Algebraic Topology of PDEs Qusay Soad Abdul-Aziz Al-Zamil 2011 MIMS EPrint

92 Pages·2017·0.67 MB·English

Algebraic Topology of PDEs
Al-Zamil, Qusay Soad Abdul-Aziz
2011
MIMS EPrint: 2011.109
Manchester Institute for Mathematical Sciences
School of Mathematics
The University of Manchester
Reports available from: http://eprints.maths.manchester.ac.uk/
And by contacting: The MIMS Secretary
School of Mathematics
The University of Manchester
Manchester, M13 9PL, UK
ISSN 1749-9097
ALGEBRAIC TOPOLOGY OF PDES
A THESIS SUBMITTED TO THE UNIVERSITY OF MANCHESTER
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
IN THE FACULTY OF ENGINEERING AND PHYSICAL SCIENCES
2011
QusaySoadAbdul-AzizAl-Zamil
SchoolofMathematics
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Contents
Abstract 5
Declaration 6
Copyright 7
Publications 8
Acknowledgements 9
1 Introduction 10
2 Preliminaries 14
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Groupactionsonsmoothmanifolds . . . . . . . . . . . . . . . . . . . . 14
2.2.1 AveragingwithrespecttoacompactLiegroupaction . . . . . . . 15
2.3 Hodgetheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.1 Hodgetheoremformanifoldswithoutboundary . . . . . . . . . . 16
2.3.2 Witten’sdeformationofHodgetheoremwhen∂M =0/ . . . . . . 17
2.3.3 Hodge-Morrey-Friedrichstheoremformanifoldswithboundary . 18
2.3.4 ModifiedHodge-Morrey-Friedrichstheorem . . . . . . . . . . . 19
2.4 TheDirichlet-to-Neumann(DN)operatorfordifferentialforms . . . . . . 21
2.4.1 DN-operatorΛandcohomologyringstructure . . . . . . . . . . 23
3 Witten-Hodgetheoryandequivariantcohomology 25
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2 Witten-Hodgetheoryformanifoldswithoutboundary . . . . . . . . . . . 26
3.3 Witten-Hodgetheoryformanifoldswithboundary . . . . . . . . . . . . 31
3.3.1 Thedifficultiesiftheboundaryispresent . . . . . . . . . . . . . 32
3.3.2 Ellipticboundaryvalueproblem . . . . . . . . . . . . . . . . . . 33
3.3.3 Decompositiontheorems . . . . . . . . . . . . . . . . . . . . . . 37
3.3.4 RelativeandabsoluteX -cohomology . . . . . . . . . . . . . . 41
M
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3.4 Relationwithequivariantcohomologyandsingularhomology . . . . . . 43
3.4.1 X -cohomologyandequivariantcohomology . . . . . . . . . . . 43
M
3.4.2 X -cohomologyandsingularhomology . . . . . . . . . . . . . . 46
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3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4 InteriorandboundaryportionsofX -cohomology 51
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4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.2 TheintersectionofH± (M)andH± (M) . . . . . . . . . . . . . . . 51
X ,N X ,D
M M
4.3 X -cohomologyinthestyleofDeTurck-Gluck . . . . . . . . . . . . . . 54
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4.3.1 RefinementoftheX -Hodge-Morrey-Friedrichsdecomposition . 55
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4.3.2 Interiorandboundaryportionsanddecompositiontheorems . . . 55
4.3.3 Interiorandboundaryportionsofequivariantcohomology . . . . 57
4.4 Conclusionsandgeometricopenproblem . . . . . . . . . . . . . . . . . 60
5 GeneralizedDN-operatoroninvariantdifferentialforms 62
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Preparingtothegeneralizedboundarydata . . . . . . . . . . . . . . . . . 62
5.3 X -DNoperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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5.4 Λ operator,X -cohomologyandequivariantcohomology . . . . . . . 72
X M
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5.5 RecoveringX -cohomologyfromtheboundarydata(∂M,Λ ) . . . . . 73
M X
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5.5.1 RecoveringthelongexactX -cohomologysequenceof(M,∂M) 73
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5.5.2 RecoveringtheringstructureoftheX -cohomology . . . . . . . 76
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5.6 Conclusionsandtopologicalopenproblem . . . . . . . . . . . . . . . . . 80
6 X -Harmoniccohomologyonmanifoldswithboundary 83
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6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 X -Harmoniccohomologyisomorphismtheorem . . . . . . . . . . . . . 84
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6.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Bibliography 89
(Wordcount: 12034)
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The University of Manchester
QusaySoadAbdul-AzizAl-Zamil
DoctorofPhilosophy
AlgebraicTopologyofPDES
December15,2011
Weconsideracompact,oriented,smoothRiemannianmanifoldM (withorwithoutbound-
ary) and we suppose G is a torus acting by isometries on M. Given X in the Lie algebra of
GandcorrespondingvectorfieldX onM,onedefinesWitten’sinhomogeneouscobound-
M
ary operator d = d+ι : Ω± → Ω∓ (even/odd invariant forms on M) and its adjoint
XM XM G G
δ .
