Differential Geometry of Projective Limits of Manifolds

Transcription

1 Unversty of Colorado, Boulder CU Scholar Mathematcs Graduate Theses & Dssertatons Mathematcs Sprng Dfferental Geometry of Projectve Lmts of Manfolds Matthew Gregory Krupa Unversty of Colorado at Boulder, Follow ths and addtonal works at: Part of the Geometry and Topology Commons Recommended Ctaton Krupa, Matthew Gregory, "Dfferental Geometry of Projectve Lmts of Manfolds" (2016). Mathematcs Graduate Theses & Dssertatons Ths Dssertaton s brought to you for free and open access by Mathematcs at CU Scholar. It has been accepted for ncluson n Mathematcs Graduate Theses & Dssertatons by an authorzed admnstrator of CU Scholar. For more nformaton, please contact cuscholaradmn@colorado.edu.

2 DIFFERENTIAL GEOMETRY OF PROJECTIVE LIMITS OF MANIFOLDS by MATTHEW GREGORY KRUPA B.S., Calforna State Polytechnc Unversty, Pomona, 2009 M.S., Unversty of Colorado, Boulder, 2013 A thess submtted to the Faculty of the Graduate School of the Unversty of Colorado n partal fulfllment of the requrement for the degree of Doctor of Phlosophy Department of Mathematcs 2016

3 Ths thess enttled: Dfferental Geometry of Projectve Lmts of Manfolds wrtten by Matthew Gregory Krupa has been approved for the Department of Mathematcs Dr. Markus Pflaum Dr. Jeanne N. Clelland Date The fnal copy of ths thess has been examned by the sgnatores, and we fnd that both the content and the form meet acceptable presentaton standards of scholarly work n the above mentoned dscplne.

4 Krupa, Gregory Matthew (Ph.D., Mathematcs) Dfferental Geometry of Projectve Lmts of Manfolds Thess drected by Professor Markus Pflaum The nascent theory of projectve lmts of manfolds n the category of locally R-rnged spaces s expanded and generalzatons of dfferental geometrc constructons, ntons, and theorems are developed. After a thorough ntroducton to lmts of topologcal spaces, the study of lmts of smooth projectve systems, called promanfolds, commences wth the ntons of the tangent bundle and the study of locally cylndrcal maps. Smooth mmersons, submersons, embeddngs, and smooth maps of constant rank are ned, ther theores developed, and counter examples showng that the nverse functon theorem may fal for promanfolds are provded along wth potental substtutes. Subsets of promanfolds of measure 0 are ned and a generalzaton of Sard s theorem for promanfolds s proven. A Whtney embeddng theorem for promanfolds s gven and a partal unqueness result for ntegral curves of smooth vector felds on promanfolds s found. It s shown that a smooth manfold of dmenson greater than one has the fnal topology wth respect to ts set of C 1 - arcs but not wth respect to ts C 2 -arcs and that a partcular class of promanfolds, called monotone promanfolds, have the fnal topology wth respect to a class of smooth topologcal embeddngs of compact ntervals termed smooth almost arcs.

5 Contents Lst of Fgures x 1 Introducton Notaton and Termnology for Elementary Concepts Lfts, Factorzatons, Fbratons, and Sequences Notaton for Indexed Collectons Germs, and Submersons and Immersons of Germs Dervatons Lmts n Set and Top Introducton to Systems, (Co)Cones, and (Co)Lmts Systems Cones and Cocones Lmts and Colmts Cofnal Subsystems Examples Systems Havng Propertes Locally/Eventually/Cofnally Non-empty Lmts and Surjectvty of Projectons Generalzed Mttag-Leffler Lemma Surjectvty of Projectons A Suffcent Condton for a Non-empty Lmt Subsets of Inverse Systems Inverse Systems of Subsets Lmt of a System of Subsets Relatonshps Between a Subset and ts Projectons Increasng and Decreasng Representatons of Subsets Canoncal Representaton of Open and Closed Subsets of a Lmt Topologes of Lmts of Drected Systems Connectedness and Local Connectedness Path-connectedness and Local Path-connectedness Partal µ -Sectons Local Compactness v

6 3 Inverse System Morphsms Lmts of System Morphsms Propertes of Inverse System Morphsms and Ther Lmts Characterzaton of Closed Vector Subspaces of R N Equvalence Transformatons Equvalent Systems Have Isomorphc Lmts Examples Canoncal Lmt Map nto a Subsystem Surjectvty Counter Examples Convergent Seres n R Identty Maps and Lmts Injectvty and Cofnalty of Order Morphsms The Canoncal Sheaf The Restrcton Sheaf Bump Functons Profnte Dmensonal Manfolds Some Basc Propertes of Promanfolds Subpromanfolds Neghborhood Bass at a Pont Products of Promanfolds Smooth and Locally Cylndrcal Maps Locally Cylndrcal Maps Canoncal Maps Induced by a Map Between Promanfolds Smoothness and Local Cylndrcty Smoothness at a Pont Nowhere Roughly Cylndrcal Maps A Suffcent Condton for Rough Cylndrcty Suffcent Condtons for Local Cylndrcty Trvally Cylndrcal Maps and Inverse System Morphsms Cylndrcty and Compactness Smooth Maps that are Not Lmts of Inverse System Morphsms A Characterzaton of Smooth Maps that Arse as Lmts of Inverse System Morphsms Smooth Parttons of Unty Partal Generalzaton of the Boman Theorem for Promanfolds The Tangent Space at a Pont The Tangent Map at a Pont Dmenson of a Promanfold at a Pont Canoncal Identfcatons of the Tangent Space at a Pont Propertes of the Tangent Map at a Pont Identfyng Tangent Spaces va Smooth Maps v

