Spectral Sequences of Operad Algebras

A spectral sequence is a device for relating something computable to something desirable via a sequence of homological operations.  The existence of a product structure can be extremely useful in spectral sequence computations.  What makes this possible is that it is well-understood when a spectral sequence is compatible with the underlying algebraic structure.  This was done, for example, by Massey as early as 1954.  However, Massey's conditions are tedious and require the verification of one condition for each natural number.  Other conditions are available which are inductive in nature, and can be found in standard references such as the  "User's Guide to Spectral Sequences" by McCleary.

 Product structures can come in many forms - one can require that a product is associative, commutative, satisfies a Jacobi condition or other conditions, almost without limit.  Operads are designed to encode a system of algebraic operations and their compositions.  These  provide the structure governing all such products.  In this talk, I will give conditions that guarantee  a spectral sequence is compatible with the action of an operad.  A spectral sequence satisfying these compatibility conditions converges to an algebra over the same operad.  I will focus on the topological setting, where spectral sequences will arise from a tower of topological spaces and maps.  This talk will end with an enticing open problem. 

Although the construction I will present is the most natural generalization of products arising in topology, many of the interesting examples of operads acting on towers of spaces arise in another manner.  I will explain these examples, and discuss to what extent our criteria apply to this situation.

 This is joint work with Laura Scull.