By Hassler Whitney

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Without loss of generality we may assume that E 0 = (*<> £»), A0 is a (k , A:)-matrix and |. l0| # 0. Then there exists an ε > 0 such that \A\ / 0 if the absolute value of each entry of A - A0 is less than ε . Now let U C M(p , η ; R) consist of all [ρ , /z)-matrices of the form Ε = (a o) • where A is a (k , /c)-matrix such that the absolute value of each entry of A - A0 is less than ε . Then we have Ε e M(p , n \ k) if and only if D = CA~ lB. because the rank of (h 0 \(A B\ =( \ A - I p_ k)\C D) Α Β \ νΛ'Λ + C xb + D) .

N) . , of R" is both the right and left unit of J'(n , n) with respect to this product. Denote by L'(n) a subset of J r(n,n) consisting of all invcrtiblc elements. 4. ( i ) l' (n) = GL(T/ , R ) . (ii) Z/(//) = (7Tr t)-\L\n)). (iii) L'(n) is a Lie group for each r. I < /· < x . (iv) L r(n) has the homotopv ivpe of GL(ZJ. R) (cf. Thom and Lcvinc [Bll]). B. Singular sets. /'(//, p) —* j'(n , ρ). ,, , ('K),/ /-(<•), , ,(0 . 41. A jet f >r ) e J'(n. p) is regular if its representative J in C'(n, p) has the maximal rank at 0.

The main problem here is the existence of embeddings of Γ in Μ and their classification up to isotopy for a given pair of manifolds Γ and Μ . Throughout this chapter we shall work in the ( " v category. §1. Embeddings and isotopies We first define isotopies of embeddings and then state the theorems of Whitney and Haefiiger. Let V be an /(-dimensional c x manifold and lei Μ be an /(/-dimensional manifold. 2. Let / and g be embeddings. We say that /' and g are isotopic and write f ~ g if there exists a C map /·' : I ' χ / — · M .

### Complex analytic varieties by Hassler Whitney

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