The goal of this work is the construction of the analogue to the Adams spectral sequence in cobordism theory, calculation of the ring of cohomology operations in this theory, and also a number of applications: to the problem of computing homotopy groups and the classical Adams spectral sequence, fixed points of transformations of period p, and others.

We give tools to compare different maps from algebraic K-theory. The results apply in particular to group morphisms from K-theory to any suitable cohomology theory or to K-theory. In particular, we obtain a characterization of the Chern character and of the Adams operations on higher K-theory.

The Virasoro operations in Witten’s theory of two-dimensional topological gravity have a homotopy-theoretic interpretation as endomorphisms of an ordinary cohomology theory with coefficients in a localization of I. Schur’s ring ∆ of Q-functions. The resulting theory has many of the features of a vertex operator algebra.

We describe a vanishing result on the cohomology of a cochain complex associated to the moduli of chains of finite subgroup schemes on elliptic curves. These results have applications to algebraic topology, in particular to the study of power operations for Morava E-theory at height 2.

§0 Introduction. The Morava K-theories K(n)∗( ) (for p a prime and 0 < n < ∞) form a collection of multiplicative cohomology theories whose central rôle in homotopy theory is now well established. However, even though they have been intensively studied there are still many aspects of their structure which remain undeveloped. In this paper we give a construction for families of operations remini...

We prove the overholonomicity of overconvergent F-isocrystals over smooth varieties. This implies that the notions of overholonomicity and devissability in overconvergent F-isocrystals are equivalent. Then the overholonomicity is stable under tensor products. So, the overholonomicity gives a p-adic cohomology stable under Grothendieck’s cohomological operations.

The indicated operation algebra is studied by methods dual to the usual ones for studying the Steenrod algebra. In particular, the operations are constructed using higher symmetries of the shuffle map and their "Adem relations" are computed using the transfer map in the cohomology of symmetric groups.

Algebraic topology is a young subject, and its foundations are not yet firmly in place. I shall give some history, examples, and modern developments in that part of the subject called stable algebraic topology, or stable homotopy theory. This is by far the most calculationally accessible part of algebraic topology, although it is also the least intuitively grounded in visualizable geometric obj...

Steenrod defined in 1947 the squares on mod 2 cohomology of spaces using explicit cochain formulae for cup-$i$ products; a family coherent homotopies derived from broken symmetry Alexander--Whitney's chain approximation to diagonal. He later his homonymous operations all primes homology symmetric groups. This approach enhanced conceptual understanding and allowed many advances, but lacked concr...