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Sullivan'sMinimal Models
RATIONALHOMOTOPYTHEORYSEMINAR
3 MARCH2012, CPR, RABAT
Professeur Agrégé-Docteur en Math
Master 1 en Sc de l'éducation, Univ. Rouen
mamouni.new.fr mamouni.myismail@gmail.com
My Ismail Mamouni, CPGE-CPR, Rabat
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 1 / 14
Aim of the talk
?Rational Homotopy Theory: Brief description ?Modèle minimal de Sullivan: Definition and main proprieties My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 2 / 14
Aim of the talk
?Rational Homotopy Theory: Brief description ?Modèle minimal de Sullivan: Definition and main proprieties My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 2 / 14
Prerequisites
Algebraic Topology basic knowledge of what is :
?Homology ?Homotopy My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 3 / 14
Prerequisites
(A,d)that means : ?A, module, vector space, algebra, ... ?A:=? kA k ?dk:Ak-→Ak+1such thatdk+1◦dk=0 ?We writed:A-→A, wheredk=d|Akandd2=0
Graded & differential
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 3 / 14
Prerequisites
(A,d)that means : ?Hk(A,d) :=kerdk+1Imdk ?H?(A,d) :=? kH k(A,d)
Cohomology
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 3 / 14
Prerequisites
?Two continuous mapsf,g:Sk-→Xare calledhomotopic when there is continuous deformationH:Sk×[0,1]-→X , such thatH(.,0) =f,H(.,1) =g. We obtain an equivalence relation≂ ?The homotopy groups :πk(X) :=C(Sn,X)/≂and
π?(X) :=?
kπ k(X). ?Type of homotopy :XandYare called with thesame type of homotopy whenπ?(X)≂=π?(Y).
Homotopy
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 3 / 14
Glossary
A simply connected spaceXis calledrationalif the following is satisfied.
π?(X)is aQ-vector space.
N.B :π?(X)?Qis aQ-vector space
Rational Space
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 4 / 14
Glossary
LetXbe a simply connected space. Arationalization ofXis simply connected and rational spaceY, such that :
π?(X)?Q≂=π?(Y)
H ?(X;Q)≂=H?(Y;Q)
Rationalization
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 4 / 14
Glossary
Any simply connected spaceXadmits an unique (up to homo- topy) CW-complex rationalization
Theorem, [FHT]
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 4 / 14
Glossary
Therational homotopy typeof a simply connected space X is the homotopy type of its rationalization.
Definition
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 4 / 14
Rational Homotopy Theory
Rationalhomotopytheory is the study of rational homotopytypes of spaces and of the properties of spaces and maps that are invariant under rational homotopy equivalence.
What it is it
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 5 / 14
Founders in 1967
work in topology, both algebraic and geomet- ric, and on dynamical systems
Doctoral advisor : William Browder
Wolf Prize in Mathematics (2010)
Leroy P. Steele Prize (2006)
National Medal of Science (2004)
Denis Sullivan (1941- ), CUNY-SUNY, USA
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 6 / 14
Founders in 1967
the "prime architect" of higher algebraic K- theory
Doctoral advisor : Raoul Bott
Fields Medal (1978)
Cole Prize (1975)
Putnam Fellow (1959)
Daniel Quillen (1940-2011), Oxford
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 6 / 14
Model of Sullivan
Acommutative graded differential algebraover the rational num- bers is a gradedQ- algebra(A,d)such that ?ab= (-1)|a||b|ba (?)d(ab) = (da).b+ (-1)|a|b.dafor alla,b?A
In particular :
?y2=0when|y|odd ?xy=yxwhen|x|even CGDA My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
From any differential and gradedQ-vector spaceV, we define the cgda
ΛVdenotes defined by
ΛV=TV??v?w-(-1)|v||w|w?v?
whereTVdenotes the tensor algebra overV. The differential onΛVis naturally extended from that ofVwith respecting the condition (*) called of nilpotence or of Leibniz
How to build it
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
Our cgda is called amodel of Sullivanwhen there exists some well ordered basis(vα)α?IofVsuch that dvα?Λ{vβ, β < α}
Model of Sullivan
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
The model of Sullivan is calledminimalwhen
Minimal model
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
The minimal model is calledellipticwhen bothVandH?(ΛV,d) are finite dimensional , in this case dx 1=0 anddxj?Λ(x1,...,xj-1) forj≥2
Elliptic model
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
Any simply connected space have a minimal model of Sullivan, (ΛV,d)(unique up to isomorphism of cgda), who models its co- homology and homotopy as follows :
Hk(X;Q)≂=Hk(ΛV,d)
k(X)?Q≂=Vk
D. Sullivan, [Su]
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
For the odd sphere :S2k+1, the model is the form(Λ{x},0)with |x|=2k+1. So
πn(S2k+1)≂=Zifn=2k+1≂=0 if not
Basic Examples
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Model of Sullivan
For the even sphere :S2k, the model is the form(Λ{x,y},d)with |x|=2k,|y|=4k-1,dy=x2. So πn(S2k)≂=Zifn=2k≂=Zifn=4k-1≂=0 if not
Basic Examples
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 7 / 14
Special Denotations
?In general for anyx?ΛV, we have dx=?k≥0β klenght=k????y1...yk? |yi|odd? |xi|evenx
αii??
