1 The difference construction.- 2 Umkehr maps and inner product spaces.- 3 Stable homotopy theory.- 4 Z_2-equivariant homotopy and bordism theory.- 5 The geometric Hopf invariant.- 6 The double point theorem.- 7 The -equivariant geometric Hopf invariant.- 8 Surgery obstruction theory.- A The homotopy Umkehr map.- B Notes on Z2-bordism.- C The geometric Hopf invariant and double points (2010).- References.- Index.