14/06/07 16:40:19.47
>>569 補足
この本の最終章が特に面白いね
Chapter 11
From Differential Structures to Operator Algebras and Geometric Structures
This chapter surveys some of the interesting interplay of exotic smoothness
with other areas of mathematics and physics. In the first section we consider
the “change” of a differential structure on a given TOP manifold to a differential
structure on a second manifold homeomorphic but not diffeomorphic
to the first one. Harvey and Lawson introduced the notion of singular
bundle maps and connections to study this problem. This leads to speculations
that such a process could give rise to singular string-like sources to
the Einstein equations of General Relativity, including torsion. The next
section deals with formal properties of a connection change and its relation
to cyclic cohomology, providing a relationship between Casson handles
and Ocneanus string algebra. This approach motivates introduction of the
hyperfinite II1 factor C* algebra T leading to the conjecture that the differential
structures are classified by the homotopy classes [M, BGl(T)+]. This
conjecture may have some significance for the the 4-dimensional, smooth
Poincark conjecture. The last section introduces a conjecture relating differential
structures on 4-manifolds and geometric structures of homology 3-spheres naturally embedded in them.