Products on 𝑀𝑈-modules

Abstract

Elmendorf, Kriz, Mandell and May have used their technology of modules over highly structured ring spectra to give new constructions of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M upper U"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">MU</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-modules such as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper B upper P"> <mml:semantics> <mml:mrow> <mml:mi>B</mml:mi> <mml:mi>P</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">BP</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>K</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">K(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M upper U left-bracket one half right-bracket Subscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>U</mml:mi> <mml:mo stretchy="false">[</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mo>∗</mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">MU[\frac {1}{2}]_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula> that are concentrated in degrees divisible by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="4"> <mml:semantics> <mml:mn>4</mml:mn> <mml:annotation encoding="application/x-tex">4</mml:annotation> </mml:semantics> </mml:math> </inline-formula>; this guarantees that various obstruction groups are trivial. We extend these results to the cases where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2 equals 0"> <mml:semantics> <mml:mrow> <mml:mn>2</mml:mn> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">2=0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> or the homotopy groups are allowed to be nonzero in all even degrees; in this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity, and that the obstructions to commutativity are given by a certain power operation; this was inspired by parallel results of Mironov in Baas-Sullivan theory. We use formal group theory to derive various formulae for this power operation, and deduce a number of results about realising <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="2"> <mml:semantics> <mml:mn>2</mml:mn> <mml:annotation encoding="application/x-tex">2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-local <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M upper U Subscript asterisk"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:msub> <mml:mi>U</mml:mi> <mml:mo>∗</mml:mo> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">MU_*</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-modules as <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper M upper U"> <mml:semantics> <mml:mrow> <mml:mi>M</mml:mi> <mml:mi>U</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">MU</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-modules.