Rational homotopy and unique factorization

Abstract

Decompositions of various kinds of mathematical objects as products or coproducts are considered, and the uniqueness of these decompositions is discussed. For instance, the following topological result is proved. Let A be the set of formal, simply-connected, rational homotopy types having finitely generated rational homotopy. Of course, A is a commutative semigroup with respect to the usual product space construction. Then A is a free commutative semigroup. This is a statement about "unique factorization" in A, and it follows from our main result (Theorem 4) concerning the unique factorization of certain differential graded Q -algebras called formal minimal algebras. 332 R. BODY AND R. DOUGLAS together with a degree one derivation d\M-*M (the differential) such that: (i) d -d = 0. (ii) d(M) C D(M) = the ideal of decomposable elements of M. (iii) H*(M), the cohomology of M (with respect to d) is a finitely generated graded Q -algebra.