Contents

1 Algebraic preliminaries 1

§1.1 Clifford algebras 1

§1.2 Clifford modules 2

2 Topological preliminaries 5

§2.1 Karoubi's A'^-theory 5

§2.2 Some elementary examples 8

§2.3 Classifying spaces 9

3 (p,q)-lagrangians and classifying spaces for K-theory 12

§3.1 (p,q)-lagrangian subspaces 12

§3.2 Graphs of linear operators and lagrangians 17

§3.3 The components of TLv* 20

§3.4 The homotopy type of TL™ 22

4 Symplectic reductions 24

§4.1 Generalized symplectic geometries 24

§4.2 Homotopic properties of the symplectic reduction process 28

§4.3 The generalized Maslov index 30

§4.4 Comparison with the traditional Maslov index 34

5 Clifford symmetric Fredholm operators 35

§5.1 The space 7VA 35

§5.2 Hamiltonian (p,q)-modules 36

§5.3 The graph map and generalized Floer operators 38

§5.4 Examples 43

6 Families of boundary value problems for Dirac operators 46

§6.1 (p,q)-Dirac operators and their Calderon projections 46

§6.2 The index of families of boundary value problems 49

§6.3 The cobordism invariance of the families index 56

§6.4 Gluing formulae 60

A Gap convergence of linear operators 64

B Gap continuity of families of BVP's for Dirac operators 68

C Pseudodifferential Grassmanians and BVP' s for Dirac operators 71

D The proof of Proposition 6.1 74

References 78

vii