Martin Bendersky

The -theory of toric manifolds

Toric manifolds, the topological analogue of toric varieties, are determined by an n-dimensional simple convex polytope and a function from the set of codimensionone faces into the primitive vectors of an integer lattice. Their cohomology was determined by Davis and Januszkiewicz in 1991 and corresponds with the theorem of DanilovJurkiewicz in the toric variety case. We use the Adams spectral sequence to compute the KO-theory of all toric manifolds and certain toric varieties.

Cup-products for the polyhedral product functor

Davis–Januszkiewicz introduced manifolds which are now known as moment-angle manifolds over a polytope [ 6 ]. Buchstaber–Panov introduced and extensively studied moment-angle complexes defined for any abstract simplicial complex K [ 4 ]. They completely described the rational cohomology ring structure in terms of the Tor-algebra of the Stanley-Reisner algebra [ 4 ]. Subsequent developments were given in work of Denham–Suciu [ 7 ] and Franz [ 9 ] which were followed by [ 1 , 2 ]. Namely, given a family of based CW-pairs X , A ) = {( X i , A i )} m i =1 together with an abstract simplicial complex K with m vertices, there is a direct extension of the Buchstaber–Panov moment-angle complex. That extension denoted Z ( K ;( X , A )) is known as the polyhedral product functor, terminology due to Bill Browder, and agrees with the Buchstaber–Panov moment-angle complex in the special case ( X , A ) = ( D 2 , S 1 ) [ 1 , 2 ]. A decomposition theorem was proven which splits the suspension of Z ( K ; ( X , A )) into a bouquet of spaces determined by the full sub-complexes of K . This paper is a study of the cup-product structure for the cohomology ring of Z ( K ; ( X , A )). The new result in the current paper is that the structure of the cohomology ring is given in terms of this geometric decomposition arising from the “stable” decomposition of Z ( K ; ( X , A )) [ 1 , 2 ]. The methods here give a determination of the cohomology ring structure for many new values of the polyhedral product functor as well as retrieve many known results. Explicit computations are made for families of suspension pairs and for the cases where X i is the cone on A i . These results complement and extend those of Davis–Januszkiewicz [ 6 ], Buchstaber–Panov [ 3 , 4 ], Panov [ 13 ], Baskakov–Buchstaber–Panov, [ 3 ], Franz, [ 8 , 9 ], as well as Hochster [ 12 ]. Furthermore, under the conditions stated below (essentially the strong form of the Kunneth theorem), these theorems also apply to any cohomology theory.

₁-periodic homotopy groups of exceptional Lie groups: torsion-free cases

The v 1 -periodic homotopy groups v 1 #751 π. (X; p) are computed explicitly for all pairs (X, p), where X is an exceptional Lie group whose integral homology has no p-torsion. This yields new lower bounds for p-exponents of actual homotopy groups of these spaces. Delicate calculations with the unstable Novikov spectral sequence are required in the proof

The complex bordism of groups with periodic cohomology

Is is proved that if BG is the classifying space of a group G with periodic cohomology, then the complex bordism groups MU*(BG) are obtained from the connective K-theory groups ku*(BG) by just tensoring up with the generators of MU. as a polynomial algebra over ku* . The explicit abelian group structure is also given. The bulk of the work is the verification when G is a generalized quaternionic group. 1. STATEMENT OF RESULTS It is well known [CE] that a finite group has periodic cohomology if and only if its Sylow subgroups are all cyclic or generalized quaternionic. Another characterization [Sw] is that these are precisely the finite groups which can act freely on a finite simplicial homotopy sphere. In [Wo], it was shown exactly which of these (the spherical space-form groups) admit a free orthogonal action on a standard sphere. Let MU.( ) denote (reduced) complex bordism and bu*( ) connective Ktheory homology. It is well known [CF1] that if BG denotes the classifying space of a finite group G, then MUn(BG+) is isomorphic to the group of bordism classes of stably almost complex n-manifolds with free G-action. Here and elsewhere X+ is the space obtained from X by adjoining a disjoint basepoint. The coefficient rings are MU. --MU*(S?) = Z[X2 : i > 1] and bu* --bu*(S0) = Z[x2] where x21 is a generator of degree 2i in a polynomial algebra. Our main result proves an extension of a conjecture of Gilkey [G, BD]. Theorem 1.1. If G is any finite group with periodic cohomology, then there is an isomorphism of graded abelian groups MU*(BG) bu*(BG) 0 Z[x2i: i > 2]. Received by the editors May 1, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 57R85; Secondary 55R35, 55N22.

Unstable towers in the odd primary homotopy groups of spheres

The unstable elements in filtration 2 of the unstable Novikov spectral sequence are computed. These elements are shown to survive to elements in the homotopy groups of spheres which are related to Im J. The computation is applied to determine the Hopf invariants of compositions of Im / and the exponent of certain sphere bundles over spheres.

ON THE COALGEBRAIC RING AND BOUSFIELD–KAN SPECTRAL SEQUENCE FOR A LANDWEBER EXACT SPECTRUM

We construct a Bousfield–Kan (unstable Adams) spectral sequence based on an arbitrary (and not necessarily connective) ring spectrum

A stable approach to an unstable homotopy spectral sequence

E

On the rational type of moment-angle complexes

with unit and which is related to the homotopy groups of a certain unstable

Stable geometric dimension of vector bundles over even-dimensional real projective spaces

E

A GENERALIZATION OF THE DAVIS-JANUSZKIEWICZ CONSTRUCTION AND APPLICATIONS TO TORIC MANIFOLDS AND ITERATED POLYHEDRAL PRODUCTS

completion