E∞-operad

In the theory of operads in algebra and algebraic topology, an E_∞-operad is a parameter space for a multiplication map that is associative and commutative "up to all higher homotopies. (An operad that describes a multiplication that is associative but not necessarily commutative "up to homotopy" is called an A_∞-operad.)

Definition

For the definition, it is necessary to work in the category of operads with an action of the symmetric group. An operad "A" is said to be an E_∞-operad if all of its spaces "E"("n") are contractible; some authors also require the action of the symmetric group Σ_"n" on "E"("n") to be free. In other categories than topological spaces, the notion of "contractibility" has to be replaced by suitable analogs, such as acyclicity in the category of chain complexes.

"E"_"n"-operads and "n"-fold loop spaces

The letter "E" in the terminology stands for "everything" (meaning associative and commutative), and the infinity symbols says that commutativity is required up to "all" higher homotopies. More generally, there is a weaker notion of "E"_"n"-operad ("n" ∈ N), parametrizing multiplications that are commutative only up to a certain level of homotopies. In particular,

* "E"₁-spaces are "A"_∞-spaces;
* "E"₂-spaces are homotopy commutative "A"_∞-spaces.

The importance of "E"_"n"- and "E"_∞-operads in topology stems from the fact that iterated loop spaces, that is, spaces of continuous maps from an "n"-dimensional sphere to another space "X" starting and ending at a fixed base point, constitute algebras over an "E"_"n"-operad. (One says they are "E"_"n"-spaces.) Conversely, any connected "E"_"n"-space "X" is an "n"-fold loop space on some other space (called "B^nX", the "n"-fold classifying space of X).

Examples

The most obvious, if not particularly useful, example of an "E"_∞-operad is the "commutative operad" "c" given by "c"("n") = *, a point, for all "n". Note that according to some authors, this is not really an _∞-operad because the Σ_"n"-action is not free. This operad describes strictly associative and commutative multiplications. By definition, any other E_∞-operad has a map to "c" which is a homotopy equivalence.

The operad of little "n"-cubes or little "n"-disks is an example of an "E"_"n"-operad that acts naturally on "n"-fold loop spaces.

See also

* operad
* A-infinity operad
* loop space

References

* cite journal
last = Stasheff
first = Jim
title = What Is...an Operad?
journal = Notices of the American Mathematical Society
year = 2004
month = June/July
volume = 51
issue = 6
pages = pp.630–631
url = http://www.ams.org/notices/200406/what-is.pdf
format = PDF
accessdate = 2008-01-17

*cite book
author = J. P. May
year = 1972
publisher = Springer-Verlag
title = The Geometry of Iterated Loop Spaces
url = http://www.math.uchicago.edu/~may/BOOKSMaster.html

*cite book
author = Martin Markl, Steve Shnider, Jim Stasheff
year = 2002
title = Operads in Algebra, Topology and Physics
publisher = American Mathematical Society
url = http://www.ams.org/bookstore?fn=20&arg1=survseries&item=SURV-96