7 7.5 Knematc Tangent Vectors Identfyng Lnear Independence The Tangent Bundle Vector Felds Integral Curves Example: Infnte and Hgher Order Tangent Bundles Fber Bundles Generalzed Cones and Generalzed Inverse System Morphsms Generalzed Cones Constructng Maps Between Subsets of Lmts Strongly Defned Lmt Weakly Defned Lmt Generalzed Inverse System Morphsms A Smooth Map s Almost the Lmt of an Inverse System Morphsm Submersons, Immersons, and Isomersons Pontwse Immersons from Smooth Manfolds Sectonal Submersons Rank ν -Constant Rank ν -Regular Immersons The Canoncal Form at a Pont of a ν -Regular Immerson Smoothness of Maps Composed on the Rght Usng Composton to Determne Equalty of Contnuous Maps Components of Fbers of ν -Regular Immersons Contnuty of Maps Composed on the Rght Smooth Embeddngs of Promanfolds The Whtney Embeddng Theorem for Promanfolds Pontwse Submersons Sard s Theorem Subsets of Measure Sard s Theorem for Promanfolds The Inverse Functon Theorem Counterexamples Sub-Promanfold Inverse Functon Theorem for Fbrated Promanfolds Local Injectvty and Vector Feld Germ Submersons and Immersons Substtute Inverse Functon Theorems v

8 14 Coherence wth C p -Paths and C p -Embeddngs of Intervals Coherence wth C p -arcs (p 1) Non-Coherence wth C p -Embeddngs (p > 1) of Intervals Non-Coherence of R d (d nfnte) wth C p -Arcs (p 1) Coherence of R d (d < ) wth Smooth Almost Arcs Coherence of R d (d < ) wth C 1 -Embeddngs of Intervals Characterzaton of Local Path-Connectedness Coherence and 0-Dmensonalty Coherence wth C p -Arcs (p > 0) Constructons of Curves nto Monotone Promanfolds Smooth Almost Arcs Fast Convergng Sequences Curves Through Sequences Through Possbly Non-Convergent Sequences Arcs Through Convergent Sequences If the Sequence s Eventually n Every µ -Fber of m If No Subsequence s Eventually n Every µ -Fber of m Smooth Topologcal Embeddngs of R Through Sequences Monotone Promanfolds Coherence wth Smooth Almost Arcs Fnte Dmensonal Monotone Promanfolds Connectedness of Monotone Promanfolds A Characterzaton of Dom F a Suffcent Condtons for Openness at a Pont Bblography 337 A Topology 342 A.1 Sequental Spaces A.2 Open Mappng Suffcent Condtons A.3 Local Homeomorphsm Condtons A.4 Propertes of Contnuous Open Maps A.5 Coherence of Topologes wth Collectons of Subsets A.6 Characterzaton of Ponts n a Map s Image A.7 Mscellaneous Lemmata B Analyss 372 B.1 Functonal Analyss B.2 Dfferentaton n Topologcal Vector Spaces B.3 Fréchet-Urysohn Hausdorff LCTVSs are Coherent wth Arcs v

9 C Dfferental Geometry 381 C.1 Tubular Neghborhood Constructons C.2 Canoncal Form C.3 Lfts of Curves and Monotoncty C.4 Partal Replacement of Lfts C.5 Mscellaneous Lemmata v

10 Lst of Fgures 2.1 Illustraton of example s nton of Sys N Example constructon of Sys mx The canoncal coordnate representaton x

11 Chapter 1 Introducton A promanfold ((M, C M ), µ ) s a projectve lmt n the category of commutatve locally R- rnged spaces of a projectve system Sys M = ((M, C M ) µ j, N) consstng smooth manfolds smooth and bondng maps that are smooth surjectve submersons. A functon f U R from an open subset U of M s smoothly locally cylndrcal at a pont m U f there exsts some N, some U Open (M ), and some functon f U R such that m µ 1 (U ) U and f = f µ on µ 1 (U ). The sheaf CM of contnuous real valued functons on M conssts of all those contnuous functons f ned on open subsets of M that are smoothly locally cylndrcal at every pont of ther doman. As was done n [20], we wll use the sheaf C M of contnuous real valued functons n leu of a smooth atlas to extend many basc notons, constructons, and results from smooth manfolds to promanfolds. Before ntatng a study of the dfferental geometry of promanfolds, we provde a thorough ntroducton to lmts of projectve systems n the category Set and Top. In addton to contanng a revew of lmts n Top, ths ntroducton also contans many new examples and results. We fnd, for nstance, suffcent condtons for a lmt to be connected, locally connected (prop ), path-connected, and locally path-connected (prop ). The study of promanfolds then begns wth a revew of [20], whch s the artcle that ntated the theory of the dfferental geometry on projectve lmts of manfolds. We formulate and 1

12 prove a generalzed Whtney embeddng theorem for promanfolds (thm ). We ne and study subsets of a promanfold that have measure 0, whch then allows us to formulate and prove Sard s theorem (thm ) for promanfolds. Addtonally, we show that a large class of fnte-dmensonal promanfolds have locally trval tangent bundles (prop ). We show that the usual nverse functon theorem (theorem ) fals to generalze from smooth manfolds to promanfolds. Whle nvestgatng potental substtutes for the nverse functon theorem, we are led to a partcularly well-behaved class of promanfolds, called monotone promanfolds ( ), and to study the noton of coherence (. A.5.1), where we say that a space s coherent wth a collecton of contnuous maps nto t f ts topology s equal to the fnal topology nduced by these maps. We prove that every monotone promanfold s coherent wth ts set of smooth almost arcs at 0, whch are those smooth topologcal embeddngs of [0, 1] whose frst dervatves do not vansh on ]0, 1] and all of dervatves vansh at 0. Knowng that monotone promanfolds are coherent wth ther smooth almost arcs at 0 allows us to prove theorem , whch provdes a smple suffcent condton for a smooth map between monotone promanfolds to be open. We prove some substtute nverse functon theorems wth the frst man result beng theorem , whch gves a verson of the nverse functon theorem where the requrement of havng a dffeomorphsm between open subsets has been relaxed to merely havng a dffeomorphsm between subpromanfolds. The second man result, theorem , s a characterzaton of when a smooth map nto a monotone promanfold s, at some gven pont, a local dffeomorphsm between open subsets. Theorem leads to a conjecture about a verson of the nverse functon theorem for promanfolds that could potentally characterze local dffeomorphsms n terms of germs of vectors felds. 2

