nLab Combinatorial Homotopy and 4-Dimensional Complexes

Combinatorial Homotopy and 4-Dimensional Complexes.

This book by Baues handles the classification of 4-dimensional complexes. Henry Whitehead commented in his famous paper, Combinatorial homotopy I, (1949), ‘What has been achieved so far is a purely algebraic description of the homotopy type of any 3-dimensional complex and of any finite, simply connected 4-dimensional complex.’ (This case of simply connected 4-dimensional complexes had been treated by him in another paper in that same year of 1949.) This book by Baues gives a solution in the non-simply connected case.


Preface by Ronnie Brown.


I Homotopy, homology, and Whitehead’s classification of simply connected 4-dimensional CW-complexes.

II The CW-tower of categories

III Crossed modules and homotopy systems of order 3

IV Quadratic modules and homotopy systems of order 4

V Cohomological invariants

VI The cohomology of categories and the calculus of tracks


List of Symbols



  • H.-J. Baues, Combinatorial Homotopy and 4-Dimensional Complexes , de Gruyter Expositions in Mathematics 2, Walter de Gruyter, (1991).
category: reference

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