kΛ kVodd?ΛVeven ?HenceΛVis bi-graded as followsΛV=? p,q(ΛpVodd?ΛVeven)q. pword-lengthgraduation andq: degreegraduation. ?Λ≥kV:=? p≥kΛpVandΛ+V:= Λ≥1V ?When(ΛV,d)is a simply elliptic minimal model, we have dV?Λ≥2V= Λ+V.Λ+V My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 8 / 14
Special Denotations
?Λ(V?W) = ΛV?ΛW ?When|y|odd,Λy={ay+b;a,b?Q}=Q1[y] ?When|x|even,Λx={? ka kxk;a,b?Q}=Q[x]
Simple Conclusions
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 8 / 14
Special Denotations
WhendVeven=0
dV odd?ΛVeven
Pure Model
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 8 / 14
Special Denotations
When dVeven=0
Hyperelliptic Model
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 8 / 14
Special Denotations
WhenV=U?W
dU=0 dW?ΛU
Two Stage Model
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 8 / 14
Example of Algebraization
For anyellipticandsimply connectedtopolog-
ical spaceX, we have
Hilali Conjecture (1990)
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 9 / 14
Example of Algebraization
For anyellipticmodel of Sullivan,(ΛV,d)we have
Algebraic version
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 9 / 14
Example of Algebraization
For the sphereSnwe have seen that
dimV=1 or 2 , and its well known that
H0(Sn;Q) =Hn(Sn;Q) =QandHi(X;Q) =0
for all other i.
Simple example in which it holds
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 9 / 14
Euler-Poincaré characteristic
For any 1-connected elliptic model(ΛV,d)we define two invari- ants. One cohomological : c:=?k≥0(-1)kdimHk(ΛV,d) and anotherhomotopic :
π:=?k≥0(-1)kdim(Vk)
Definition
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 10 / 14
Euler-Poincaré characteristic
we have the following :
Morever,
χc>0??χπ=0
In this case
H?(ΛV,d) =Heven(ΛV,d)
S. Halperin, [Ha83]
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 10 / 14
Euler-Poincaré characteristic
For any graded vector spaceA, the Euler-Poincaré characteristic is defined as follows
χ(A) :=?k≥0(-1)kdimAk
So,
χc=χ(H?(ΛV,d)), χπ=χ(V)
N.B
χ(H?(A,d)) =χ(A)
Generalisation
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 10 / 14
Euler-Poincaré characteristic
?Asχπ=dimVeven-dimVodd, we putdimVeven=pand dimVodd=n+p, so ?χπ=-panddimV=2n+p ?p=0??H?(ΛV,d) =Heven(ΛV,d) ?p?=0??dimH?(ΛV,d) =2dimHeven(ΛV,d)
Util Remark
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 10 / 14
Toral Rank
rk0(X) :=The largest integern≥1 for whichXadmits an almost-freen-torus action
Definition
The equality holds whenXis pure
C. Allday & Halperin,[AH78]
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 11 / 14
Toral Rank
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 11 / 14
Toral Rank
For anyellipticandsimply connectedtopolog-
ical spaceX, we have dimH?(X;Q)≥2rk0(X)
Toral Rank Conjecture (TRC), S.Halperin (1986)
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 11 / 14
Toral Rank
TRC : dimH?(X;Q)≥2p-ε
Conj. H : dimH?(X;Q)≥2n+p
?Conj H+2n+p≥2p-ε=?CRT
The link between ConjH & TRC
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 11 / 14
Formal dimension
For an elliptic spaceX, we put
fd(X) :=max{k,Hk(X,Q)?=0}
Definition
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 12 / 14
Formal dimension
IfXis a 1-connectedand elliptic space of minimal Sullivan model (ΛV,d), then fd(X)≥dimV
J. Friedlander and S. Halperin, [FH79]
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 12 / 14
Formal dimension
IfXis a 1-connected and elliptic manifold, thenfd(X) =dimX
Best known result, losed source
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 12 / 14
Formal dimension
There exists a special homogeneousbasisx1,...,xnofVeven and a basisy1,...yn+pofVoddsuch that : n? n+p? n+p? i=1|yi| -n? i(|xi| -1) =fd(X)
J. Friedlander and S. Halperin, [FH79]
My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 12 / 14
Main References of RHT
C. Allday & S. Halperin,Lie group actions on espace s of finite rank,
Quar. J. Math. Oxford28(1978), 69-76.
D.E. Blair and S.I. Goldberg,Topology of almost contact manifolds, Journal of Differential GeometryVol.1(1967), Intelpress, 347-354. J. Friedlander and S. Halperin,An arithmetic characterization of the rational homotopy groups of certain espace s,
Invent. Math.53
(1979), 117-133. Y. Félix, S. Halperin & J.-C. Thomas,Rational Homotopy Theory,
Graduate Texts in Mathematics, vol. 205,
Springer-Verlag, 2001.
S. Halperin,Finitness in the minimal models of Sullivan,Transc. AMS
230(1983), 173-199.
D. Sullivan,Infinitesimal computations in topology,Publications
Mathématiques de l'IHÉS
, 47 (1977), 269-331 My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 13 / 14 My Ismail Mamouni (CPGE-CPR Rabat)RHT Seminar, CPR RabatSullivan's Minimal Models 14 / 14quotesdbs_dbs11.pdfusesText_17
Sullivans Minimal Models - RHT Seminar, CPR Rabat