RTG Mini-Course Perspectives in Geometry Series

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1 RTG Mii-Course Perspectives i Geometry Series Jacob Lurie Lecture III: The Cobordism Hypothesis (1/27/2009) A quick review of ideas from previous lectures: The origial defiitio of a -dimesioal tqft provided a diffeomorphism ivariat for a closed -maifold, which we could compute by cuttig the maifold ito pieces This cuttig was restricted, however, to closed submaifolds of codimesio 1 We would like cut alog a triagulatio of a maifold, but this requires the otio of a exteded tqft We discussed the otio of higher categories, amely (, )-categories, where all k- morphisms are ivertible, for k > There were several motivatios First was the eed to hadle more complicated ivariats obtaied from a tqft, eg cohomology classes Secod is that some diffeomorphism ivariats are quite subtle, eg, we eed more refied iformatio regardig cohomology classes that are Diff(M)-equivariat A example of a (, )-category is Bord, which is comprised of the followig data: objects: 1-morphisms: 2-morphisms: -morphisms: 0-maifolds bordisms of 0-maifolds (ie, 1-maifolds with boudary) bordisms of bordisms (ie, 2-maifolds with corers) bordisms of bordisms of of bordisms (ie, -maifolds with corers) ( + 1)-morphisms: diffeomorphisms of -maifolds ( + 2)-morphisms: isotopies of diffeomorphisms of -maifolds These otes are by o meas a substitute for the lectures, foud at Scribes: Braxto Collier, Parker Lowrey, Michael B Williams 1

2 I a (, )-category, we ca thik of the collectio of -morphisms as formig a topological space For example, i the case of Bord, the collectio of -morphisms is the classifyig space for -maifolds with corers Note that the category Bord is a symmetric mooidal category (ie, a -category) with = Also, there are variats of this category depedig o the type of maifolds uder cosideratios: maifolds without orietatio, with a framig, with a spi structure, etc We will be iterested i Bord fr where we cosider -framed maifolds Recall the Cobordism Hypothesis (CH): Theorem Let C be a symmetric mooidal (, )-category The there is a bijectio Fu (Bord fr, C) = {fully dualizable objects of C}, where Fu (Bord fr, C) is the collectio of symmetric mooidal fuctors Bord fr C The correspodece is give by Z Z( ) We will address the followig questios i this lecture: What does it mea to be fully dualizable? What happes i the uframed case? What is the relatio of ch to the Mumford cojecture? Note that the statemet of ch is otrivial, eve whe = 1, whe we cosider a exteded tqft (although it is easy if we cosider a ordiary tqft) Example Cosider Hom Bord1 (, ), the classifyig space for closed 1-maifolds (which are disjoit uios of copies of S 1 ) This cotais as a compoet the classifyig space for oe copy of S 1 This compoet classifies orieted circle budles, givig a atural iclusio CP Hom Bord1 (, ) Give a exteded tqft we d have Z : Bord 1 C, Hom Bord1 (, ) Hom C (1 C, 1 C ) ψ CP I the case of a ormal 1-dimesioal TQFT, we had Z( ) = X C, ad previous computatio showed that Z(S 1 ) = dim X via coev 1 C X X ev 1 C 2

3 I the exteded TQFT case, we have more iformatio Namely, the map ψ gives us a object dim X ad a actio o dim X by the symmetry group of the circle Our calculatio of Z(S 1 ) i the o-exteded case broke the symmetry group actio by cuttig apart the circle Breakig symmetry is aki to pickig a poit of CP, thus our previous method gave the value of our exteded TQFT o that poit This method is isufficiet i this ew world Example Let C be the (, 1)-category whose objects are associative C-algebras, ad whose morphisms Hom C (A, B) are chai complexes of A, B-bimodules (details o how to make this a (, 1) category are omitted) Here, every object is fully dualizable (defied below) For ow it suffices to say that each has a dual, amely the opposite algebra If we believe the Cobordism Hypothesis, the to each algebra A C we ca assig a field theory Z Sice our tesor product is the ormal tesor product, the C is the uit object ad Z(S 1 ) is a chai complex of C, C-bimodules, which are C-vector spaces This is the Hochschild chai complex CH (A) It is a classical fact that CH (A) carries a actio by a circle We ow tur to the otio of full dualizability Defiitio If C = Vect C, the a object V C is fully dualizable iff dim V < We d like to geeralize this otio to arbitrary categories, but what does fiite dimesioal mea? To aswer this, we recall the otio of dual objects Defiitio If V C = Vect C, we ca defie the dual V = Hom(V, C), ad there exists a pairig, called evaluatio: V V ev C, which defies a uiversal property This relatioship is symmetric iff dim V < There is also a caoical map, called coevaluatio: C coev V V These two maps are compatible, i that the followig compositios are the idetity: { id coev V V V ev id V V, V coev id id ev V V V V (1) This formalism makes sese i ay symmetric moodial category We exted this aively to ay symmetric moodial (, )-category (or -(, )-category for short): Defiitio Let C be a -(, )-category A object X C has a dual if it is dualizable i its homotopy category, ie if there is aother object (called its dual object) X ad 1-maps ev ad coev satisfyig the above relatios up to homotopy Defiitio Let C be a -(, 1)-category dualizable A object X C is fully dualizable if it is 3

