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Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80 (Paperback)
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Smoothings of Piecewise Linear Manifolds. (AM-80), Volume 80 (Paperback)
Series: Annals of Mathematics Studies
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The intention of the authors is to examine the relationship between
piecewise linear structure and differential structure: a
relationship, they assert, that can be understood as a homotopy
obstruction theory, and, hence, can be studied by using the
traditional techniques of algebraic topology. Thus the book attacks
the problem of existence and classification (up to isotopy) of
differential structures compatible with a given combinatorial
structure on a manifold. The problem is completely "solved" in the
sense that it is reduced to standard problems of algebraic
topology. The first part of the book is purely geometrical; it
proves that every smoothing of the product of a manifold M and an
interval is derived from an essentially unique smoothing of M. In
the second part this result is used to translate the classification
of smoothings into the problem of putting a linear structure on the
tangent microbundle of M. This in turn is converted to the homotopy
problem of classifying maps from M into a certain space PL/O. The
set of equivalence classes of smoothings on M is given a natural
abelian group structure.
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