13 Notaton and Termnology for Elementary Concepts The followng table lsts notaton for some the basc concepts that the reader s assumed to be famlar wth. When usng any of the followng notaton, we may omt wrtng a symbol f t s clear from context. Let x S Open (X) Let S be an open subset of X contanng x. Ths notaton wll only be used when x X s already known. We may also replace Let wth for some, or for all, etc. or replace Open(X) wth Closed(X), Compact(X), etc. L = R L s by nton equal to R or f the symbol L s free then t s shorthand for let L = R. X Y Set subtracton: X Y = {x X x Y }. X Y Mnkowsk set subtracton, where X and Y are subsets of some {0} n (resp. {c} N, etc.) addtve group: X Y = {x y x X, y Y }. For n N and c any object, {0} n = (0,..., 0) (resp. {c} N = (c, c,...), etc.) s the n-tuple of 0 s (resp. the constantly c sequence, etc.). {c} N also denotes the sngleton set of all maps N {c}, whch we dentfy wth ths tuple. (I, ) A set I wth a partal order on I. (I, op ) The dual order of (I, ), where for any, j I, op j j. I 0, I < 0, etc. Defned as I 0 = { I 0 }, where 0 I. The sets I < 0, I > 0, and I 0 are ned analogously. ϕ j, ϕ >j, etc For ϕ = (ϕ ) I, ϕ j = (ϕ ) I s ϕ s frst j I coordnates. ϕ >j, j etc. are ned analogously. If ϕ = (ϕ ) I then we wll nstead wrte ϕ j, ϕ >j, etc. 3

14 Pr I J = Pr J, Pr 0, etc. For J I, Pr J S S j s the canoncal projecton onto the I j J coordnates n J ned by (s ) I (s j ) j J. For 0 I, Pr 0 = Pr I 0, Pr >0 = Pr I > 0, etc. Id A The dentty morphsm of an object A. In X S = In S The natural ncluson In X S S X, where S X. C k (X Y ) C k -maps from X to Y, where k Z 0 { }. Smlarly, C k ((X, x) (Y, y)) denotes the C k -ponted maps from X to Y. C k X Y (resp. Ck X ) Sheaf of Ck -maps from X nto Y (resp. nto R). [G] x = G x The set of germs at x X, where G s a collecton of maps ned on neghborhoods of x n X. (X, τ X ) A set X and a topology τ X on X. τ X R For R X, the subspace topology nherted from (X, τ X ) by R. Cl X (R) = R The closure of R n X, where R X. Int X (R) The nteror of R n X, where R X. Fr X (R) The fronter or topologcal boundary of R n X, where R X. dm z Z The dmenson of a (pro)manfold or vector space Z at z Z. If z s omtted then ths ndcates that dm z Z s ndependent of z Z and dm Z s ths common value. T M T M T M M s the canoncal projecton from the tangent bundle T M of a smooth manfold (or promanfold) M onto M. dam(s) The dameter of S M n a metrc space (M, d). Defned as dam(s) = sup d(s, ŝ). s,ŝ S d(s, T ) Dstance between S M and T M: d(s, T ) = nf d(s, t). s S,t T B d r(m 0 ) And for m 0 M we wll wrte d(m 0, T ) = d({m 0 }, T ). Closed ball of radus r > 0 around m 0 M n (M, d), where B d r(m 0 ) = {m M d(m, m 0 ) r} should not be confused wth the notaton B d r(m 0 ) for the open ball s B d r(m 0 ) closure n M. 4

15 B d r(s) Open ball of radus r > 0 around S M: B d r(s) = f D X Y f s a map on D wth codoman Y where D X. s S Bd r(s). We may also wrte D Open (X) or D Closed (X) n place of D X. f (X, R) (Y, S) f s a map f X Y, R X, S Y, and f(r) S. If R = {x} or S = {y} are sngleton sets then we omt wrtng { }. co(s) The convex hull of a subset S of some vector space over R. carr f The carrer of a map f X R, where R s contaned n some gven addtve group: carr f = {x X f(x) 0}. supp f The support of f X R. Defned as supp f = carr(f). Im f The mage or range of a map f. f(r) f(r) = {f(x) x Dom(f) R}, where R s any set. f R R D S Restrcton of f D Y to D R consdered as a map wth codoman S, where R and S are any sets such that f (R D) S. f R Set, Top, Man Denotes f R R D Y where f D Y and R s any set. The category of sets (resp. topologcal spaces, smooth manfolds) and maps (resp. contnuous maps, smooth maps). N, Z N = {1, 2,...} and Z = {..., 1, 0, 1,...}. S (resp. S) For S a set of sets, S = (X) The power set of a set X. S S S (resp. S = S). S S Lst of abbrevatons: LCTVS LHS (resp. RHS) resp. TVS Locally Convex Topologcal Vector Space Left (resp. Rght) Hand Sde respectvely Topologcal Vector Space 5

16 Defnton, Notaton, and Conventon Let P be a set, S P, and be a bnary relaton on a set P, whch we wll dentfy the relaton as a set of ordered pars n the usual way. When we wrte (S, ) then we mean the restrcted relaton (S, S S ) and we call S an deal of (P, ) ([12, p. 36]) f for all s S and p P, p s p S, or equvalently, f S = s S P s. The bnary relaton on P a preorder (on P ) f t s reflexve and transtve and t s called a partal order (on P ) f t s an antsymmetrc (.e. p q and q p mples p = q) preorder on P. If S P then an element p P s called an upper (resp. lower) bound of S (n P ) f s p (resp. p s) for every s S and we say that drects P and that P s drected (by ) f s a preorder and every par of elements n P has an upper bound n P. Notaton and Conventon Unless ndcated otherwse, (I, ) and (A, ) wll henceforth denote partal orders. Elements of I wll be denoted by h,, j, and k whle elements of A wll be denoted by a, b, c, and possbly d, whch wll prevent the rse of any ambguty from usng the same symbol (.e. ) to denote both order relatons. Defnton A map ι A I s called order-preservng or an order morphsm (from (A, ) to (I, )) f for all a, b A, whenever a b then ι(a) ι(b). We wll say that the order-preservng map ι A I s cofnal (resp. strct or ncreasng) f ts mage s cofnal n I (resp. f a < b mples ι(a) < ι(b)). Notaton and Mnemoncs If the symbol ι (resp. α) represents a map between the sets I and A then ts prototype wll be ι A I (resp. α I A) where the symbol ι (resp. α) was chosen so that one may mmedately determne that a value ι(a) (resp. α()) s an element of I (resp. A). Gven two order-preservng maps ι A I and α I A and an element a A, we may wrte αι(a) = α(ι(a)) to prevent an abundance of parentheses. Remark If ι A I s a cofnal order morphsm between two preorders then A beng drected mples that I s drected. Assumpton All categores wll be assumed to be concrete categores. Each of Group, Top, Man, etc. wll be pared wth ts usual forgetful functor. 6

