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 S_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,

• E₁-spaces are A_∞-spaces;
• E₂-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 BⁿX, 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 E_∞-operad because the S_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