X
M
First,Witten[35]showedthattheresultingcohomologyclasseshaveX -harmonicrep-
M
resentatives (forms in the null space of ∆ =(d +δ )2), and the cohomology groups
X X X
M M M
are isomorphic to the ordinary de Rham cohomology groups of the set N(X ) of zeros
M
of X . The first principal purpose is to extend Witten’s results to manifolds with bound-
M
ary. In particular, we define relative (to the boundary) and absolute versions of the X -
M
cohomology and show the classes have representative X -harmonic fields with appropri-
M
ate boundary conditions. To do this we present the relevant version of the Hodge-Morrey-
Friedrichs decomposition theorem for invariant forms in terms of the operators d and
X
M
δ ;theproofinvolvesshowingthatcertainboundaryvalueproblemsareelliptic. Wealso
X
M
elucidatetheconnectionbetweentheX -cohomologygroupsandtherelativeandabsolute
M
equivariant cohomology, following work of Atiyah and Bott. This connection is then ex-
ploited to show that every harmonic field with appropriate boundary conditions on N(X )
M
has a unique corresponding an X -harmonic field on M to it, with corresponding bound-
M
ary conditions. Finally, we define the interior and boundary portion of X -cohomology
M
and then we define the X -Poincare´ duality angles between the interior subspaces of X -
M M
harmonicfieldsonM withappropriateboundaryconditions.
Second,In2008,BelishevandSharafutdinov[9]showedthattheDirichlet-to-Neumann
(DN) operator Λ inscribes into the list of objects of algebraic topology by proving that the
deRhamcohomologygroupsaredeterminedbyΛ.
In the second part of this thesis, we investigate to what extent is the equivariant topology
ofamanifolddeterminedbyavariantoftheDNmap?. Basedontheresultsinthefirstpart
above, we define an operator Λ on invariant forms on the boundary ∂M which we call
X
M
theX -DNmapandusingthiswerecoverthelongexactX -cohomologysequenceofthe
M M
topologicalpair(M,∂M)fromanisomorphismwiththelongexactsequenceformedfrom
the generalized boundary data. Consequently, This shows that for a Zariski-open subset
of the Lie algebra, Λ determines the free part of the relative and absolute equivariant
X
M
cohomology groups of M. In addition, we partially determine the mixed cup product of
X -cohomology groups from Λ . This shows that Λ encodes more information about
M X X
M M
theequivariantalgebraictopologyofM thandoestheoperatorΛon∂M. Finally,weeluci-
datetheconnectionbetweenBelishev-Sharafutdinov’sboundarydataon∂N(X )andours
M
on∂M.
Third, based on the first part above, we present the (even/odd) X -harmonic cohomol-
M
ogy which is the cohomology of certain subcomplex of the complex (Ω∗,d ) and we
G XM
provethatitisisomorphictothetotalabsoluteandrelativeX -cohomologygroups.
M
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Declaration
No portion of the work referred to in this thesis has been
submitted in support of an application for another degree or
qualification of this or any other university or other institute
oflearning.
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Copyright
i. The author of this thesis (including any appendices and/or schedules to this thesis)
owns certain copyright or related rights in it (the “Copyright”) and s/he has given
The University of Manchester certain rights to use such Copyright, including for
administrativepurposes.
ii. Copies of this thesis, either in full or in extracts and whether in hard or electronic
copy, may be made only in accordance with the Copyright, Designs and Patents
Act 1988 (as amended) and regulations issued under it or, where appropriate, in
accordance with licensing agreements which the University has from time to time.
