ABSTRACT
This note announces some results on the relationship between global invariants and local topological structure. The first section gives a local-global formula for Pontrjagin classes or L-classes. The second section describes a corresponding decomposition theorem on the level of complexes of sheaves. A final section mentions some related aspects of "singular knot theory" and the study of nonisolated singularities. Analogous equivariant analogues, with local-global formulas for Atiyah-Singer classes and their relations to G-signatures, will be presented in a future paper.
ABSTRACT
Let f be a periodic differentiable map from a sphere to itself. A well-known conjecture of Smith asserts that in many cases (e.g., when the fixed points are isolated) the derivatives of f at its fixed points, regarded as Jacobian matrices, are linearly similar. Here we give counterexamples to this conjecture. The results show that, in many cases, these Jacobian matrices are only nonlinearly similar. This uses our recent discovery of orthogonal matrices which are nonlinearly similar without being linearly similar. Some results on general smooth actions of finite groups on differentiable manifolds are presented; the topological equivalence of their tangential representations at the fixed points is studied.
ABSTRACT
The main result of this note (Theorem A) is that the set of piecewise linear (P.L.) manifolds of the same homotopy type as the n-torus, T(n), n >/= 5, is in one-to-one correspondence with the orbits of A(n-3)(pi(1)T(n)) [unk] Z(2) under the natural action of the automorphism group of pi(1)T(n). Every homotopy torus has a finite cover P.L. homeomorphic to T(n); hence the generalized annulus conjecture holds in dimension >/=5 (Kirby, R. C., "Stable homeomorphisms," manuscript in preparation). The methods of this classification are also used to study some conjectures of R. C. Kirby (manuscript in preparation) related to triangulating manifolds.
