Pure Maths Colloquium: Cubes of chain complexes, multi-complexes and totalisation
27 October 2014 16:00 in CM221
While the notion of a commutative cubical diagram of
chain complexes is quite obvious, it is far less clear what a homotopy
commutative cubical diagram should be. One possible definition is
obtained by demanding that its "totalisation", a graded module
equipped with a degree-shifting endomorphism, actually is a chain
complex. The resulting theory is used to define higher-dimensional
mapping tori, which will be employed to obtain a homological
characterisation of finite domination of chain complexes.
In some more detail, a chain complex is finitely dominated if it is
homotopy equivalent to a bounded complex of finitely generated
projective modules. Finite domination of a chain complex can be
characterised by vanishing of its "Novikov homology", that is, homology
with coefficients in certain formal Laurent series rings.
In the talk I will present joint work with David Quinn, relating the
homological machinery of homotopy commutative cubes with Novikov
homology and finite domination.
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