Thispagemustformpartofanysuchcopiesmade.
iii. TheownershipofcertainCopyright,patents,designs,trademarksandotherintellec-
tual property (the “Intellectual Property”) and any reproductions of copyright works
in the thesis, for example graphs and tables (“Reproductions”), which may be de-
scribed in this thesis, may not be owned by the author and may be owned by third
parties. Such Intellectual Property and Reproductions cannot and must not be made
available for use without the prior written permission of the owner(s) of the relevant
IntellectualPropertyand/orReproductions.
iv. Further information on the conditions under which disclosure, publication and com-
mercialisation of this thesis, the Copyright and any Intellectual Property and/or Re-
productions described in it may take place is available in the University IP Policy
(seehttp://www.campus.manchester.ac.uk/medialibrary/policies/
intellectual-property.pdf), in any relevant Thesis restriction declarations de-
posited in the University Library, The University Library’s regulations (see http:
//www.manchester.ac.uk/library/aboutus/regulations)andinTheUniver-
sity’spolicyonpresentationofTheses.
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Publications
Thelastfourchaptersofthisthesiswhichcontainthenewresultsarebasedonthefollowing
publications:
• Chapter 3 is based on the paper “Witten-Hodge theory for manifolds with boundary
and equivariant cohomology.” (with J. Montaldi) [2], Differential Geometry and its
Applications(2011),doi:10.1016/j.difgeo.2011.11.002.
• Chapter4isbasedon[2]andthepaper“GeneralizedDirichlettoNeumannoperator
on invariant differential forms and equivariant cohomology.” (with J. Montaldi) [3],
TopologyanditsApplications,159,823–832,2012,doi:10.1016/j.topol.2011.11.052.
• Chapter5isbasedonthepaper[3].
• Chapter6isbasedonthePreprint“X -HarmonicCohomologyandEquivariantCo-
M
homologyonRiemannianManifoldsWithBoundary”[4].
Itwasagreathonourformetopresenttheresultsof[2]andthemaingoalof[3]inthe25th
BritishTopologyMeetingwhichtookplaceinMertonCollageattheUniversityofOxford,
6th−8th September2010(see,http://www.maths.ox.ac.uk/groups/topology/btm2010).
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Acknowledgements
Firstandforemost,mygreatthanksgotothealmightyAllahwhomadethisworkpossible.
I am heartily thankful to my supervisor, Professor James Montaldi, whose encourage-
ment, guidance and support from the initial to the final level enabled me to develop an
understanding of this huge subject. He provided enthusiasm, inspiration, sound advice,
perfectteaching,excellentsupervision,andlotsofgreatideas. Healwaysseemedtoknow
what I needed and I was shown through him how a mathematician should be. I offer him
mydeepestthanks.
I highly appreciate the valuable comments given by the examiners on the whole thesis
whichimprovedtheexpositionofthethesis. Thankyou.
I wish to express my gratitude to Professor Bill Lionheart from University of Manch-
ester for his suggestion of the references [11] and [33]. I am also thankful to Dr Clay-
ton Shonkwiler from University of Georgia in the USA for his valuable discussions with
me about the cohomology ring structures. In addition, I would like to thank Dr Marta
Mazzocco from Loughborough University in the UK who introduced me to the field of
isomonodromicdeformationsystemsin2008.
I humbly thank all my old and new friends and all the exceptional people involved in
mylifewhowishgoodthingsonmeandinthisoccasionIwouldliketomentionmywife’s
parentswhoalwayskeepsupportingme.
Aspecialthanksshouldgotomywifeforherpatienceinthepastfouryearsandforher
continuous support to achieve my best. I should not forget to give warm hug with sweet
thanks to my beloved Shadan (my daughter) and Muhammad (my son) who have meant a
lottomylife.
I offer my sincerest thanks to my brother who has sacrificed a lot in the past years to
push me to be better and encourage me towards my best. Thank you for looking after our
parents during my absence. In addition, I would like to thank my sister for her constant
supporttomeandIwishherabrightandpromisingfutureinMedicalCollege.
Lastly, and most importantly, I wish to express my deep and sincere gratitude to my
mother and father. They bore me, raised me, supported me, taught me, and loved me.
Thank you for reading to me as a kid. I graciously thank them for everything they have
done.
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