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Introduction

Since its first version, EAT has been designed to determine homology groups which couldn't be reached by using alternative methods of computation (even theoretic and non-algorithmic methods).

But the aim of the implementors is now to expand the application field of the system and to make easier the use of the program for a larger community of algebraic topologists. For this, a 240 pages long user guide has been written which contains in particular many examples, ranging from elementary surfaces to sophisticated operations on iterated loop spaces.

EAT can deal with the following mathematical structures:

• chain complexes,
• simplicial sets,
• differential graded coalgebras and comodules
and their corresponding versions with effective homology (roughly speaking, an object with effective homology is a structure on which an algorithm to compute its homology groups can be applied, see [4]).

The operations which have been implemented in the current version of EAT are:

• tensor product of two chain complexes,
• cartesian product of two simplicial sets,
• twisted cartesian product of two simplicial sets, in the particular case of loop spaces,
• wedge of two simplicial sets,
• disk pasting,
• suspension,
• iterated loop space of a simplicial set
and the corresponding operations on objects with effective homology (in particular, the construction of a loop space with effective homology involves the implementation of an effective version of the Eilenberg-Moore spectral sequence, see [1]).

The examples in the user guide cover:

• finite simplicial sets,
• simple surfaces and the real projective planes in each dimension,
• the n-dimensional spheres and Moore spaces (a Moore space is a connected space which has only one non-null homology group, other than the -dimensional homology group; so, the spheres are particular cases of Moore spaces),
• wedge, supension and disk pasting over iterated loop spaces of the previous examples.
EAT is organised in a tree of eleven (Common Lisp) modules:
• CC.LISP, chain complexes.
• SS.LISP, simplicial sets.
• TPR.LISP, tensor products of chain complexes.
• TWPR.LISP, loop spaces and twisted (tensor and cartesian) products.
• CCEH.LISP, objects with effective homology.
• EZ.LISP, a module devoted to the Eilenberg- Zilber theorem (the bridge from geometry to algebra).
• HOMOLOGY-GROUPS.LISP, implementations of the final (elementary and well-known) algorithms to compute homology groups of finite type chain complexes or, more generally, of objects with effective homology.
• DISK-PASTE.LISP, disk pasting.
• SUSPENSION.LISP, suspensions.
• WEDGE.LISP, wedges.
• EILENBERG-MOORE.LISP, effective homology of iterated loop spaces.
This set of modules enables the topologist to work on a computer with the objects and operators which are the matter of his papers and textbooks, in a way which is very close to its usual practice.

A new version of the system is currently under development and could be available in 1998.