Nicolaas H. Kuiper

Manifolds which are like projective planes

© Publications mathématiques de l’I.H.É.S., 1962, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Minimal total absolute curvature for immersions

Immersions or maps of closed manifolds in Euclidean space, of minimal absolute total curvature are called tight in this paper. (They were called convex in [25].) After the definition in Chapter 1, many examples in Chapter 2, and some special topics in Chapter 3, we prove in Chapter 4 that topological tight immersions ofn-spheres are only of the expected type, namely embeddings onto the boundary of a convexn+1-dimensional body. This generalises a theorem of Chern and Lashof in the smooth case. In Chapter 5 we show that many manifolds exist that have no tight smooth immersion in any Euclidean space.

The topology of holomorphic flows with singularity

© Publications mathématiques de l’I.H.É.S., 1978, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Topological Classification of Linear Endomorphisms

Letf=f +⊕f −⊕f ∞⊕f 0 be a linear direct sum decomposition of a real endomorphismf∈ End ℝ n , where the components correspond respectively to absolute values of eigenvalues |λ|: 0<|λ|<1(f +); |λ|>1(f −); |λ|=0(f ∞); and |λ|=1(f 0). We conjecture that a complete set of invariants with respect to topological equivalence consists of the dimensions and orientations off + and off −, together with the linear isomorphy classes off ∞ andf 0. We prove that this is true in case ℝ n contains nof-periodic points of periods 5 or ≧7. The conjecture is also true in case it is true for all periodic rotations. In §9 we make some comments on this unsolved case. The main interest of the paper is in §6 and 7.

Tight Embeddings and Maps. Submanifolds of Geometrical Class Three in E N

Differential geometry is a field in which geometry is expressed in analysis, algebra, and calculations, and in which analysis and calculations are sometimes understood in intuitive steps that could be called geometric.

A new knot invariant

Recall that a torus knot is an honest knot (not isotopic to a circle in a plane) which lies in an unknot ted embedded torus M c I R 3. An unknot ted torus M has two embedded oriented circles ~ and q, representing homology classes 14[ and Ir/I of a preferred basis for the homology group over 7/, Ha (M), namely bounding in the interior and in the exterior o f M c l g 3 respectively. If 7:S1--~M~]R 3 is a torus knot then the absolute values o f the intersection numbers in Ha(M) of 7 with ~ and r/, are denoted p and q respectively and one can assume 2 < p < q. We may have to modify M (exchange inside and outside) wi thout moving 7, to see this. The integers p and q are coprime and they characterize the isotopy class of the torus knot completely but for reflection in a plane. There is the s tandard model ~p,q for t hep q t o r u s knot defined in Sect. 2, (2.2). For suitable M, ~, r/, 7, the intersection numbers are [~ l~ l r / [= I r / [ c~ ]~ l= l ~H0(M) , [7[ c-~l~l =p , tTl~l~l=q, and I71=ql~l-pl~/[.