CONTRIBUCIONES CIENT ´ IFICAS EN HONOR DE MIRIAN ANDR ´ ES G ´ OMEZ (Laureano Lamb´an, Ana Romero y Julio Rubio, editores), Servicio de Publicaciones, Universidad de La Rioja, Logro˜ no, Spain, 2010. OPEN QUESTIONS RELATED TO A ∞ -STRUCTURES IN A COMPUTATIONAL FRAMEWORK A. BERCIANO, M. J. JIM ´ ENEZ, AND R. UMBLE Dedicado a la memoria de Mirian Andr´es G´omez Resumen. En este art´ıculo presentamos una breve revisi´on sobre las A∞- (co)´algebras, comenzando con su origen, y continuando con algunas l´ıneas actuales de investigaci´on en varias disciplinas acad´emicas y con algunos pro- blemas abiertos relacionados con su c´alculo, con una atenci´on especial al papel de la Teor´ıa de Perturbaci´on Homol´ogica. Abstract. In this paper we give a brief review of A∞-(co)algebras, begin- ning with their origin, some actual lines of research in various academic disci- plines, and some open questions related to their computation with particular attention to role of Homological Perturbation Theory. 1. Introduction In the early sixties, J. Stasheff introduced the notion of an A n -space [29], which is a topological space X whose singular chains A = C * (X) come equipped with operations  m i : A ⊗i → A  1≤i≤n that relate to one another in a systematic way. The operation m 1 is a degree -1 differential, m 2 is a multiplication, and m 1 is a derivation of m 2 ; thus, m 1 m 1 = 0 and m 1 m 2 = m 2 (m 1 ⊗ 1 - 1 ⊗ m⊗ 1 ). If m 2 is associative up to homotopy, there is a chain homotopy m 3 , called the associator, that relates the two associations in three variables, i.e., m 1 m 3 + m 3 (m 1 ⊗ 1 ⊗ 1+ 1 ⊗ m 1 ⊗ 1+1 ⊗ 1 ⊗ m 1 )= m 2 (1 ⊗ m 2 ) - m 2 (m 2 ⊗ 1) . In this case, (A, m i ) 1≤i≤3 is an A 3 -algebra. If there is a chain homotopy m 4 relating the five chain homotopic associations in four variables, the tuple (A, m i ) 1≤i≤4 is an A 4 -algebra, and so on. Stasheff’s definition of an A n -space was motivated by the space ΩX of base pointed loops on a topological space (X, *), whose points range over all continuous maps α : ([0, 1] , {0, 1}) → (X, *). Given α, β ∈ ΩX, the product αβ is defined by αβ (t)=  α (2t) , 0 ≤ t ≤ 1 2 β (2t - 1) 1 2 ≤ t ≤ 1. Thus the product (αβ) γ is not associative, but is associative up to homotopy, as indicated by the following linear change of parameter diagram: Key words and phrases. Bar construction, Cobar construction, A∞-(co)algebra, A∞- bialgebra, contraction, Basic Perturbation Lemma. This work was partially supported by “Computational Topology and Applied Mathematics” PAICYT research project FQM-296, Spanish MEC project MTM2006-03722. 171