Daniel Quillen

Rational homotopy theory

For i ≥ 1 they are indeed groups, for i ≥ 2 even abelian groups, which carry a lot of information about the homotopy type of X. However, even for spaces which are easy to define (like spheres), they can be very hard to compute. Even in low dimensions it is difficult to see a clear pattern among the homotopy groups of spheres; especially the torsion shows a seemingly wild behaviour. This suggests that in a first step it might be a good idea to ignore the torsion in the homotopy groups and to just consider the rational homotopy groups (the homotopy groups tensored with Q).

The Spectrum of an Equivariant Cohomology Ring: II

Let G be a compact Lie group (e.g., a finite group) and let HG= H*(BG, Z/pZ) be its mod p cohomology ring. One knows this ring is finitely generated, hence upon dividing out by the ideal of nilpotent elements it becomes a finitely generated commutative algebra over the field Z/pZ. It is the purpose of this series of papers to relate the invariants attached to such a ring by commutative algebra to the family of elementary abelian p-subgroups of G. For example we prove a conjecture of Atiyah and Swan to the effect that the Krull dimension of the ring equals the maximum rank of an elementary abelian p-subgroup. Another result, which will appear in part II, asserts that the minimal prime ideals of the ring are in one-one correspondence with the conjugacy classes of maximal elementary abelian p-subgroups. Actually the theorems of the series are formulated more generally for the equivariant cohomology ring of a G-space X, defined by the formula

Homotopy properties of the poset of nontrivial p-subgroups of a group

Let G be a group and let Y,(G) be the set of nontrivial p-subgroups of G ordered by inclusion, where p is a prime. To the poset (= partially ordered set) YD(G) one can associate a simplicial complex 1 9JG)l in a well-known way. This simplicial complex appears in the work of Brown [2, 31 on Euler characteristics and cohomology for discrete groups. One of his results is an interesting variant of the Sylow theorem on the number of Sylow groups; it asserts that for G finite the Euler characteristic of 1 YD(G)I is congruent to 1 modulo the order of a Sylow p-subgroup. Our aim in this paper is to investigate various homotopy invariants of the simplicial complex ] YJG)l, such as its homology, connectivity, etc. We now review the contents of the paper. In Section 1 we review properties of the functor X ++ 1 X 1. Throughout the paper we use this functor to assign topological concepts to posets. For example, we call two posets homotopy equivalent when the associated simplicial complexes are. In Section 2 we first show that Y@(G) is homotopy equivalent to the poset z

On the formal group laws of unoriented and complex cobordism theory

(G) consisting of elementary abelian p-groups (called p-tori for short). Hence the remainder of the paper is mostly concerned with the smaller poset dJG). We show &‘,(Gr x G,) is h omotopy equivalent to the join of .z

Elementary Proofs of Some Results of Cobordism Theory Using Steenrod Operations

(G1) and &JGa). We prove that dV(G) is contractible when G has a nontrivial normal p-subgroup, and state a conjecture to the effect that the converse holds when G is finite. The conjecture is proved for solvable groups in Section 12; the proof served as motivation for most of the second half of the paper. In Section 3 we show for a finite Chevalley group that -0lJG) has the homotopy type of the Tits building associated to G, and hence it has the homotopy type of a bouquet of spheres. Section 4 contains a proof of the aforementioned result of Brown. The next two sections relate the connected components of &(G) to topics in finite group theory, e.g., in Section 6, Puig’s analysis [S] of the Alperin fusion theorem is described.

Algebra extensions and nonsingularity

In this note we outline a connection between the generalized cohomology theories of unoriented cobordism and (weakly-) complex cobordism and the theory of formal commutative groups of one variable [4], [5]. This connection allows us to apply Carriers theory of typical group laws to obtain an explicit decomposition of complex cobordism theory localized at a prime p into a sum of Brown-Peterson cohomology theories [ l ] and to determine the algebra of cohomology operations in the latter theory.

