Elements of codimension theory

Let us assume that V RN + R is a smooth function defined near x = (xxw) = 0 (R stands for the set of real numbers — the real axis, while RN denotes the set of vectors consisting of N real numbers — the real N-dimensional space). 

En will stand for the set of all such functions. We shall define an expansion of the function VeEN in a Taylor series as the jet jV 

For example, for the function cos(x) its Taylor series and the four-order can be written in the form 

The function V is said to be /c-determined if every function U meeting the condition jkU = jkV (i.e. having an identical Taylor expansion up to and including terms of order k) is equivalent, in the sense of definition (2.12), to the function V. In other words, a local character of the function V in the neighbourhood of x = 0 is fully determined by k first terms of the Taylor series (modification of higher-order terms of the Taylor series, i.e. replacement of V with U, does not change a local shape of V). 

a /c-determined function is equivalent to some polynomial in 

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From the point of view of the algebraic theory of codimension 1 surgery the above decompositions of the elements sGL (A x,x 1,E) may be deduced from the following... 

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