Group actions on finite homotopy spheres

• Main Author: Clarkson, James Price
• Language: English
• Published: University of British Columbia 2010
• Online Access: http://hdl.handle.net/2429/27134

Recently, Grodal and Smith [7}have developed a finite algebraic model to study hG-spaces where G is a finite group. The procedure associates to each G-space X with finite F_p homology a perfect chain complex of functors over the orbit category. When X has the homotopy type of a sphere, this construction is particularly well behaved. The reverse construction, building an hG-space from the algebraic model, generally produces an infinite dimensional space. In this thesis, we construct a finiteness obstruction for hG-spheres working one prime at a time. We then begin the development of a global finiteness obstruction. When G is the metacyclic group of order pq, we are able to go further and express the global finiteness obstruction in terms of dimension functions. In addition, we relate the work of tom Dieck and Petrie [19] concerning homotopy representations to the newer model of Grodal and Smith, and compute the rank of V_w(G). We conclude with some new examples of finite Σ₃-spheres.