Paulo Lima-Filho

Completions and fibrations for topological monoids

We show that, for a certain class of topological monoids, there is a homotopy equivalence between the homotopy theoretic group completion M + of a monoid M in that class and the topologized Grothendieck group M associated to M. The class under study is broad enough to include the Chow monoids effective cycles associated to a projective algebraic variety and also the infinite symmetric products of finite CW-complexes. We associate principal fibrations to the completions of pairs of monoids, showing the existence of long exact sequences for the naive approach to Lawson homology [Fri91, LF91 a]. Another proof of the Eilenberg-Steenrod axioms for the functors X → SP(X) in the category of finite CW-complexes (Dold-Thom theorem [DT56]) is obtained

Algebraic cycles and infinite loop spaces

SummaryIn this paper we use recent results about the topology of Chow varieties to answer an open question in infinite loop space theory. That is, we construct an infinite loop space structure on a certain product of Eilenberg-MacLane spaces so that the total Chern map is an infinite loop map. An analogous result for the total Stiefel-Whitney map is also proved. Further results on the structure of stabilized spaces of alebraic cycles are obtained and computational consequences are also outlined.

Algebraic cycles and the classical groups. I: real cycles

Abstract The groups of algebraic cycles on complex projective space P (V) are known to have beautiful and surprising properties. Therefore, when V carries a real structure, it is natural to ask for the properties of the groups of real algebraic cycles on P (V) . Similarly, if V carries a quaternionic structure, one can define quaternionic algebraic cycles and ask the same question. In this paper and its sequel the homotopy structure of these cycle groups is completely determined. It turns out to be quite simple and to bear a direct relationship to characteristic classes for the classical groups. It is shown, moreover, that certain functors in K-theory extend directly to these groups. It is also shown that, after taking colimits over dimension and codimension, the groups of real and quaternionic cycles carry E∞-ring structures, and that the maps extending the K-theory functors are E∞-ring maps. This gives a wide generalization of the results in (Boyer et al. Algebraic cycles and infinite loop spaces, Invent. Math. 113 (1993) 373.) on the Segal question. The ring structure on the homotopy groups of these stabilized spaces is explicitly computed. In the real case it is a simple quotient of a polynomial algebra on two generators corresponding to the first Pontrjagin and first Stiefel–Whitney classes. These calculations yield an interesting total characteristic class for real bundles. It is a mixture of integral and mod 2 classes and has nice multiplicative properties. The class is shown to be related to the Z 2 -equivariant Chern class on Atiyahs KR-theory.

Quaternionic algebraic cycles and reality

In this paper we compute the equivariant homotopy type of spaces of algebraic cycles on real Brauer-Severi varieties, under the action of the Galois group Gal(C/R). Appropriate stabilizations of these spaces yield two equivariant spectra. The first one classifies Dupont/Seymours quaternionic K-theory, and the other one classifies an equivariant cohomology theory 3*(-) which is a natural recipient of characteristic classes KH*(X) → 3*(X) for quaternionic bundles over Real spaces X.

Holomorphic K-Theory, Algebraic Co-cycles, and Loop Groups

In this paper we study the “holomorphic K -theory” of a projective variety. This K theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups. This theory has been introduced in various places such as [12], [9], and a related theory was considered in [11]. This theory is built out of studying algebraic bundles over a variety up to “algebraic equivalence”. In this paper we will give calculations of this theory for “flag like varieties” which include projective spaces, Grassmannians, flag manifolds, and more general homogeneous spaces, and also give a complete calculation for symmetric products of projective spaces. Using the algebraic geometric definition of the Chern character studied by the authors in [6], we will show that there is a rational isomorphism of graded rings between holomorphic K theory and the appropriate “morphic cohomology” groups, defined in [7] in terms of algebraic co-cycles in the variety. In so doing we describe a geometric model for rational morphic cohomology groups in terms of the homotopy type of the space of algebraic maps from the variety to the “symmetrized loop group” ΩU(n)/Σn where the symmetric group Σn acts on U(n) via conjugation. This is equivalent to studying algebraic maps to the quotient of the infinite Grassmannians BU(k) by a similar symmetric group action. We then use the Chern character isomorphism to prove a conjecture of Friedlander and Walker stating that if one localizes holomorphic K theory by inverting the Bott class, then rationally this is isomorphic to topological K theory. Finally this will allows us to produce explicit obstructions to periodicity in holomorphic K theory, and show that these obstructions vanish for generalized flag manifolds.

Algebraic cycles and the classical groups II: quaternionic cycles.

In part I of this work we studied the spaces of real algebraic cycles on a complex projective space P(V), where V carries a real structure, and completely determined their homotopy type. We also extended some functors in K-theory to algebraic cycles, establishing a direct relationship to characteristic classes for the classical groups, specially Stiefel-Whitney classes. In this sequel, we establish corresponding results in the case where V has a quaternionic structure. The determination of the homotopy type of quaternionic algebraic cycles is more involved than in the real case, but has a similarly simple description. The stabilized space of quaternionic algebraic cycles admits a nontrivial infinite loop space structure yielding, in particular, a delooping of the total Pontrjagin class map. This stabilized space is directly related to an extended notion of quaternionic spaces and bundles (KH-theory), in analogy with Atiyahs real spaces and KR-theory, and the characteristic classes that we introduce for these objects are nontrivial. The paper ends with various examples and applications.

Algebraic families of E∞-spectra

Abstract We construct three different spectra, and hence three generalized cohomology theories, associated to an algebraic variety X : the morphic spectrum Z X —an E ∞ - ring spectrum—related to intersection theory when X is smooth; the multiplicative morphic spectrum M X and the holomorphic K -theory spectrum K X . The constructions use holomorphic maps from X into appropriate moduli spaces, and are functorial on X . The coefficients for the corresponding cohomology theories reflect algebraic geometric and topological invariants for the variety X . In the morphic case, the coefficients are given in terms of Friedlander–Lawsons morphic cohomology for varieties (Friedlander and Lawson, 1992). The theory carries total Chern class maps and total cycle maps , extending to the stable category classical constructions in algebraic geometry.