17 Defnton and Conventon If we say that M s a manfold then we mean that t s a Hausdorff second-countable locally-eucldean topologcal space wth a smooth structure. By usng the canoncal dentfcaton descrbed n remark C.0.3, we wll dentfy the smooth structure as ether a smooth maxmal atlas or equvalently by CM, ts sheaf of smooth R- valued functons. Remark Observe that our nton of a manfold does not requre that t be connected nor that t have homogeneous dmenson. Ths wll be advantageous snce even the study of projectve systems of connected manfolds may requre us to work wth nduced systems of dsconnected manfolds whose dmensons dverge to nfnty (e.g. f we wsh to apply lemma 3.2.1(6)). Defnton Suppose M s a d-dmensonal manfold wth no dstngushed metrc. Call a chart (U, ϕ) on M a (coordnate) ball (resp. box, cube) f ϕ(u) s an open ball (n the Eucldean norm) n R d (resp. a product of d-open ntervals, a product of d-open ntervals of the same length). If (U, ϕ) s a chart on M and B M then we wll say that B s an (proper) open ball (resp. box, cube) n (U, ϕ) f Cl M (B) = B U and (B, ϕ B ) s a coordnate ball (resp. box, cube). A subset B M wll be called an (proper) open ball (resp. box, cube) n M f there exsts some chart (U, ϕ) on M such that B s an open ball (resp. box, cube) n (U, ϕ). It should be clear what s meant f we replace the word open n (proper) open ball (resp. box, cube) wth closed or f we add the words centered at p, of radus r, wth sdes of length l, etc. Conventon and Remark Although callng a map f open f t maps open sets to open sets s not controversal, one fnds n the lterature that some authors call a map f X Y open f t maps every open subset of X to an open subset of Y, whch s the nton used n ths paper (see. A.0.6), whle others requre merely that these mages be open n Im f. To prevent ths as well as other smlar msunderstandngs (and ther consequences), we wll often rewrte the map s prototype (e.g. we wll usually wrte 7

18 f X Y s open or f X Im f s open nstead of smply f s open ) so that the reader may henceforth safely assume that the topologcal terms used n any nton (e.g. maps open sets to open sets ) are relatve to the topologes of the doman and codoman that presented n the prototype. The analogous assumptons can also be safely made f X and Y are endowed wth structures other than topologes, such as algebrac structures or sheaves. Lfts, Factorzatons, Fbratons, and Sequences Defnton An ndexed collecton of objects x has nfnte range or s nfnte-ranged f {x s an ndex } s nfnte, that t s njectve f whenever and j are dstnct ndces then x x j and by a net (resp. sequence) of dstnct ponts n a set X we mean an njectve net (resp. sequence). If x s a net n a space X and x X then x x s njectve n X means that x s njectve, x x for all ndces, and x x n X; the meanngs of let x x be njectve n X and suppose x x has an njectve subsequence n X should be clear. Let f X Y be a map between spaces and let (y ) I be a net n Y. If (y ) I s convergent n Y then by an f-lft of (y ) I we mean an I-drected convergent net (x ) I n X such that f(x ) = y for all I. If there exsts a net (x ) I n X that s an f-lft of a convergent net (y ) I then we ll say that f lfts (y ) I to (x ) I, that (y ) I s f-lftable, and that f can lft (y ) I. When we wrte (x ) I x s an f-lft of (y ) I y then we mean that (y ) I converges to y n Y, f(x) = y, and (x ) I s an f-lft of (y ) I that converges to x n X. Defnton summarzes the termnology related to expressng a gven morphsm n terms of other morphsms. Most of the termnology s ether based on or taken drectly from the termnology found n [5]. Defnton Let E, B, X, and Z be objects and let π E B be a morphsm (n some 8

19 gven category). If f Z B and f Z E are morphsms such that f = π f then we wll call f a π-lft of f (on Z to E) and call f the π-drop of f (on Z to B). If for some gven morphsm f Z B there exsts some π-lft of f then we wll say that f s π-lftable (to E) and that f arses as a π-drop (from E) (ned on Z). If p E X s a morphsm for whch there exsts a morphsm p B X such that p = p π then we wll say that p factorzes through π, p s factorzed through π (by p ), and that p descends (through π) to p. E f π p Z f B p X It s apparent from the unversal property of lmts that the ablty to fnd approprate lfts of a maps can be useful when workng wth nverse systems and ther lmts. Ths naturally leads us to consder the homotopy lftng property, whch we now generalze to a nton that s well-suted to nverse systems n the sense that t wll make the statement of lemma both concse and smple. Defnton Let A, B, E, and Z be objects n a concrete category C such that A Z and the natural ncluson In A Z s a morphsm, and let π E B be a morphsm. Say that π has the extenson lftng property from A to Z (n C) f for any morphsm f A E, whenever a morphsm H Z B extends π f A B to all of Z (.e. H In = π f) then there exsts some π-lft, H Z E, of H extendng f to Z (.e. H In = f): E A In Z f H H B π In Top, f ths s true wth Z = X [0, 1] and A = X {0} then we say that π has the homotopy lftng property wth respect to X and π s called an X-fbraton. If r Z 0 then call 9