THE ADAMS CONJECTURE

In this paper I give new proofs of the structure theorems for the unoriented cobordism ring [12] and the complex cobordism ring [8, 131. The proofs are elementary in the sense that no mention of the Steenrod algebra or Adams spectral sequence is made. In fact, the only result from homotopy theory which is used in an essential way is the Serre finiteness theorem in order to know that the complex cobordism group of a given dimension is finitely generated. The technique used here capitalizes on the fact that there are two rather different approaches to defining operations in the complex and unoriented cobordism generalized cohomology theories. The first proceeds via characteristic classes and leads to the LandweberNovikov operations [6, 91, while the second is the analog of the Steenrod power method due to tom Dieck [14]. Using the technique of “localization at the fixpoint set” (Atiyah-Segal [l], tom Dieck [15, 16]), it is possible to derive an equation expressing the Steenrod operation in terms of the LandweberNovikov operations in which the Steenrod operation is zero modulo terms of high filtration. One thereby obtains nontrivial relations involving the action of the Landweber-Novikov operations on the cobordism ring which can be used to show that the cobordism ring is generated by the coefficients of the formal group law expressing the behavior of cobordism Euler classes of line bundles under tensor product. From this, Lazard’s results [7] on formal group laws can be applied to neatly prove that the two cobordism rings are polynomial rings. The paper also contains two new results of interest. The main theorem of the paper shows that the reduced complex cobordism o*(X) of a finite complex is generated by its elements of positive degree as a module over the complex cobordism ring. By duality this implies that * Supported by the Alfred I’. Sloan Foundation, the National Science Foundation, and the Institute for Advanced Study.

On the associated graded ring of a group ring

This paper is concerned with a notion of nonsingularity for noncommutative algebras, which arises naturally in connection with cyclic homology. Let us consider associative unital algebras over the complex numbers. We call an algebra A quasi-free, when it behaves like a free algebra with respect to nilpotent extensions in the sense that any homomorphism A -+ R/I, where I is a nilpotent ideal in R, can be lifted to a homomorphism A -+ R. If we restrict to the category of finitely generated commutative algebras, then this lifting property characterizes smooth algebras, the ones corresponding to nonsingular affine varieties. In this way quasi-free algebras appear as noncommutative analogues of smooth algebras. Stretching the analogy, we can even regard quasi-free algebras as analogues of manifolds. One of the aims of this paper is to develop the analogy further by showing that quasi-free algebras provide a natural setting for noncommutative versions of certain aspects of manifolds. To give an example, let us consider the analogue of an embedding: an extension A = R/I, where A and R are quasi-free algebras playing the role of the submanifold and ambient manifold respectively. In the manifold situation, I/I2 is the module of linear functions on the nor2 mal bundle, and the symmetric algebra SA(III ) is the algebra of polynomial functions. Now in passing from commutative to noncommutative algebras, the symmetric algebra of a module is replaced by the tensor algebra of a bimodule.

Cyclic homology and nonsingularity

THIS paper contains a demonstration of the Adams conjecture [l] for real vector bundles. Unlike an old attempt of mine [12], which has recently been completed by Friedlander [S], and the proof of Sullivan [1.5], no use is made of the etale topology of algebraic varieties. The proof uses only standard techniques of algebraic topology together with some basic results on the representation rings of finite groups, notably the Brauer induction theorem and one of its well-known consequences: the fact that modular representations can be lifted to virtual complex representations.

Algebra cochains and cyclic cohomology

In this paper we show that if the group ring KG of a group G over a field K is filtered by the powers of its augmentation ideal, then the associated graded ring is isomorphic to the universal enveloping algebra of the p-Lie algebra gr? G oz K, where gr? G is the graded p-Lie algebra associated to the p-lower central series of G and where p is the characteristic exponent of K. The proof uses ideas of Lazard’s thesis [I].