Topology 39 (2000) 89}94 Complexity in rational homotopy Luis Lechuga, Aniceto Murillo* Algebra, Geometrn H a & Topologn H a, Universidad de Malaga, Ap. 59, 29050 Malaga, Spain Received 24 July 1998; in revised form 14 September 1998 Abstract The computation of classical invariants of the rational homotopy type of simply connected spaces is shown to be an NP-hard problem.  1999 Elsevier Science Ltd. All rights reserved. 1. Introduction In this note we prove that computing most of the numerical invariants of the rational homotopy type of simply connected spaces are NP-hard problems, even if we restrict ourselves to very particular classes of spaces. For readers not very familiar with complexity, let us brie#y recall some of the basic terms we shall use (see [2] or [12]). Formally, a problem "I     is just a family of "nite subsets of non-negative integers, each I  being an instance of the problem. Often, each instance may be viewed as a single integer which carries the codi"cation of the sequence of integers which it represents. Therefore, it is also common in computer sciences to see a problem, together with its solution, as a function f : P in which  is also a subset of . If I 3, f (I ) is the solution to the instance I. A decision problem is a problem f with just two possible values, usually 0, 1 (Yes or No). The language of a decision problem is the set of instances I for which the answer is yes, i.e. f (I)"1. A decision problem f : P (or simply  for convenience) belongs to the class P (polynomial) if there is an algorithm A that solves the problem in polynomial time, i.e., there is a polynomial p such that for each instance I 3  of length n, A produces f (I) in a number of steps bounded by p (n). On the other hand, a problem  belongs to the class NP (non-deterministic polynomial) if there is an algorithm A and a polynomial p such that: given an instance I 3 of length n and a certi"cate * Corresponding author.  Partially supported by a DGICYT grant (PB97-1095). 0040-9383/99/$ - see front matter  1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 4 0 - 9 3 8 3 ( 9 8 ) 0 0 0 5 9 - 7