Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 1 Towards Non Commutative Algebraic Topology Ronnie Brown University College London, May 7, 2003. This is a slightly edited version of the transparencies presented. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 2 Acknowledgements to: work of Henry Whitehead; many collaborators, particularly Philip Higgins, Jean-Louis Loday, Tim Porter, Chris Wensley; 21 Bangor research students; Alexander Grothendieck, for correspondence a`baton rompu 1982-93 and Pursuing Stacks (1983, 600 pages). References: http://www.bangor.ac.uk/~mas010 http://www.bangor.ac.uk/~mas010/fields-art3.pdf Current support: Leverhulme Emeritus Fellowship `crossed complexes and homotopy groupoids' to produce a book with now agreed title: Title of this seminar! Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 3 Start (1965): My attempt to get a form of the Van Kampen Theorem (VKT) for the fundamental group which would also calculate the fundamental group of 1 the circle S . Discovered Higgins' work on groupoids! Motivation: expository and aesthetic, thinking about anomalies. Trying to understand the algebraic structures underlying homotopy theory. Homotopy and deformation underly notions of classification in many branches of mathematics. Aim: explore the situation (so can not be directed at other peoples' problems). NOT mainstream algebraic topology - we are digging a new and additional channel. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 4 Overall plan is underlying number geometry ____//_ ______//_algebra _____________// processes | theory | | | | fflffl| computer algorithms ____//_ implementation | | ffl* *ffl| sums Example: Extensions A æ E i T of groups A, T are determined by classes of factor systems 1 2 k : T ! Aut (A), k : T x T ! A. 3 -3 But if T = gp is the Trefoil group, then T is infinite, so what can you do? Our result : Extensions are determined by elements 3 -2 a 2 A, ax , ay 2 Aut (A) such that (ax ) (ay ) is the inner automorphism determined by a. Example: f : P ! Q a morphism of groups. Form the cofibration sequence Bf BP - ! BQ ! C (Bf ). Find ß2 and first k-invariant of C (Bf ). E.g. (P = C3 6 Q = S4 ) =) ß2 (C (Bf )) ~= C6 . Use non commutative methods (and computers). Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 5 Major themes in 20th century mathematics: o non commutativity o local to global o higher dimensions o homology o K -theory (Atiyah, Bull LMS, 2000) The VKT for ß1 is a classical example of a non commutative local-to-global theorem. So any tools which are developed for generalisations and for higher dimensional forms of it could be generally useful. Non commutative algebraic topology conveniently combines all the above major themes, and has yielded substantial new calculations, new understanding, new prospects, of which the last is possibly the most important. Applications to concurrency (GETCO). Work with Tony Bak, Tim Porter on Tony's `global actions'. Peter May's interest in higher categorical structures. Recent EPSRC Grant on Higher dimensional algebra and differential geometry. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 6 Why think of non commutative algebraic topology? Back in history! Topologists of the early 20th century knew well that: 1) Non commutative fundamental group ß1 (X, a) had applications in geometry, topology, analysis. 2) Commutative Hn (X ) were defined for all n > 0. 3) For connected X , ab H1 (X ) ~= ß1 (X, a) . So they dreamed of higher dimensional, non commutative versions of the fundamental group. Gut feeling: dimensions > 1 need invariants which are `more non commutative' than groups. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 7 1932: ICM at Z ürich: C~ech: submits a paper on higher homotopy groups ßn (X, x); Alexandroff, Hopf: prove commutativity for n > 2; persuade C~ech to withdraw his paper; only a small paragraph appears in the Proceedings. Reason for commutativity (in modern terms): group objects in groups are commutative groups. ß2 (X, x), even considered as a ß1 (X, x)-module, is only a pale shadow of the 2-type of X . What is going on? Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 8 Overall philosophy: look for algebraic models of homotopy types topological _______________________//algebraic oo_____________________ * *(*) data II B w data III wwww III www III www U IIII www B I$$I --www Top 1) U is a forgetful functor and B = U O B; 2) is defined homotopically; 3) (local to global, allowing calculation!): preserves certain colimits; 3) (algebra models the geometry) O B ' 1; 4) (capture homotopical information): 9 natural transformation 1 ! B O with good properties. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 9 Some examples: ____________________________________________________________________________* *_____ |||| | | |||| | | |||| topological data algebr|aic data | ||||_________________________________________|____________________________|_* *______ |||| | | |||| | | |||| spaces with base point groups| | ||||_________________________________________|____________________________|_* *_____ |||| | | |||| | | |||| spaces with a set of base | | |||| | | |||| groupo|ids | |||| | | |||| points | | ||||_________________________________________|____________________________|_* *_____ |||| | | |||| | | |||| based pairs crosse|d modules | ||||_________________________________________|____________________________|_* *_____ |||| | | |||| | | |||| filtered spaces crosse|d complexes | ||||_________________________________________|____________________________|_* *_____ |||| | | |||| n | | |||| n-cubes of spaces cat -|groups | ||||_________________________________________|____________________________|_* *_____ So on the blue side we have various generalisations of groups. Here are some more! || * * | ____________________________________________________________________________* *____|| || * * | || cubical * * | || polyhedral * * | || ks__+3_ cubical ks__+3__ * * | || !-groupoids * * | || T -complexes * * | || T -complexes * * | || with connections * * | || ":B" | * * | || """" | * * | || """" | * * | || """"" | * * | || """"" | * * | || simplicial """" | * * | || """" | * * | || """"" | * * | || J -groupoids """" | * * | || ckO """"" | * * | || OOOOOOOO """" | * * | || OOOOOO """"" | * * | || OOOOOOO """" | * * | || #+ z" fflffl| * * | || * * | || simplicial crossed globular * * | || ks_____+3_ ks________+3_ * * | || * * | || T-complexes complexes !-groupoids * * | ||__________________________________________________________________________* *____ | The equivalence of the red structures is required for the proof of the Brown-Higgins GVKT. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 10 Features of groupoids: structure in dimension 0 and 1; composition operation is partially defined; allows the combination of groups and graphs, or groups and space. Higher dimensional algebra (for me) is the study and application of algebraic structures whose domains of the operations are given by geometric conditions. This allows for a vast range of new algebraic structures related to geometry. Why so many structures? More compact convex sets in dimension 2 than dimension 1! The algebra has to express and cope with structures defined by different geometries. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 11 Easiest example: Cubes. Cubical methods are used in order to express the intuitions of 1) Multiple compositions (algebraic inverses to subdivisions); 2) Defining a commutative cube. 3) Proving a multiple composition of commutative cubes is commutative (Stokes' Theorem?!). 4) Construction and properties of higher homotopy groupoids. 5) Homotopies and tensor products: m n ~ m+n I x I = I Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 12 Category FTop of filtered spaces: X* : X0 X1 X2 . . . X1 of subspaces of X1 . Homotopical quotient: p : R(X* ) - ! %(X* ) = R(X* )= where n R(X* )n = FTop (I* , X* ), n I* = n-cube with its skeletal filtration, n : homotopy through filtered maps rel vertices of I . Then R(X* ), %(X* ) are cubical sets with connections. (Connections are extra degeneracy operations.) But R(X* ) has standard partial compositions: + - for i = 1, . . ., n, if a, b 2 R(X* )n and @i a = @i b we can define a Oi b 2 R(X* )n by 8 < 1_ a(. . ., 2ti, . . .) ti 6 , 2 (a Oi b)(t1 , . . ., tn ) = : b(. . ., 2t 1_ i - 1, . . .) ti > 2 . Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 13 Major result 1): The standard compositions on R(X* ) are inherited by %(X* ) to make it the fundamental cubical !