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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:
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 ).
differential graded coalgebras and comodules
The operations which have been implemented in the current version of
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 ).
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,
iterated loop space of a simplicial set
The examples in the user guide cover:
EAT is organised in a tree of eleven (Common Lisp) modules:
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
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.
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.
EILENBERG-MOORE.LISP, effective homology of iterated loop spaces.
A new version of the system is currently under development and could
be available in 1998.