David J. Anick

Thin algebras of embedding dimension three

Let R = R0 ⊕ R1 ⊕ R2 ⊕ … be a finitely generated commutative graded k-algebra, with Hilbert series HR(λ) = ∑∞n = 0dimk(Rn)λn. Suppose we require that R have a presentation as R ≈ k[x1,…, xg][α1,…,αr], where g and r and the degrees d1,…, dr of the homogeneous polynomials α1,…, αr are specified in advance. No one knows precisely which Hubert series such an R may have. A lower bound is the coefficient-wise inequality HR(λ) ⩾ ¦(1-λ) gПi=1r(1-λdi) ¦, absolute value symbols denoting the initial non-negative segment of a power series. Can this lower bound always be attained? This deceptively simple-sounding question has been answered affirmatively only when g = 2 (R. Froberg, Report No. 37, Department of Mathematics, University of Stockholm, Stockholm, Sweden, 1982; A. Iarrobino, Memoir No. 188, Amer. Math. Soc., Providence, R. I., 1977), when r ⩽ g, and when r = g + 1 and char(k) = 0 (A. Iarrobino, Compressed Algebras, Trans. Amer. Math. Soc. 285 (1984), 337–378. This paper considers the case g = 3, settling it affirmatively whenever k is an infinite field.

On the homology of quotients of path algebras

We extend the results and techniques of [Al] to find a method of constructing projective resolutions for certain simple modules over homomorphic images of path algebras. We provide a number of applications in the case when the image algebra is finite dimensional.

Inert sets and the Lie algebra associated to a group

Abstract Independently, J. Labute ( Trans. Amer. Math. Soc. 288 (1985), 51–57) has given a sufficient condition for the Lie algebra associated with a discrete groups lower central series to have a certain presentation, and S. Halperin and J.-M. Lemaire ( Math. Scand ., in press) have discussed the conditions under which some homogeneous elements in a graded algebra constitute an “inert set.” As it turns out, these conditions essentially coincide. As a consequence, various theorems about graded algebras may now be translated into results into group theory. A nontrivial application to link groups is offered.

An R-Local Milnor-Moore Theorem

Over a subring R of the rationals, we explore the properties of a new functor. K from spaces to differential graded Lie algebras. We prove a Hurewicz theorem which identifies H∗K(X) with π∗(ΩX) ⊗ R in a range of dimensions. Using it, we prove an R-local version of the Milnor-Moore theorem.

Rational dependence among Hilbert and Poincaré series

Abstract For a certain collection of sets of formal power series, we show that a series belonging to any one set is related by a rational formula to some series in any other set. The collection includes the set of Poincare series of loop spaces on finite CW complexes; the subset obtained when we restrict to complexes of dimension four; the set of Hilbert series of finitely presented graded algebras; the set of Poincare series of Noetherian local rings; and the subset corresponding to those rings whose maximal ideal cubed vanishes.

The Adams-Hilton model for a fibration over a sphere

Abstract Let F→E→Sm+1 be a fibration, where E is 1-connected and m≥2. We show how the Adams-Hilton model for the total space can be expressed succinctly in terms of the Adams-Hilton model for the induced map μ : SmxF→F. We also explore how the Adams-Hilton model for F may be deduced from knowledge of the model for E. Lastly we examine certain iterated relative Samelson products in the mod p homotopy of F, and we extend the work of Cohen. Moore, and Neisendorfer concerning the fibers of pinch maps.

Commutative rings, algebraic topology, graded lie algebras and the work of Jan-Erik Roos

The classic book, Homological Algebra, by Cartan and Eilenberg begins in the following way: ‘During the last decade the methods of algebraic topology have invaded extensive- ly the domain of pure algebra . . . The invasion has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras and associative algebras . . . We present here a single cohomology (and also a homology) theory which embodies all three.’ A few lines later the authors mention Hilbert’s syzygy theorem as an application; this is one instance of the usefulness of these methods when applied to the study of commutative rings. A central role in Homological Algebra is played by the functors Tor and Ext. Ap- plied to a ring homomorphism R + k (k a field) they give a graded k-vector space Tor

Homotopy exponents for spaces of category two

(k, k) whose graded dual Ei = Ext,*(k, k) is, equipped with the Yoneda product, a graded algebra. With the introduction of differential homological algebra by Eilenberg and Moore [14] the functor Tor was extended to differential graded algebras (DGA’s). They showed, in particular, that for the cochain algebra C* of a pointed l-connected CW complex of finite type: Torc*(k,