-groupoid of X* . This is quite difficult to prove, and is non trivial even in dimension 2. The result is precise in that there is just enough filtration room to prove it. Major result 2): The quotient map p : R(X* ) ! %(X* ) is a Kan fibration of cubical sets. This result is almost unbelievable. Its proof has to give a systematic method of deforming a cube with the right faces `up to homotopy' into a cube with exactly the right faces, using the given homotopies. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 14 Here is an application which is essential in many proofs. Theorem: Lifting multiple compositions Let [ff(r)] be a multiple composition in %n (X* ). Then representatives a(r) of the ff(r) may be chosen so that the composition a(r) is well defined in Rn (X* ). Explanation: To say that [ff(r)] is well defined says representatives a(r) agree with neighbours up to homotopy, and these homotopies are arbitrary. All these homotopies have to be used to obtain the representatives which actually agree with their neighbours. This is an example of why setting up higher homotopy groupoids is not straightforward. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 15 Proof: The multiple composition [ff(r)] determines a cubical map A : K ! %(X* ) where the cubical set K corresponds to a subdivision of the geometric cube. Consider the diagram * _____________//_R(X* ) . | -== | - | | - | | - | | - | | 0 p| | -A | | - | | -- | | - | fflffl|- fflffl| _____________// K A %(X* ) Then K collapses to *, written K & *. By the fibration result, 0 A lifts to A , which represents a(r), as required. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 16 Major result 3): If X1 = U [W V with U, V, W open, and the induced filtrations U* , V* , W* are connected then C) X* is connected; I) The following diagram %(W* ) _____//%(V* ) | | | | fflffl| fflffl| %(U* ) ____//_%(X* ) is a pushout of cubical !-groupoids with connection. Proof Outline: Verify the universal property with regard to maps to G. Take a 22 %(X* )n . Subdivide a = [a(r)] so that each a(r) lies in U or V . Use connectivity to deform a(r) to 0 0 0 a(r) 2 R(Y* ), Y = U, V, W such that a = [a(r)] is defined. Map the pieces over to G and recombine. Analogy with email. You have to prove independence of choices. This needs a technology of commutative cubes. Applications: Translate to crossed complexes. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 17 Down to earth and explain crossed modules JHC Whitehead in 1939-50 abstracted properties of @ : ß2 (X, X1 , a) ! ß1 (X1 , a) * *(*) to define a Crossed Module: morphism of groups p ~ : M ! P and action M x P ! M, (m, p) 7! m of the group P on the group M such that: p -1 -1 ~n o ~(m ) = p (~m)p o n mn = m for all m, n 2 M, p 2 P. Now a key concept in non commutative algebraic topology and homological algebra. Simple consequences of the axioms: o Im ~ is normal in P o Ker ~ is central in M and is acted on trivially by Im ~, so that Ker ~ inherits an action of M= Im ~. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 18 Standard algebraic examples: (i) normal inclusion M / P ; (ii) inner automorphism map Ø : M ! Aut M ; (iii) the zero map 0 : M ! P where M is a P -module; (iv) an epimorphism M ! P with kernel contained in the centre of M . Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 19 Theorem (Mac Lane-Whitehead, 1950 Crossed modules classify all connected weak based homotopy 2-types. Crossed modules as candidates for 2-dimensional groups? 1974 (published 1978): Brown and Higgins proved that the functor 2 : (based pairs of spaces) ! (crossed modules) preserves certain colimits. This allows totally new 2-dimensional homotopical calculations. One can compute with crossed modules in a similar, but more complicated, manner to that for groups. Recent work with Chris Wensley uses symbolic computation to do more sums. The aim of these new calculations is to prove (i.e. test) the power of the machinery. Grothendieck's aim in Pursuing Stacks was Non Abelian Homological Algebra. The real aim is an extension of method, in the belief that methods last longer than theorems. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 20 Induced crossed modules (Brown-Higgins, 1978). f : P ! Q a group morphism. 