A Simpler Proof of Atiyah-Segal Completion Theorem

Lihuang Ding

My research gives an explanation of a simple proof of the Atiyah-Segal completion theorem, explaining the abstract notations in the original proof.

My research focuses on the Atiyah-Segal completion theorem. This is a theorem in equivariant K theory in mathematics. It relates completion of K group with K group of completion space. To be explicit, let me give some standard notations in topology and algebra first. Suppose G is a finite group for simplicity. The spaces we study are CW complexes. Here a CW complex means a space that can be written as a sequence of attaching disks. And a G-vector bundle means attaching disk orbits (G/H)×Dk. K theory studies the vector bundles over a space, which is a locally trivial projection with each fiber a vector space. The direct sum of two vector bundles gives an addition on the set of vector bundles and it forms a monoid. The K group of a space is defined by making this direct sum a group by a construction called Grothendieck group. For any group G, there exists a universal bundle EG→BG such that every bundle can be obtained by pull-backs. Then the completion space is defined as X×EG. This work follows from a 1988 paper, and my research is simply explaining the proof in the paper which was abstract and unclear to make it simple and clear.