Generalized Steenrod-Hopf invariants for stable homotopy theory

Author:
Warren M. Krueger

Journal:
Proc. Amer. Math. Soc. **39** (1973), 609-615

MSC:
Primary 55H15

MathSciNet review:
0385860

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Abstract: In his paper *On the groups* . IV, Adams suggested that one might try to continue his and invariants to a sequence of higher homotopy invariants, each defined upon the vanishing of its predecessors and each taking its value in a certain Ext group. Recently he pointed out the efficacy of relocating his and invariants in Ext groups formed over a certain abelian category of comodules. It is the purpose of this note to carry out the program suggested above in a setting of the sort just mentioned. More specifically, for each homology theory which is representable by a comutative ring spectrum and whose ring of cooperations is flat over the coefficient ring, a sequence of higher homotopy invariants is constructed whose first term is Adams' invariant for this theory.

**[1]**J. F. Adams,*On the groups 𝐽(𝑋). IV*, Topology**5**(1966), 21–71. MR**0198470****[2]**-,*Lectures on generalized cohomology*, Category Theory, Homology Theory and their Applications, III (Battelle Inst. Conf., Seattle, Wash., 1968), Springer, New York, 1969, pp. 1-138. MR**40**#4943.**[3]**J. F. Adams, A. S. Harris, and R. M. Switzer,*Hopf algebras of cooperations for real and complex 𝐾-theory*, Proc. London Math. Soc. (3)**23**(1971), 385–408. MR**0293617****[4]**Samuel Eilenberg and John C. Moore,*Homology and fibrations. I. Coalgebras, cotensor product and its derived functors*, Comment. Math. Helv.**40**(1966), 199–236. MR**0203730****[5]**Franklin P. Peterson,*Functional cohomology operations*, Trans. Amer. Math. Soc.**86**(1957), 197–211. MR**0105679**, 10.1090/S0002-9947-1957-0105679-9

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DOI:
https://doi.org/10.1090/S0002-9939-1973-0385860-9

Keywords:
Stable homotopy theory,
spectrum,
coalgebra,
comodule,
homology operation,
exact couple,
canonical injective resolution,
extended comodule,
unitary spectrum

Article copyright:
© Copyright 1973
American Mathematical Society