20 π an r-fbraton f t s a n -fbraton for all 0 n r, where n s the standard n-smplex, and where n the partcular case that π s a 0-fbraton we wll follow [34] and say that π has the path-lftng property. If π s an r-fbraton for all postve ntegers r then we wll call π a weak fbraton, a Serre fber space, or a Serre fbraton. If C B then π s a fbraton over C f π π 1 (C) π 1 (C) C s a fbraton. Notaton for Indexed Collectons The nature of nverse or drect systems regularly causes a prolferaton of ndces, where t s unfortunately often the case that the more plentful the ndces then the more dffcult a proof or statement becomes to dgest whle smultaneously ncreasng the rsk that the (usually relatvely smple) deas or ntuton underlyng t s obscured or even entrely mssed. So to help avod wrtng unnecessary ndces we wll now ntroduce some notaton, conventons, and ntons. Defnton, Notaton, and Conventon We wll denote a collecton (M ) I (resp. (m ) I, (H α() ) I, etc.) of objects ndexed by some ndexng set I by M (resp. m, H α( ), etc.), where for each ndex, the th component of M s the object M, whch we may also denote by (M ). If L s any set then M L = (M ) I L denotes the restrcton of M to L. If f s a map and M are sets then by f (M ) (resp. f 1 (M ), etc.) we mean the collecton (f(m )) I (resp. (f 1 (M )) I, etc.). If S = (S l ) l Λ then by S M we mean that S M for all I L whereas f we ntroduce S by sayng let S M wthout specfyng S s ndexng set then t should be assumed that S s ndexed by M s ndexng set. By F M N (resp. G N M, H α( ) N α( ) M, etc) we mean a collecton of morphsms whose th -component has prototype F M N (resp. G N M, H α() N α() M, etc.) and f S = (S l ) l Λ M then we ll use F (S ) (resp. G 1 (S ), (H α( ) ) 1 (S ), etc.) to denote (F (M )) I Λ (resp. (G 1 (S )) I Λ, ((H α() ) 1 (S )), etc.). The meanng of M, S M, and all other smlar notaton I Λ should now be easy to deduce. 10

21 Mnemoncal Notaton and Conventon Gven sets Z (resp. Z ) and some ndex λ, ths ndex wll always appear as a subscrpt (resp. superscrpt) to any element or subset of Z λ (resp. Z λ ) (e.g. we wll wrte let S λ Z λ and never wrte let S Z λ ). Remark Although ths conventon does occasonally ntroduce an unnecessary ndex, by applyng t consstently the net effect (n the author s opnon) wll be to ncrease the overall readablty of ths paper f the reader adopts the perspectve that ths ndex s nothng more than a persstent remnder of whch set (.e. whch component of Z ) ths element or subset s contaned wthn. Defnton and Conventon Gven a collecton of sets or maps (S λ ) λ Λ ndexed by some subset Λ I, f we ne S = for all I Λ then we wll call the I-ndexed collecton of sets (S ) I the canoncal -extenson (of (S λ ) λ Λ ) (to I) and we may dentfy ths I-ndexed collecton of sets wth (S λ ) λ Λ and thereby denote both by S. Notaton We wll henceforth always use M = (M ) J and µ = (µ ) J (resp. N = (N a ) a B and ν = (ν a ) a B ) to denote, respectvely, a collecton of objects and a collecton of morphsms ndexed by some subset J I (resp. B A) where f the subset J (resp. B) s omtted or not clear from context then t s to be assumed that J = I (resp. B = A). Furthermore, all µ (resp. all ν ) wll share the same doman (usually denoted by M (resp. N)) and each µ (resp. ν a ) wll have codoman M (resp. N a ). Defnton, Notaton, and Conventon Gven any map ι A I, by a collecton of sets (resp. morphsms, maps, etc.) ndexed by ι or an ι-ndexed collecton of sets (resp. morphsms, maps, etc.) we wll mean the par consstng of an A-ndexed collecton of sets (resp. morphsms, maps, etc.) together wth ι, where f ι s understood then we may also refer to the A-ndexed collecton (rather than the par) as an ι-ndexed collecton. If (S a ) s ι(a) a A an ι-ndexed collecton and ι s njectve then we wll drop the redundant ndex and nstead wrte ths collecton as S ι( ) = (S ι(a) ) a A and furthermore, we may use ι to dentfy ths A-ndexed collecton as an (Im ι)-ndexed collecton. 11

22 On the other hand, f ι s not njectve, say = ι(a 1 ) = ι(a 2 ) for a 1 a 2, then regardless of whether or not S a 1 ι(a 1 ) and Sa 2 ι(a 2 ) are equal, we wll stll frequently wrte S ι(a 1 ) (resp. S ι(a2 )) n place of S a 1 (resp. ι(a 1 ) Sa 2 ι(a 2 ) snce t s easy to deduce from the symbols present whch ) of these sets we are referrng to; however, we would not wrte S snce there s no way to deduce from the symbols present f S s referrng to S a 1 or S a 2. We further extend ths conventon to the notaton used when ntroducng the collecton such a collecton n the followng way: f t s the case (say n a proof or remark) that we wll always be able to wrte S ι(a) nstead of S a ι(a) then rather than ntroducng ths A-ndexed collecton by wrtng let S ι( ) = (Sa ι(a) ) a A... we wll nstead wrte let S ι( ) = (S ι(a) ) a A.... Gven any collecton of sets R = (R ) I ndexed by I we wll call an ι-ndexed collecton of sets S ι( ) = (S ι(a) ) a A an (ι-ndexed) collecton of subsets (of R ) f S ι(a) R ι(a) for all a A. If S ι(a) = R ι(a) for all a A then we wll call S ι( ) a subcollecton of R. Defnton If O s a collecton of subsets of a set Z then f we say that G s a presheaf of maps on O we mean that G s presheaf on O that assgns to each O O a non-empty set of maps G(O) such that Dom γ = O for all γ G(O) and that G s restrctons are the canoncal restrctons of maps. If H s a collecton of maps ned on subsets of Z and f O s a collecton of subsets of then say that H s closed under restrctons to O f for all h H and O O, whenever O Dom h then h O H. If H s closed under restrctons to Dom H = {Dom h h H} then we wll say that H s closed under restrctons and call H a closed collecton of maps (from subsets of Z). Remark and Conventon To every presheaf G of maps on a collecton of subsets O of a set Z we can form the set of maps H = G(O) that s closed under restrctons O O whle f H s a collecton of maps ned on subsets of Z that s closed under restrctons then the assgnment ned by sendng O Dom H to G(O) = {γ H Dom γ = O} forms a presheaf of maps on Dom H. It s clear that above constructons are nverses of each other so we wll henceforth dentfy collectons of maps ned on subsets of Z that are closed under restrctons wth presheaves of maps on collectons of subsets of Z. Consequently, ths 12

