.028
[5] S. Klaus , A category of higher cohomology operations and finite spectra , Preprint Mathematisches Forschungsinstitut Oberwolfach ( 2000 ) [7] S. Klaus , Cochain operations and subspace arrangements , Preprint Mathematisches Forschungsinstitut Oberwolfach ( 2000 ) MR 1961552
[6] L. Kristensen , On secondary cohomology operations , Math. Scand. 12 , 57 - 82 ( 1963 ) MR 159333 | Zbl 0118.18303 · Zbl 0118.18303
[7] L. Kristensen , On a Cartan formula for secondary cohomology operations , Math. Scand. 16 , 97 - 115 ( 1965 ) MR 196740 | Zbl 0151.31203 · Zbl 0151.31203
[8] L. Kristensen , A secondary product structure in cohomology theory , Math. Scand. 17 , 113 - 149 ( 1965 ) MR 202135 | Zbl 0148.43202 · Zbl 0148.43202
[9] L. Kristensen , Massey products in Steenrod’s algebra , Proc. Advanced Study Inst. on Algebraic Topology , Aarhus ( 1970 ), Vol. II , 240 - 255 MR 279807 | Zbl 0246.55012 · Zbl 0246.55012
[10] L. Kristensen , I. Madsen , On evaluation of higher order cohomology operations , Math. Scand. 20 , 114 - 130 ( 1967 ) MR 222886 | Zbl 0166.18902 · Zbl 0166.18902
[11] L. Kristensen , I. Madsen , On the structure of the operation algebra for certain cohomology theories , Conf. on Algebraic Topology, University of Illinois at Chicgo Circle, Proceedings ( 1968 ), 134 - 160 MR 250301 | Zbl 0212.28102 · Zbl 0212.28102
[12] A. Kock , L. Kristensen , I. Madsen , Cochain functors for general cohomology theories I, II , Math. Scand. 20 , 131 - 176 ( 1967 ) MR 214056 | Zbl 0166.18903 · Zbl 0166.18903
[13] J.P. May , Simplicial objects in algebraic topology, Chicago Lectures in Mathematics , Univ. of Chicago Press ( 1992 ), Chicago MR 1206474 | Zbl 0769.55001 · Zbl 0769.55001
[14] L. Smith , Secondary cohomology theories , Indiana Univ. Math. J. 23 , 899 - 923 ( 1974 ) MR 339135 | Zbl 0285.55005 · Zbl 0285.55005 · doi:10.1512/iumj.1974.23.23074
[15] E. Spanier , Secondary operations on mappings and cohomology , Ann. of Math. 75 , 260 - 282 ( 1962 ) MR 133823 | Zbl 0105.17003 · Zbl 0105.17003 · doi:10.2307/1970174
[16] N.E. Steenrod , D.B.A. Epstein , Cohomology operations , Annals of Math. Studies 50 , Princeton Univ. Press ( 1962 ), Princeton MR 145525 | Zbl 0102.38104 · Zbl 0102.38104
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