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• W2021058801 abstract "ON THE EXISTENCE AND CLASSIFICATION DIFFERENTIABLE ANDRE OF EMBEDDINGS HAEFLICER and MORRIS W. HIRSCH (Receked 3 Jnnunry 1963) sl. INTRODUCTION LET A4 be a compact /c-connected differential to prove, under suitable restrictions on k and the Euclidean space R’“-“-’ (Theorem (2.3)), of embeddings of 1M in R *“-’ if 1M is orientable Theorems (2.1) and (2.2) which reduce the immersions, and then applying A particular THEOREM n-manifold without boundary. Our object is II, an existence theorem for embedding IV in and a classification theorem for isotopy classes (Theorem (2.4)). This is done by first proving embedding problems to questions involving the theory of immersions case of (2.3) is the following: (I. 1). If n > 4, M is embeddable Whitney class p- in R 2n- 1 if and only if its normal Stiefel- ’ vanishes. Massey [5, 6, 71 has shown power of 2. Thus we obtain: that if P-l # 0, then M is non-orientable TIIEOREM (I .2). If n > 4 and A4 is orientable, M is embedable in R2”-‘. This is also true if n = 3; see [4]. The case n = 4 is unsolved, connected. embeddable However, in R5. Smale has proved (unpublished) and ?I is a that every even if M is simply homotopy 4-sphere is It should be remarked that the existence Theorems (2.1) and (2.3) apply to both orientable and non-orientable manifolds, but the classification Theorems (2.2) and (2.4) apply only to orientable manifolds. (1.3). DEFINITIONS AND NOTATION. All manifolds boundary of a manifold X is 2X. considered here are differential. The We put X - IYX = int X. An immersion of an n-manifold X in Euclidean r-space R” is a differentiable map f: A’-+ R” of rank n everywhere. An embedding is an immersion which is l- 1. If f and g are immersions of X in X”, a regular homotopy connectingfto g is a differentiable homotopy F: X x I+ R” such that F, = f, F, = g, and each F, is an immersion. If in addition each F, is an embedding, then F is an isotopy." @default.
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• W2021058801 title "On the existence and classification of differentiable embeddings" @default.
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