In Part I of this paper (6) we proved various index theorems for manifolds with boundary including an extension of the Hirzebruch signature theorem. We now propose to investigate the geometric and topological implications of these theorems in a variety of contexts. In §2 we consider a generalization of the signature theorem involving unitary representations of the fundamental group. This leads to a new differential invariant of manifolds with given fundamental group G independent of any Riemannian metric, which generalizes previously known invariants for finite G studied in (4). In more detail the situation is as follows. We consider an oriented Riemannian manifold Y of odd dimension and a unitary representation oc:n1(Y)-^U(n). On the space of all exterior differential forms of even degree there is a natural self-adjoint operator B denned by B<j> = i(-l)e+(*d-d*)<f> (deg<f> = 2p, d i m 7 = 2Z-l) .
[1]
V. K. Patodi,et al.
On the heat equation and the index theorem
,
1973
.
[2]
G. Wilson.
K-THEORY INVARIANTS FOR UNITARY G-BORDISM
,
1973
.
[3]
M. Atiyah,et al.
Vector bundles and homogeneous spaces
,
1961
.
[4]
S. Chern,et al.
Characteristic forms and geometric invariants
,
1974
.
[5]
M. Atiyah,et al.
The Index of Elliptic Operators: IV
,
1971
.
[6]
V. K. Patodi,et al.
Spectral asymmetry and Riemannian Geometry. I
,
1973,
Mathematical Proceedings of the Cambridge Philosophical Society.
[7]
M. Atiyah,et al.
Compact lie groups and the stable homotopy of spheres
,
1974
.