8 9 >< M ______//_f* (M ) > | | = crossed ~ || | crossed | |@| P -module >: fflffl|| fflffl|> Q-module ____f_____// ; P Q f* : crossed P -modules ! crossed Q-modules Universal property: left adjoint to pullback by f . Construction: generated by symbols q m , m 2 M, q 2 Q q -1 with @(m ) = q (f m)q and rules p q (f p)q (m ) = m , CM2 for @. Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 21 Example 1) Let f : P ! Q be a morphism of groups, inducing a cofibration sequence BP ! BQ ! C (Bf ). Algebraic description of the 2-type of C (Bf ) as an induced crossed module f* (P ! P ), so we can calculate specific examples. 1) (Brown, Wensley, 1995) M, P, Q finite =) f* (M ) finite. Hence computations of homotopy 2-type of B(C (Bf )) when ~ = 1P : P ! P and f : P / Q; more generally of a homotopy pushout __________// BP BQ | | | | fflffl| fflffl| B(M ! P ) _____//_X Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 22 0 2) ~ = 1 : F (R) ! F (R), ! : F (R) ! Q defined by ! : R ! Q. Then 0 @ : C (!) = !* (F (R)) ! Q is the free crossed Q-module on !. (Defined directly by Whitehead). Corollary is a major result: Theorem W (1949) 2 ß2 (X1 [ {er }r2R , a) ! ß1 (X1 , a) is isomorphic to the free crossed ß1 (X1 , a)-module on the classes of the attaching maps of the 2-cells. This is important for relating combinatorial group theory and 2-dimensional topology. (Identities amon relations. ) Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 23 Some Computer Calculations (C.D. Wensley using GAP): [m, n] is the nth group of order m in GAP. M / P ; f : P 6 S4 . Calculate f* M . 0 2 Set C2 = <(1, 2)>, C2 = <(1, 2)(3, 4)>, C2 = <(1, 2), (3, 4)>. _________________________________________________________________ | | | | | | | | | | | | | | | | | | M | P | f* M | ker @ |Aut (f* M ) | | |_|____|_________|__________________|__________|______________|_|__ | | | | | | | | | | | | | | | | | | C2 | C2 | GL(2, 3) | C2 | S4 C2 | | | | | | | | | | | | | | | | | | | | | | | | | | | | C3 | C3 | C3 SL(2, 3) | C6 | [144, 183] | | | | | | | | | | | | | | | | | | | | | | | | | | | | C3 | S3 | SL(2, 3) | C2 | S4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | S3 | S3 | GL(2, 3) | C2 | S4 C2 | | | | | | | | | | | | | | | | | | | | 0 | 0 | | 3 | | | | | C2 | C2 | [128, ?] | C4 C2 | | | | | | | | | | | | | | | | | | | | | 0 | 2 | + | | | | | | C2 |C2 , C4 | H | C4 | S4 C2 | | | | | | 8 | | | | | | | | | | | | | | 0 | | 3 | | | | | | C2 | D8 | C2 | C2 | SL(3, 2) | | | | | | | | | | | | | | | | | | | | 2 | 2 | | | | | | | C2 | C2 | S4 C2 | C2 | S4 C2 | | | | | | | | | | | | | | | | | | | | 2 | | | | | | | | C2 | D8 | S4 | I | S4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | C4 | C4 | [96, 219] | C4 | [96, 227] | | | | | | | | | | | | | | | | | | | | | | | | | | | | C4 | D8 | S4 | I | S4 | | | | | | | | | | | | | | | | | | | | | | | | | | | | D8 | D8 | S4 C2 | C2 | S4 C2 | | |_|_____|________|_________________|__________|_______________|_|_ ker @ ~= ß2 (C (Bf )). Need the non commutative structure to find this. Hard to determine the first k-invariant in 3 H (Coker @, ker @). Geometric significance of the table? Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 24 Conclusion: Key inputs: VKT for the fundamental groupoid ß1 (X, X0 ) on a set X0 of base points (RB: 1967). CLAIM: all of 1-dimensional homotopy theory is better presented using groupoids rather than groups. Substantiated in books by Brown (1968) and Higgins (1971). Ignored by most topologists! Hint as to higher dimensional prospects: (Group objects in groupoids) , (crossed modules). (Grothendieck school, 1960s). Generalising: (congruences on a group) , (normal subgroups). Further outlook: Generalise this to other algebraic structures than groups. See work of Fr öhlich, Lue, Tim Porter. Groupoids in Galois Theories (Grothendieck, Magid, Janelidze). Towards Non Commutative Algebraic Topology: UCL May 7, 2003 * * 25 So look for higher homotopy groupoids. And applications of groupoids, multiple groupoids, and higher categorical structures in mathematics and science. Hence the term higher dimensional algebra (RB, 1987). Web search shows many applications. Work strongly uses categorical methods, and also uses computational tools for some calculations. Pursuing Stacks has been a strong international influence. I gave an invited talk in Delhi in February to an International Conference on Theoretical Neurobiology! It is still early days!