Pinching and concordance theory
type: Item
instance of: scholarly article
title: Pinching and concordance theory
zbMATH Open document ID: 0783.53027
published in: Journal of Differential Geometry
publication date: 1994-03-13T00:00:00Z
review text: It is known that a complete simply connected manifold \(M\) whose sectional curvature \(sec(M)\) satisfies \(1/4 < sec(M) \leq 1\) is homeomorphic to a sphere (Rauch-Berger-Klingenberg). Beyond that, the diffeomorphism type of \(M\) is subject to the following condition (from ``Theorem A'' and ``Remark 1.8''). Let \(W(M)\) denote the space of smooth Morse functions on \(M\) having exactly two critical points. With the above hypotheses on \(M\), the map from \(W(M)\) to \((M\times M)\setminus \Delta\) (where \(\Delta =\)diagonal) which sends \(f\) to the ordered pair of its critical points has a section respecting the canonical involutions. (The involutions send \(f\) to \(-f\), and \((x,y)\) to \((y,x)\).) Methods from concordance theory and algebraic \(K\)-theory show that many exotic spheres do not satisfy the condition.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 57R60
zbMATH DE Number: 488228
zbMATH Keywords: sphere
zbMATH Keywords: Morse functions
zbMATH Keywords: exotic spheres
author: Michael S. Weiss
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454299
OpenAlex ID: W1488421603
Wikidata QID: Q115180135
DOI: 10.4310/JDG/1214454299
description: scientific article; zbMATH DE number 488228