Equivariant Homotopy Theory

Some thirty years ago, D. Quillen introduced his higher algebraic K-groups. This construction was the culmination of attempts by many different mathematicians to arrive at a definition which generalized the known constructions in dimensions 0, 1, and 2, and which had good formal properties extending those already known in low dimensions. These groups are of interest for at least three reasons.

  • A complete understanding of the groups would permit the computation of the cohomology of arithmetic groups and more generally the automorphism groups of free modules over rings, at least in a stable range.
  • The complete understanding of the algebraic K-theory of group rings of infinite discrete groups would allow the resolution of two fundamental conjectures in high dimensional geometric topology, namely the conjectures of Novikov and Borel.
  • It is anticipated that the higher K-groups will contain number theoretic and algebraic geometric information.

    Quillen's groups have excellent formal properties, which allow a number of reductions of computational problems. Foremost among these is the localization sequence, which permits the reduction of most calculational problems to calculations for fields together with the usual kinds of extension and differential problems in spectral sequences. The calculations for fields, though, is not reduced by Quillen's methods. Quillen and Lichtenbaum did formulate a conjecture on what the K-theory of fields should look like. The idea is that the K-theory of fields should be computable in terms of the algebraic K-theory of the algebraic closure of the field, and the action of the absolute Galois group of the field on this algebraic K-theory. Geometrically, this is formulated as an equivalence statement between the algebraic K-theory spectrum of the field and the so-called ``homotopy fixed set'' of the action of the absolute Galois group on the algebraic K-theory spectrum of the algebraic closure. Algebraically, it gives rise to a spectral sequence whose $E_2$ term involves Galois cohomology with coefficients in Tate twists of modules of roots of unity, and which if the equivalence statement above were true would converge to the K-theory of the field. Unfortunately, it turns out that the conjecture as described above is false, as one can see from elementary counterexamples. However, the counterexamples are all in low dimensions, and the conjecture can be formulated as the statement that the homotopy groups of the homotopy fixed set described above agree with the K-groups of the field in high dimensions, where "high" means higher than the cohomological dimension of the absolute Galois group of the field.

    This conjecture was made about the time Quillen's groups were introduced. There has been some progress since then, in particular Suslin's result that the algebraic K-groups of algebraically closed fields, with finite coefficients, are computable, and agree with the homotopy groups of the space $BU$ with finite coefficients. There has also been work done on the conjecture by Thomason and Dwyer-Friedlander-Snaith-Thomason which shows that the conjecture is in a sense true after inverting the so-called "Bott element" and that the map from the K-theory to the homotopy fixed set is surjective on homotopy groups above the cohomological dimension of the field.

    There is a second family of conjectures concerning specific algebraic descriptions of portions of the algebraic K-theory. Prior to the work of Quillen, Milnor had introduced a purely algebraic theory by studying the relations under cup product of units in the field. Milnor K-theory is the quotient of the ring generated freely by the units in the field by certain relations. This ring maps to that part of Quillen K-theory which is generated by elements in dimension one, and Milnor conjectured that these groups could also be computed in terms of Galois cohomology. This conjecture was also proposed nearly thirty years ago, and was recently resolved in the affirmative by the fundamental work of V. Voevodsky. There is also an odd primary analogue of the Milnor conjecture called the Block-Kato conjecture.

    Both the Quillen-Lichtenbaum and Bloch-Kato conjectures are currently the subject of extremely active research. In particular, Voevodsky's work has allowed the affirmative resolution of the Quillen Lichtenbaum conjecture at the prime 2 for number fields. The work of Madsen-Hesselholt has verified it for the case of finite extensions of the p-adic numbers. The Bloch-Kato conjecture is currently being worked on intensively by Voevodsky and M. Rost, and it appears that it may be close to resolution.

    G. Carlsson has developed a modified version of the Quillen-Lichtenbaum conjecture which potentially is correct in all dimensions, not just the sufficiently large ones. This conjecture and program for its proof involve methods from stable homotopy theory, including particularly equivariant stable homotopy theory and the methods from the theory of ring spemodule spectra. The actual formulation involves a notion of a derived version of completion of the representation ring of the absolute Galois group of the field. The method has been verified for a number of examples, and by comparison with the motivic machinery of Voevodsky appears to be correct for fields with abelian absolute Galois group. Much work remains, in particular work clarifying the nature of the completed representation ring for non-abelian profinite groups.

    It appears that the time is now ripe for a concerted effort to resolve the conjectures of Bloch-Kato and Quillen-Lichtenbaum. Such collaborations have already begun. I. Madsen spent a month at the American Institute of Mathematics (AIM) and Stanford and laid the groundwork for comparing the results of Madsen-Hesselholt on p-adic fields with the modified descent result. V. Voevodsky and M. Rost spent two months at AIM and Stanford in 1999 working on Bloch-Kato, and in collaboration with Carlsson on comparing the modified descent procedure with the motivic theory. A great deal of progress was made during both of these visits.

    In the course of these collaborations, it became clear that one extremely important element in the resolution of these conjectures would be the further development of equivariant stable homotopy theory in some particular directions. Equivariant stable homotopy theory has been a key ingredient in the description of various homotopy fixed point spectra under finite group actions, in particular in the Atiyah-Segal completion theorem and in the affirmative resolution of Segal's Burnside ring conjecture. It will serve as a model for development of the algebraic K-theory of varieties introduced by Voevodsky. What has also become clear from the above mentioned collaborations is that various particular directions of study will play a key role in the outstanding questions in algebraic K-theory. Those particular directions are as follows

  • The detailed study of Eilenberg-MacLane spaces in the equivarent stable setting. Here we mean the study of analogues of the Adams spectral sequence, analogues of the Dold-Thom theorem, and the investigation of the homological algebra of Mackey functors. This understanding will be extremely important in the further development of the homotopy theory of varieties.
  • The study of equivariant versions of connective complex K-theory, and the completion conjecture for connective K-theory. There are various versions of complex connective K-theory. We are looking for one which has Thom isomorphisms for equivariant complex bundles. Part of the desired information is the computation of the homotopy type of the fixed point spectrum. This program has been carried out for specific groups by Bruner and Greenlees. This will be necessary to extend the modified descent machinery from fields to rings.
  • The study of analogues of equivariant stable homotopy theory for profinite groups, including the development of an Atiyah-Segal type completion theory in this setting. This is expected to be important in understanding the homotopy type predicted by the modified descent procedure.
  • Clarification of the current state of equivariant complex bordism, and development of a Novikov spectral sequence in this setting. This is an important ingredient in the ongoing work on the homotopy and bordism of varieties, and in the work on the Bloch-Kato conjecture.

    In order to promote the development of equivariant stable homotopy theory in these directions, we will hold a week long workshop at Stanford University on these topics during the summer of 2000. The format of the workshop would include two morning talks, one afternoon talk, and a substantial amount of time for informal interaction.