23 dentfcaton allows us to treat any presheaf of maps on a collecton of subsets of Z as f t was a closed collecton of maps ned on subsets of Z and vce-versa, whch we shall henceforth do wthout comment. Defnton Let h Z M and h Z M be maps, S Z, and z Z. By the h-saturaton of S we mean the set h 1 (h(s)) and we ll say that S s h-saturated f S = h 1 (h(s)). Call S an h -fber of z or an h -fber contanng z f there s some ndex such that S = h 1 (h (z)). Germs, and Submersons and Immersons of Germs Many of the ntons below, ncludng that of trace, generalzes that of Bourbak ([11]). Defnton and Notaton Let M be a set, Φ a collecton of M-valued maps, Z a space, z Z, z = (z λ ) λ Λ a net, and let R and S be two collectons of sets. The trace of R n M s tr M (R) = {R M R R} whle Φ R wll denote the set of all restrctons ϕ R as ϕ ranges over Φ and R ranges over R. We wll denote the trace of Nhd z (Z) n M by [M] Z z and by [z ] Z z we mean [{z λ λ Λ}] Z z, where we may omt Z from the notaton f t s understood. If the doman of each map n Φ contans z and all of ther values agree at z then we ll denote ths common value by Φ(z) and call t the value of Φ at z. If m s any pont then let Φ z (resp. Φ z m ) denote the set of all maps n Φ whose domans contan a neghborhood of z (resp. and map z to m). Say that R s fner than S and that S s coarser than R f for all S S there s some R R such that R S. Many of the followng ntons consst of ntons from Bourbak ([11]) or ther generalzatons. Defnton and Notaton Let Z be a set, F be a flter base on Z, z Z, G a set of maps, and m any object. If R, S Z then R and S have the same germ (wth respect to F) f there exsts some F F such that R F = S F. Ths forms an equvalence relaton 13

24 on (Z), the power set of Z, and the equvalence class contanng a set R Z s denoted by [R] F and called the germ of R (wth respect to F) where f Z s a space and F = Nhd z (Z) then we wll use the notaton [R] z and call t the germ of R at z n Z. Say that two maps γ and η have the same germ wth respect to F ( resp. at z) or that they have the same F-germ ( resp. at z) f there exsts some F F such that F Dom γ Dom η and γ = η on F (resp. and z F ), where f Z s a topologcal space and F s a flter bases for Nhd S (Z) then we ll say that γ and η have the same germ at z (n Z). Ths forms an equvalence relaton on G and the equvalence class contanng a map g G, denoted by [g] G F, s called the germ of g (n G) (wth respect to F) and the set of all germs n G wll be denoted by [G] F. If Z s a space, g G, and Dom g s a neghborhood of z n Z then call the germ of g G wth respect to Nhd z (Z) the G-germ of g at z n Z, denote t by [g] G z, and let [G] z = [G z ] z and [G] z m = [G z m ] z If any of Z, F, or G are understood then they may be omtted from the notaton. Remark If G s a presheaf of maps nstead of a set of maps, then the above ntons and notaton related to sets of maps generalze mmedately n the obvous way. Defnton Let F (M, m) (N, n) be a ponted map, (Z, z) be a ponted space, and G (resp. H) be a presheaf of M-valued (resp. N-valued) maps ned subsets of Z. If γ s an M-valued map then let F (γ) = F γ. If we wrte F G H then we mean that F (γ) = F γ belongs to H for all γ G, whch then allows F to descend to the followng map between germs at z: F [G] z [H] z [γ] z [F γ] z whch, by overloadng notaton, we wll also wrte as F [G] z [H] z (so F ([γ] z ) = [F γ] z ). More generally, for Φ [G] z and Ψ [H] z, f we wrte ether F (Φ) = Ψ or F (Φ) = Ψ then we mean that there exsts some γ Φ and some η Ψ such that F (γ) = F γ and η have the 14

25 same H-germ at z, where n ths case we wll overload notaton by lettng both F (Φ) and F (Φ) denote ths germ. If Φ [G] z and we wrte ether F (Φ) [H] z or F (Φ) [H] z then we mean that there exsts some Ψ [H] z such that F (Φ) = Ψ. If we wrte F [G] z [H] z or F [G] z [H] z then we mean that F (Φ) [H] z for all Φ [G] z. We wll wrte and say that (1) F [G] z [H] z s a germ submerson at m (from H germs (at z)) (to G germs) or that F lfts H germs at z (through n) to G germs (at z) through m f for all Ψ [H] z n there exsts some Φ [G] z m such that Ψ = F (Φ). (2) F [G] z [H] z can lft germs (at z) through n f for all Ψ [H] z n there exsts some Φ [G] z (not necessarly though m) such that F (Φ) = Ψ. (3) F [G] z [H] z s a germ mmerson at m f for all Φ, Φ [G] z m, F (Φ) = F ( Φ) Φ = Φ. (4) F [G] z [H] z s a germ bjecton at m f t s a germ mmerson at m and F [G] z [H] z s a germ submerson at m. where f we wrte F [G] z [H] z nstead of F [G] z [H] z then we mean that n addton to satsfyng that nton we also have F [G] z [H] z. Remark The notaton F [G] z [H] z was chosen to emphasze that none of the above ntons of germ submerson requre F [G] z [H] z. These ntons were motvated by the stuaton where Z, M, and N are smooth manfolds and both G and H consst of varous sets of contnuous maps from neghborhoods of z n Z nto M and N, respectvely (e.g. say there are no addtonal restrctons on G whle H conssts solely of smooth topologcal embeddngs). Example (Boman Theorem). Recall ([1, p. 3], [27, cor. 3.14]) that one part of the Boman theorem states that a map F M N between two manfolds s smooth f and only f F γ R N s smooth for all smooth γ R M. It s easy to see that we can express 15

