On The Homotopy Groups Of The Suspended Quaternionic Projective Plane And The Applications

Let HPⁿ denote the quaternionic projective space of dimension n.It is the orbit space(Hⁿ⁺¹-{0})/（H-{0}）.In this thesis,we determine the the 2,3-components of the homotopy groups π_r+k（Σ^kHP²）for all 7 ≤r ≤15 and all k≥0,therefore we essentially determine the whole groups π_r+k（Σ^kHP²）where 1 ≤r≤15,k≥0.We mainly use Brayton Gray’s relative James construction to obtain the homotopy information of the fibres of the pinch maps,and use Toda’s secondary composition method to decide the composition relations of generators of homotopy groups and to solve the extension problems.Successively we can determine these homotopy groups.Then,we give the applications which include the classification theorems of the 1-connected CW complexes having same types of the suspended HP3 localized at 3 and the decompositions of the suspended self smashes.This thesis is composed of five chapters and organized as follows:In Chapter 1,we introduce some fundamental knowledge of algebraic topology.In Chapter 2,we introduce Brayton Gray’s relative James construction and our methods to deal with the long exact sequence induced by the Puppe fibre sequence generated by the pinch map.In Chapter 3,we determine the 3-local homotopy groups π_r+k（ΣHP²）,7 ≤r ≤15.In Chapter 4,we determine the 2-local homotopy groups π_r+k（Σ^kP²）,7 ≤r ≤15.In Chapter 5,we give the classification theorems of the 1-connected CW complexes having same types of suspended HP3 localized at 3 and the decompositions of the suspended self smashes.