Lecture 4.
4.1. Standard pairs
We will be somewhat sketchy in the rest of the proof of Theorem 1.1. However, all
of the things we will be leaving as exercises are quite doable, and do not use any
hard results, in addition to what we have already stated.
The third step in the proof of Theorem 1.1 is the special case when h is the
characteristic function of a locally closed subset of a certain special form. We need
the following definition. Let X be a real algebraic manifold, S be a semi-algebraic
stratification of X , and d = dim X. A pair Y ::i Z of com pad subsets of X is called
standard if the following three conditions hold. (Note that we do not assume Y
and Z to be semi-algebraic.)
(1) Y contains a (non-empty) open subset of X.(2) Y is a manifold with corners of codimension two, and can be stratified with
the following four kinds of strata:
- several strata of dimension d, whose union is the open part of Y;
- several red strata of dimension d - 1, which do not meet Z, such that any
point in a red stratum has a neighborhood where Y looks (smoothly) like a manifold
with boundary;
•several green strata of dimension d - 1, each an open subset of Z, such that
any point in a green stratum has a neighborhood where Y looks like a manifold
with boundary;
•several corner strata of dimension d-2; these are colored both red and green,
and lie inside Z; each point in a corner stratum has a neighborhood where Y looks
like the product IR$, 0 x JRd-^2 near a corner point; we assume that one red and one
green wall meet along each corner stratum.
(3) Each stratum of the stratification of Y is transverse to each stratum of S.
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