26 ts equvalent formulaton n terms of germs of smooth curves as: F M N s smooth f and only f F [C (R M)] 0 [C (R N)] 0, where C (R M) (resp. C (R N)) denotes the set of all smooth curves nto M (resp. N) wth doman R. The second part of the Boman theorem states that F M N s smooth f and only f f F M R s smooth for all smooth f N R. Dervatons Defnton Let A be an algebra over a feld F and let M be an A-bmodule. Then a map D A M s an F -dervaton nto M or a dervaton (over F ) nto M f t s lnear over F and satsfes the product rule: D(fg) = D(f)g + fd(g) for all f, g A. If ev A F s an F -algebra homomorphsm then we may make F nto an A-bmodule by nng A F (a, α) F ev(a)α wth the rght acton of A on F ned analogously and we wll denote the set of all F - dervatons from A nto F by Der ev (A F ). In the partcular case where the F -algebra A s a collecton of ether F -valued maps or equvalence classes of such maps, all of whch may be evaluated at some pont p, then we wll let Der p (A F ) = Der evp (A F ) where ev p A F s the usual evaluaton at p map (.e. ned by ev p (a) = a(p)) and we wll call an element of Der p (A F ) an (F -)dervaton at p (on A) (nto F ). 16

27 Chapter 2 Lmts n Set and Top Just as a proper understandng of modern dfferental geometry would be made sgnfcantly more dffcult wthout a well-developed ntutve understandng of the topology of Eucldean spaces and ther constructon from smpler spaces such as R, so too would a proper understandng of promanfolds (.e. projectve lmts of manfolds) be made more dffcult wthout a well-developed ntuton about ther topology and constructon from more basc spaces. For ths reason the ntroducton to lmts n Top that follows s wrtten n a way so that, to the best of the author s ablty, the statements and ther proofs obscure as lttle of the underlyng ntuton that the author has about them. Furthermore, n addton to entrely new results and extensons of well-known results, where well-known means that they can be found n a standard reference on ths subject such [11] or [12], even many of the well-known results n ths chapter have proofs that, to the best of the author s knowledge, have not appeared elsewhere. Before contnung, t s recommended that the reader have a basc understandng of lmts and colmts of systems where ths can be obtaned by readng Dugundj [12], whch was the author s prmary reference for ths chapter, or Bourbak [11]. 17

28 Introducton to Systems, (Co)Cones, and (Co)Lmts Systems As the name suggests, lmts may be thought of as the objects that would result f one were able to do a sequence of actons forever, where the objects beng acted upon and the rules of these actons are encapsulated n the followng nton of an nverse system. Example may help clarfy how, at least n the case when the system s ordered by N, the nton of an nverse system captures these deas. The nton of nverse system that we ve adopted s based the nton gven n [12]. Defnton An nverse or projectve system (over I) (n a concrete category C ) s a quadruple (M, µ j, I, ), whch we may also denote by (M, µ j ) or Sys M, where (1) (I, ) s a partal order, (2) M = (M ) I wth M an object n C for each ndex I, (3) µ j M j M s a morphsm for all, j I wth j wth µ = Id M f = j, and where these morphsms satsfy the compatblty condton: µ j µ jk = µ k whenever j k The morphsms µ j are called the connectng maps or the bondng maps of the system. If Sys M = (M, µ j, I, ) s a quadruple consstng of objects M, morphsms µ j, and a partal order (I, ), then call the quadruple Sys op M = (M, µ j, I, op ) the dual or transpose of Sys M, where op represents the dual order of (I, ). A drect system s a quadruple (M, µ j, I, ) whose transpose (M, µ j, I, op ) s an nverse system. If Sys M s a projectve or drect system ordered by (I, ) then we wll say that Sys M s: drected f ts partal order (I, ) s a drected set. 18

29 surjectve (resp. njectve, etc.) f all connectng maps are surjectve (resp. njectve, etc.). compact (resp. connected, etc.) f all objects are compact (resp. connected, etc.) topologcal spaces. smooth (resp. smooth submersve) f all objects are smooth manfolds and all connectng maps are smooth (resp. smooth submersons). ponted f all objects and maps are ponted. Remarks Some authors (e.g. [5]) reserve the term bondng map for the partcular case of I = N and then only for connectng maps of the form µ,+1. By vewng a partal order as a category n the usual way, a system may be vewed as a functor from a partal order (I, ) nto C. The class of all nverse (resp. drect) systems n some gven category wll tself become a category f we use nton to ne ts morphsms. The same holds true of the class of all nverse (resp. drect) systems when ther orders (.e. (I, )) are requred to belong to a certan category (e.g. systems ndexed by drected partal orders, systems ndexed by N, etc). In the notaton (M, µ j, I, ), the symbol µ j n ths tuple actually represents a collecton of morphsms where there s one morphsm for each par of ndces, j I such that j; ths tuple should properly be wrtten as (M, (µ j ) (,j), I, ), where s vewed a collecton of ordered pars from I I. Conventon Snce we wll only be workng n concrete categores, whenever we refer to Sys M as a system n Set then we are actually referrng to the system that results from applyng the category s forgetful functor to all of Sys M s objects and connectng morphsms. 19

30 Assumpton and Notaton Unless ndcated otherwse, we wll henceforth assume that Sys M = (M, µ j, I, ) and Sys N = (N, ν ab, A, ) are projectve systems where both (I, ) and (A, ) are partal orders. If we declare that Sys M and Sys N are drect systems then unless ndcated otherwse, we wll assume that these systems are the tuples Sys M = (M, µ j, I, ) and Sys N = (N a, νa, b A, ). Whenever we wrte µ pq or µ q p (resp. ν pq or νp ) q then unless ndcated otherwse, t should be assumed the p and q are ndces n I (resp. A) wth p q. We wll also usually assume that all M and N a are non-empty, where t should be clear from context when ths assumpton s or s not beng made. Example and Defnton Gven any space Z and any partal order (I, ) we can form the constant or trval system over I by lettng Z = Z and µ j = Id Z for all j n I. We wll denote ths system by (Z, Id Z, I, ), (Z, Id Z, I, ), or smply ConstSys Z. Example and Defnton If J I then the restrcton of Sys M = (M, µ j, I, ) to J, denoted by Sys M J and called a subsystem of Sys M, s the system (M J, µ j, J, J J ) that conssts of all those M and µ j for whch all ndces belong to J. Example and Defnton If J I, 0 I, and S 0 M 0 then the system nduced by S 0 and J (and 0 ) s Sys M J,S0 = (S j, µ jk Sk, J) where for all j, k J wth j k, S j = µ 1 j 0,j (S j 0 ) f j 0 M j otherwse and µ jk Sk S k S j. By the system nduced by S 0, denoted by Sys M S0, we mean the system nduced by S 0 and J = I 0. Example Let M = (M ) N be any sequence of objects and let µ, +1 = (µ,+1 ) N be a sequence of morphsms where µ,+1 M +1 M for all N. Ths collecton of objects and 20

