Homology and images of semianalytic sets
Robert Hardt · Bulletin of the American Mathematical Society · 1974
The homology of semianalytic sets may be treated using chains which are themselves locally-finite integral combinations of disjoint, oriented semianalytic submanifolds.The analytic image of a relatively compact semianalytic set, though not necessarily semianalytic, admits a finite stratification into connected analytic submanifolds of various dimensions.A subset A of a (real) analytic manifold Mis called analytic (respectively, semianalytic) if M can be covered by open sets U for which there is a real-valued function ƒ (respectively, a finite family J^ of real-valued functions) analytic in U so that UnA equals/ _1 {0} (respectively, UC\A is a union of connected components of/-~1{0}~£"~1{0} for some/, g e 3F).A stratum in M is a connected (properly embedded) differentiable submanifold of M. A stratification S? of a subset A of M is a locally finite partition of A into strata Sf so that (A nClos S)~S is a union of strata in SP having dimension less than the dimension of S. It is well known [9, §13], [7, 2.8] that every semianalytic set admits a stratification into semianalytic strata.Ay-dimensional analytic chain T in M is a sum of integral multiples of oriented /-dimensional semianalytic strata belonging to some fixed stratification of M. Since the restriction to these strata of/dimensional Hausdorff measure is locally-finite by [2, 3.4.8(13)], the analytic chain T is (by oriented integration, counting multiplicities, of differential j forms of compact support in M) a /-dimensional current in M. The set spt T, being the union of the closures of the strata occurring with nonzero multiplicity, is semianalytic.Fory^l, the (y-1/dimensional current ar, defined by dT(y) =T(dy>) for y> e &~\M), is, by [2, 4.2.28], also an analytic chain in M.Suppose M^> A=>B.Using the group of real analytic cycles 2£$(A, 5)== {T.T is a /dimensional analytic chain of compact support, spt T<^A, AMS (MOS) subject classificatipns (1970).Primary 32C05, 32C25; Secondary 55B35, 55B45, 28A75.