By Alfred Barnard Basset

Initially released in 1910. This quantity from the Cornell collage Library's print collections was once scanned on an APT BookScan and switched over to JPG 2000 layout through Kirtas applied sciences. All titles scanned disguise to hide and pages could comprise marks notations and different marginalia found in the unique quantity.

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Those wanting to go deeper into the subject can look at [56, 55], or also at [171] where Milnor presents these groups from his own viewpoint, which is very elegant and closer to what we need for this text (see also [118]). Consider ﬁrst the orthogonal group O(3) of linear transformations of R3 that preserve the usual metric in R3 , given by the standard quadratic form (1, 1, 1). This is the group generated by reﬂections on all 2-planes through the origin and it can be identiﬁed with the space of orthonormal 3-frames in R3 ; its elements are matrices with determinant ±1.

The boundary of the tube N (ε, δ) is homeomorphic to Sε under a homeomorphism that leaves Tε,δ pointwise ﬁxed. In fact the same proof shows that N (ε, δ) is homeomorphic to the ball Bε . Proof. 4 in [168], which is a consequence of the Curve Selection Lemma. Let f1 , . . , fk be the components of f and deﬁne a function: r(x) = f12 (x) + · · · + fk2 (x) . Let ∇r be its gradient. The level surfaces r−1 (s), s ∈ R+ are the tubes f −1 (|t|) for |t|2 = s, and the vector ﬁeld ∇r is transversal to these tubes (away from V ).

When there is a manifold “playing the role of a Milnor ﬁbre”) it is equal to 1 − χ(C) for a smoothing C of the singularity, χ(·) is the Euler-Poincar´e characteristic. However, there exist surface singularities which have smoothings with diﬀerent Euler-Poincar´e characteristics ([204]). This does not permit us to generalize the notion of the Milnor number to higher dimensions so that for smoothable singularities it has the usual expression in terms of the Euler characteristic. In [72] is introduced a notion of Milnor number for every isolated singularity in an analytic variety.