From Homotopy to Homology through Pictures

Date of Award

2003

Document Type

Thesis

Degree Name

Bachelors

Department

Natural Sciences

First Advisor

Mullins, David

Keywords

Homotopy, Homology, Pictures, Mathematics

Area of Concentration

Mathematics

Abstract

To construct a K(G, 1)-complex for a group G, with presentation P=< X[R > one can build up skeletons by starting with a point and adding higherdimensional cells successively. It is easy to see how to attach the first two dimensions, but the third dimension and above are not clear for the general group. This paper introduces a set of tools and a series of module isomorphisms to aid in understanding the above construction. One would also like to calculate the homology groups of a group G, which is done by constructing a free resolution of Z[G]-modules, taking the tensor product of the modules with Z over Z[G] and then finding the homology groups of the resulting chain complex. This paper proves, using � among other tools � Igusa's pictures, that the map of a three cell into the 2-skeleton in the above construction is isomorphic to the kernel of the 2-dimensional map in the free resolution.

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