31 morphsms nduces a projectve system Sys M = and µ j = µ,+1 µ +1,+2 µ j 1,j M j M (M, µ j, N, ) when we ne µ = for, j N wth +1 < j. Smlarly, f gven sequences of objects M = (M ) N and morphsms Id M µ +1 = (µ +1 ) N where µ +1 M M +1 for all N then ne µ = Id M and µ j = µ j j 1 µ+2 +1 µ +1 M M j for, j N wth + 1 < j so as to obtan a drect system. It s clear that the above ntons of µ and µ j (resp. µ and µj ) are the only ones that would make (M, µ j, N, ) (resp. (M, µ j, N, )) nto an nverse (resp. drect) system. Conventon Henceforth, we may ne nverse (resp. drect) systems drected by N by specfyng only the bondng maps µ,+1 (resp. µ j ) where t should then be mmedately assumed that the bondng maps µ j (resp. µ j ) are ned as above for all j. Example and Defnton Suppose that R = (R j ) j J s a collecton of sets ndexed by some set J and gve J the partal order nduced by reverse set ncluson on R (.e. j k R k R j ). For all j, let In j R j R, In R R, and In R R denote the natural nclusons. It s straghtforward to verfy that f (J, ) s drected then ( R, In ) s a lmt of the nverse system (R, In j, J, ) whle f (J, op ) s drected then ( R, In ) s a colmt of the drect system (R, In j, J, op ). If J s drected or contans a mnmum element then let I = J where otherwse we wll stpulate that J not contan the symbol and then ne I = J { }. If I J and f there s also some dstngushed set X that contans each R j as a subset then let R = X and otherwse let R = j J R j, where n ether case we also gve I the partal order nduced by reverse set ncluson on (R ) I, whch clearly extends J s orgnal partal order and makes nto I s mnmum. By the (canoncal) nverse system nduced by R (and nclusons) we mean the nverse system Sys R = ((R ) I, In j, I, ) where each In j R j R s the natural 21

32 ncluson. The next nton generalzes the nton, found n [45, Sheaves], of a presheaf of objects of a category on a bass of a topology. Example and Defnton Let C be some category, (Z, τ Z ) be a topologcal space, and B be a bass for Z. A presheaf (of objects) n C on B s a drect system Sys M = (M(A), µ B A, B, ) n C where the partal order (B, ) s reverse set ncluson and where we wrte M(A) nstead of M A n order to conform wth the standard notaton for presheaves. If B = τ Z then we call Sys M a presheaf n C on Z and by a morphsm of presheaves we mean a drect system morphsm ( ) between presheaves. A prototypcal example of a presheaf s C Z M where for every open subset U of Z, C Z M (U) = C(U M) conssts of all contnuous maps from U nto the gven space M and where the connectng maps are the usual restrctons of domans of functons (.e. µ V U (f) = f V for V U open n X). It should now be clear that a presheaf (and ndeed any system) may be vewed as nothng more than an ndexed collecton of nformaton (e.g. maps ndexed by open subsets of Z) where nformaton between dfferent ndces are related to each other n a consstent way (.e. va the connectng morphsms, whch satsfy the consstency condton). By takng ths pont of vew, t s natural to extend the nton of a presheaf by consderng drect system ndexed by an arbtrary collecton B of subsets of Z that s partally ordered by reverse set ncluson; we wll call any such drect system a presheaf on B. Example Let (q j ) j=1 be a sequence of natural numbers greater than 1 and for all j N, ne ρ j S 1 z S 1 C z j and µ j,j+1 = ρ qj, whch gves us a projectve system Sys M = (S 1, µ j, N). Suppose that for each j N, we ve wrtten q j as the product q j = λ(j) p j,l for some λ(j) N and some natural l=1 22

33 numbers p j,1,..., p j,λ(j) greater than 1. Let = (S Sys M 1, µ ab, N) denote the system that s ned just as Sys M, except that the sequence p 1,1,..., p 1,λ(1), p 2,1,..., p 2,λ(2), p 3,1,... s used n place of q 1, q 2,.... It s clear that Sys M can be obtaned from by restrctng to some cofnal Sys M Sys M J subset of N. Once the reader has knowledge of lmts and lemma , ths example wll ental that f we replace q 1, q 2,... wth p 1,1,..., p 1,λ(1), p 2,1,..., p 2,λ(2), p 3,1,... then (up to a unque somorphsm) we would not have changed the lmt of Sys M. In partcular, ths mples that to understand the lmts of systems of the form ned above, t suffces to understand the lmts of such systems where every q j s prme. Example Suppose that M s a smooth manfold and let T 0 M = M. We can nductvely ne for every k N, the k th -order tangent bundle by T k M = T(T k 1 M), where for all k Z 0, we wll denote the canoncal projecton by T k k+1 M T k+1 M T k M or by T T k M T k+1 M. Ths gves us the followng projectve system of manfolds whose bondng maps are smooth surjectve submersons: Sys T M = (T k M, TM k k+1, Z 0 ). Cones and Cocones Defnton Let Sys M be an nverse (resp. drect) system n some category C where Sys M = (M, µ j, I, ) (resp. Sys M = (M, µ j, I, )), let Z s an object n C, and let h Z M (resp. h M Z) be a collecton of morphsms ndexed by I, whch we wll denote by h (resp. h ). We wll say that h (resp. h ) s compatble or consstent wth Sys M f µ j h j = h (resp. h j µ j = h ) whenever j,.e. f the respectve dagram commutes: 23