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June 6, 2009

Nonabelian Algebraic Topology

Posted by John Baez

Here it is — the magnum opus of cubical methods in algebraic topology:

It’s still just a preliminary version, but it’s 516 pages long, and packed with good stuff — and it’s free!

Grab a copy! I’m sure the authors will be grateful if you catch typos and other problems.

The preface explains the basic idea of the book:

Our aim for this book is to give a connected and we hope readable account of the main features of work on extending to higher dimensions the theory and applications of the fundamental group.

We describe algebraic structures in dimensions greater than 1 which develop the nonabelian character of the fundamental group: they are in some sense ‘more nonabelian than groups’, and they reflect better the geometrical complications of higher dimensions than the known homology and homotopy groups. We show how these methods can be applied to determine homotopy invariants of spaces, and homotopy classification of maps, in cases which include some classical results, and allow results not available by classical methods.

In Part I we give some history of work on the fundamental group and groupoid, in particular explaining how the van Kampen theorem gives a method of computation of the fundamental group. We are then mainly concerned with the extension of this nonabelian work to dimension 2, using the key concept, due to J.H.C. Whitehead in 1946, of crossed module.

In Part II we extend the theory of crossed modules to crossed complexes, giving applications which include many basic results in homotopy theory, such as the relative Hurewicz theorem. This Part is intended as a kind of handbook of basic techniques in this border area between homology and homotopy theory.

However for the proofs of these results, particularly of the Higher Homotopy van Kampen theorem, and of the use of the tensor product and homotopy theory of crossed complexes, i.e. monoidal closed structures, we have to introduce in Part III another algebraic structure, that of ‘cubical ω-groupoids with connections’, and to prove its equivalence with crossed complexes. This equivalence algebraicises some long standing geometric methods or intuitions in relative homotopy theory.

Posted at June 6, 2009 5:09 PM UTC

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