A Note on the Eilenberg-Mac Lane Isomorphism for Quadratic Forms
C\'esar Galindo
TL;DR
This paper provides an elementary proof of the Eilenberg-Mac Lane trace isomorphism linking third 2-abelian cohomology to quadratic forms, with explicit constructions and conditions for expressing quadratic forms as traces.
Contribution
It offers a new elementary proof and explicit constructions for the Eilenberg-Mac Lane isomorphism, enhancing understanding of quadratic forms in cohomology.
Findings
Elementary proof of the trace isomorphism
Explicit constructions of the isomorphism
Characterization of quadratic forms as traces
Abstract
We give an elementary proof of the Eilenberg-Mac Lane trace isomorphism between the third 2-abelian cohomology group and quadratic forms. Our approach yields explicit constructions and we characterize when quadratic forms can be expressed as traces of bilinear forms for arbitrary coefficient groups.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Homotopy and Cohomology in Algebraic Topology · Geometric and Algebraic Topology