4 What about for higher? Dualizability is ot eough; we eed the stricter otio of a object beig fully dualizable We approach this from a differet perspective i the followig digressio We start by aalyzig 2-categories The prototypical example of a (strict) 2-category is that of categories: objects are categories, morphisms are fuctors, ad 2-morphisms are atural trasformatios We wat to uderstad adjoit fuctors Defiitio If C ad D are categories, the two fuctors F : C D, G : D C are adjoit if there are atural bijectios Hom D (F (c), d) = Hom C (c, G(d)), for all c C ad d D We say that F is a left adjoit ad that G is a right adjoit Adjoitess works i ay 2-category: takig d = F (c), we have which gives a map ie, a atural trasformatio id d Hom D (d, d) = Hom C (c, G(d)) = Hom C (c, G F (d)), We ca use u to recover the adjuctio maps c (G F )(c), u : id C G F Hom D (d, d) = Hom C ((G F )(c), G(d)) u Hom C (c, G(d)) To guaratee that this is a bijectio, we wat iverses, that is, v : F G id D iducig iverse maps How might u ad v be related? Ideally, the followig compositios would be the idetity: { id v id u G G F G G, F u id F G F v id (2) F Note the similarity to the duality requiremet i (1) The moral is that adjoits are like duals We use the followig example to clarify this relatioship Example Let C be a -category There exists a 2-category BC whose cosistig of the followig data: objects: 1-morphisms: compositio: 4 oe object: Hom BC (, ) = C from C

5 The we have a correspodece {duals i C} {adjoits i BC} A pair of 1-morphisms F, G : are adjoit i BC iff F ad G are dual i C We ow ed our digressio ad make a defiitio Defiitio If C is a (, )-category, we say that 1-morphisms F : X Y, G : Y X are adjoit if there exist 2-morphisms u : id X G F, v : F G id Y such that the relatios (2) hold up to isomorphism We would like all our morphisms at all levels to have adjoits This turs out to be too strog sice i a (, )-category, this would imply that all -morphism are ivertible So we defie iductively Defiitio A (, )-category has adjoits if whe = 1, o requiremets; if 2, all morphisms have left ad right adjoits; if > 2, the (, 1)-category Hom C (X, Y ) has adjoits Defiitio Let C be a -(, )-category We say that C has duals if C has adjoits; each object has a dual as defied above Equivaletly, we say that BC has adjoits A modified versio of ch ow reads: Theorem If C had duals, the there is a bijectio Fu (Bord fr, C) = {objects of C} We ote that the category Bord fr has duals, the dual of a maifold is the same maifold with opposite orietatio ad read backwards For ay -(, )-category C, there is a largest subcategory C fd C such that C fd has duals (The fd stads for fully dualizable) It is obtaied by throwig out all items without adjoits I this way we obtai a equivalece of -groupoids from ch: Fu (Bord fr, C) = Fu (Bord fr, C fd ) = {objects of C} 5

6 Defiitio A object X C is fully dualizable if X C fd What does fully dualizable mea? Let C be a (, )-category = 1: A object X C is fully dualizable if C has dualizable; = 2: A object X C is fully dualizable if ad oly if X is dualizable ad ev ad coev have left adjoits; > 2: This is more complicated A object X C is fully dualizable if X is dualizable; ev ad coev have left adjoits; the adjoits have adjoits etc Defiitio Let M d be a maifold with d A -framig of M is a isomorphism T M R d = R Observe that the group O() acts o the set of -frames of ay maifold I this way we get a actio of O() o Bord fr I the statemet of ch, O() acts o Bord fr ad, by the Cobordism Hypothesis, o the objects of C Example If C has duals, the each X C has a correspodig X C There is a map X X which ca be thought of as a cotravariat fuctor C C op, a ivolutio of C, or a actio of O(1) o the objects of C Example Let C be a -groupoid For example, C = π X for some topological space X That C is a symmetric category meas that X is a E -space, meaig that X has a multiplicatio that is commutative ad associative up to (ay possible) homotopy I particular, π 0 X is a commutative mooid The C has duals iff X is grouplike, which meas π 0 X is a abelia group, iff X is a ifiite loop space, i which case X = Ω Y = Map((D, S 1 ), (Y, )) Here, O() acts o D i a atural way that preserves the boudary S 1, ad this iduces a actio o X I more sophisticated laguage, if C is a -groupoid, the O() acts via the J- homomorphism Suppose G is a topological group with a represetatio G O() The 6

7 we ca defie the otio of G-structure o a maifold M d (that is, a priciple G-budle P M such that the associated vector budle is (P R )/G = T M R d i the usual way); usig these maifolds, the category Bord G makes sese For example, Bord SO() Bord {1} Bord Spi() Bord O() = Bord ; = Bord fr ; = spi maifolds; = bordism theory for o-orieted maifolds; G acts o objects of C if C has duals We ca restate ch for G-maifolds: If C has duals, the there is a bijectio Fu (Bord G, C) = {objects of C} hg = MapG (EG, {objects of C}) The hg refers to homotopy fixed poits uder the G-actio, Map G meas G-equivariat maps, ad EG is a cotractible space with free G-actio Example Let = 1 ad G = O(1) The Fu (Bord O(1) 1, C) = {objects of C} h O(1), which is the space of symmetrically self-dual objects of C, that is, objects such that X = X There is a odegeerate pairig X X 1 If C = Vect C ad V C, this says that dim V < with a odegeerate symmetric biliear form 7