Mathematical knowledge management in algebraic topology

Abstract

The work presented in this thesis tries to particularize Mathematical Knowledge Management to Algebraic Topology. Mathematical Knowledge Management is a branch of Computer Science whose main goal consists in developing integral assistants for Mathematics including computation, deduction and powerful user interfaces able to make the daily work of mathematical researchers easier. Our application context is Algebraic Topology using the Kenzo system [1], a Common Lisp program devoted to Algebraic Topology developed by Francis Sergeraert, as an instrumental tool. We can split the work presented in this thesis into three main parts which coincide with the main goals of Mathematical Knowledge Management. Our first task has consisted in developing a system called fKenzo [2], an acronym of friendly Kenzo. This system not only provides a friendly graphical user interface to interact with the Kenzo system (the kernel of our application) but also guides the interaction of the user with the system (avoiding in this way execution errors). Moreover, fKenzo allows one to integrate other symbolic computation systems (such as GAP) and theorem prover tools (for instance, ACL2 ) by means of a plug-in system. The second part of the thesis is focussed on increasing the computational capabilities of the Kenzo system. Three new Kenzo modules have been developed which in turn extend the fKenzo system. The first one allows us to study the pushout of simplicial sets, an important construction