THE TOPOLOGICAL q-EXPANSION PRINCIPLE

Abstract

for sharing their ideas, time and knowledge with me. I am indebted to many graduate students at MIT, for friendship and mathematical discussions, particularly to Charles Rezk, Nitya Kitchloo, Dan Christiensen and, last but not least, to my soulmate and sister in crime, Jean Strachan.The Studienstiftung des deutschen Volkes and the MIT deserve thanks for a generous financial support.This work is dedicated to Christina in appreciation for her constant faith and love.

Extensions of umbral calculus II: double delta operators, Leibniz extensions and Hattori-Stong theorems

“…More generally a similar result will hold for E* (E n E A ... n E). These remarks follow closely the case of elliptic cohomology considered in Theorem 2.10 of [16]. …”

“…That is to say, if p divides ul e in E*, then p divides e. Now ul = cp-l modulo p, where Cp-i 1 is the coefficient of tP/p in the log series of the formal group law (see [15], Lemma The following generalisation closely parallels a result of Laures [16], Theorem 1.6, which applies to the case of elliptic cohomology. THEOREM 11.3. -Let E be a complex-oriented ring spectrum and X a space or spectrum such that E* (X ) satisfies the height-one Landweber exactness condition for all primes, then E* (X) is a pure s u bgro u p of K* ( E n X).…”

ON THE DOUBLE TRANSFER AND THE f-INVARIANT

“…In principle, this result determines the class p * [κ], for any primitive p. This can be made more concrete by passing to elliptic homology and appealing to the invariants introduced by Laures [12] and Behrens [4], as explained in the following sections.…”

“…is realised by a morphism of spectra q 0 : Ell → KU, which is derived from Miller's elliptic character (see [12], following Miller [14]). Hence there is an induced morphism of spectra Ell∧Ell → KU∧Ell, which induces a morphism of right Ell * -modules…”

“…This paper studies the algebraic double transfer using invariants that are derived from elliptic homology, Ell, via the f -invariant of Laures [12], which requires that 6 is inverted, and via the f -invariant of Behrens [4], which is defined when working p-locally for p ≥ 5. For the purposes of this paper, elliptic homology Ell should be taken to be TMF[ 1 6 ] (as in [5]), where TMF is the spectrum of topological modular forms.…”