Metrics of positive scalar curvature and generalised Morse functions, Part I

105 Pages·2011·1.15 MB·English

by  Mark Walsh

MEMOIRS
of the
American Mathematical Society
Number 983
Metrics of Positive Scalar Curvature
and Generalised Morse Functions,
Part I
Mark Walsh
January 2011 • Volume 209 • Number 983 (second of 5 numbers) • ISSN 0065-9266
American Mathematical Society
Number 983
Metrics of Positive Scalar Curvature
and Generalised Morse Functions,
Part I
Mark Walsh
January2011 • Volume209 • Number983(secondof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Walsh,MarkP.
MetricsofpositivescalarcurvatureandgeneralisedMorsefunctions,partI/MarkWalsh.
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ISBN978-0-8218-5304-7(alk. paper)
1.Curvature. 2.Morsetheory. 3.Riemannianmanifolds. 4.Algebraictopology. I.Title.
QA645.W35 2011
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Contents
Introduction ix
0.1. Background ix
0.2. Main results xii
0.3. The connection with generalised Morse functions and Part II xiv
0.4. Acknowledgements xvii
Chapter 1. Definitions and Preliminary Results 1
1.1. Isotopy and concordance in the space of metrics
of positive scalar curvature 1
1.2. Warped product metrics on the sphere 2
1.3. Torpedo metrics on the disk 4
1.4. Doubly warped products and mixed torpedo metrics 5
1.5. Inducing a mixed torpedo metric with an embedding 9
Chapter 2. Revisiting the Surgery Theorem 11
2.1. Surgery and cobordism 11
2.2. Surgery and positive scalar curvature 12
2.3. Outline of the proof of Theorem 2.3 14
2.4. Part 1 of the proof: Curvature formulae for the first deformation 15
2.5. Part 2 of the proof: A continuous bending argument 17
2.6. Part 3 of the proof: Isotoping to a standard product 26
2.7. Applying Theorem 2.3 over a compact family of psc-metrics 29
2.8. The proof of Theorem 2.2 (The Improved Surgery Theorem) 31
Chapter 3. Constructing Gromov-Lawson Cobordisms 35
3.1. Morse Theory and admissible Morse functions 35
3.2. A reverse Gromov-Lawson cobordism 40
3.3. Continuous families of Morse functions 41
Chapter 4. Constructing Gromov-Lawson Concordances 45
4.1. Applying the Gromov-Lawson technique over a pair of cancelling
surgeries 45
4.2. Cancelling Morse critical points: The Weak and Strong Cancellation
Theorems 47
4.3. A strengthening of Theorem 4.2 48
4.4. Standardising the embedding of the second surgery sphere 49
Chapter 5. Gromov-Lawson Concordance Implies Isotopy for Cancelling
Surgeries 51
5.1. Connected sums of psc-metrics 51
5.2. An analysis of the metric g(cid:2)(cid:2), obtained from the second surgery 51
v
vi CONTENTS
5.3. The proof of Theorem 5.1 53
Chapter6. Gromov-LawsonConcordanceImpliesIsotopyintheGeneralCase 65
6.1. A weaker version of Theorem 0.8 65
6.2. The proof of the main theorem 67
Appendix: Curvature Calculations from the Surgery Theorem 71
Bibliography 79
Abstract
It is well known that isotopic metrics of positive scalar curvature are concor-
dant. Whetherornottheconverseholdsisanopenquestion,atleastindimensions
greater than four. We show that for a particular type of concordance, constructed
usingthesurgerytechniquesofGromovandLawson,thisconverseholdsinthecase
of closed simply connected manifolds of dimension at least five.
ReceivedbytheeditorJanuary8,2009.
ArticleelectronicallypublishedonJune8,2010;S0065-9266(10)00622-8.
2000 MathematicsSubjectClassification. Primary53-02,55-02.
Keywordsandphrases. Positivescalarcurvature,MorseTheory,surgery,cobordism,isotopy,
concordance.
(cid:2)c2010 American Mathematical Society
vii

Introduction
0.1. Background
Let X be a smooth closed manifold. We denote by Riem(X) the space of
Riemannian metrics on X. Contained inside Riem(X), as an open subspace, is
the space Riem+(X) which consists of metrics on X which have positive scalar
curvature (psc-metrics). The problem of whether or not this space is non-empty
(i.e. X admits a psc-metric) has been extensively studied; see [11], [12], [25] and
[28]. In particular, when X is simply connected and dimX = n ≥ 5, it is known
that X always admits a psc-metric when X is not a spin manifold and, in the case
whenX isspin, itadmitssuchametricifandonlyiftheindexα(X)∈KO ofthe
n
Dirac operator vanishes; see [25] and [28]. Considerably less is known about the
topology of the space Riem+(X), even for such manifolds as the sphere Sn. For
example, it is known that Riem+(S2) is contractible (as is Riem+(RP2)); see [25],
but little is known about the topology of Riem+(Sn) when n≥3.
In this paper, we will focus on questions relating to the path-connectivity of
Riem+(X). Unless otherwise stated Riem+(X) is assumed to be non-empty. Met-
rics which lie in the same path component of Riem+(X) are said to be isotopic.
Twopsc-metrics g andg onX are saidtobeconcordantif thereisapsc-metric g¯
0 1
on the cylinder X×I (I =[0,1]) which near X×{0} is the product g +dt2, and
0
which near X×{1} is the product g +dt2. It is well known that isotopic metrics
1
are concordant; see Lemma 1.4 below. It is also known that concordant metrics
need not be isotopic when dimX = 4, where the difference between isotopy and
concordance is detected by the Seiberg-Witten invariant; see [26]. However, in the
case when dimX ≥ 5, the question of whether or not concordance implies isotopy
is an open problem and one we will attempt to shed some light on.
Before discussing this further, it is worth mentioning that the only known
method for showing that two psc-metrics on X lie in distinct path components of
Riem+(X), is to show that these metrics are not concordant. For example, in-
dex obstruction methods may be used to exhibit a countable infinite collection of
distinct concordance classes for X = S4k−1 with k > 1, implying that the space
Riem+(S4k−1) has at least as many path components; see [4] (or Example 0.1 be-
low for the case when k = 2). In [3], the authors show that if X is a connected
spin manifold with dimX = 2k+1 ≥ 5 and π (X) is non-trivial and finite, then
1
Riem+(X)hasinfinitelymanypathcomponentsprovidedRiem+(X)isnon-empty.
Again, this is done by exhibiting infinitely many distinct concordance classes. For
a general smooth manifold X, understanding π (Riem+(X)) is contingent on an-
0
swering the following open questions.
(i) Are there more concordance classes undetected by index theory?
(ii) When are concordant metrics isotopic?
ix