Classification of Algebraic Complexes

  • F. E. A. JohnsonEmail author
Part of the Algebra and Applications book series (AA, volume 17)


By an algebraic n-complex over Z[G] we mean an exact sequence of Z[G]-modules
$$E_* = (0 \rightarrow J \rightarrow E_n \stackrel{\partial_n}{\rightarrow} E_{n-1} \stackrel{\partial _{n-1}}{\rightarrow} \cdots\stackrel{\partial_2}{\rightarrow}E_1 \stackrel{\partial_1}{\rightarrow} E_0\rightarrow\mathbf{ Z} \rightarrow0)$$
in which each E r is finitely generated and stably free over Z[G]. The notion is an abstraction from a cell complex X with π 1(X)=G for which \(\pi_{r}(\tilde{X}) = 0\) for 0<r<n. In this chapter we use the Swan homomorphism of Chap.  7 to classify algebraic n-complexes up to homotopy equivalence.


Algebraic Complexity Swan Homomorphism Weak Homotopy Equivalence Exact Sequence Yoneda Product 
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Copyright information

© Springer-Verlag London Limited 2012

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonUK

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