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 *
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Acknowledgements to: work of Henry Whitehead;
many collaborators, particularly
Philip Higgins, JeanLouis Loday, Tim Porter,
Chris Wensley;
21 Bangor research students;
Alexander Grothendieck, for correspondence a`baton
rompu 198293 and Pursuing Stacks (1983, 600
pages).
References: http://www.bangor.ac.uk/~mas010
http://www.bangor.ac.uk/~mas010/fieldsart3.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 *
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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 *
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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 kinvariant 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 *
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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 localtoglobal 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 *
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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 *
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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 2type of X .
What is going on?
Towards Non Commutative Algebraic Topology: UCL May 7, 2003 *
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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.
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Some examples:
____________________________________________________________________________*
*_____
  
  
 topological data algebraic data 
______________________________________________________________________*
*______
  
  
 spaces with base point groups 
______________________________________________________________________*
*_____
  
  
 spaces with a set of base  
  
 groupoids 
  
 points  
______________________________________________________________________*
*_____
  
  
 based pairs crossed modules 
______________________________________________________________________*
*_____
  
  
 filtered spaces crossed complexes 
______________________________________________________________________*
*_____
  
 n  
 ncubes 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_ *
* 
 *
* 
 Tcomplexes complexes !groupoids *
* 
__________________________________________________________________________*
*____ 
The equivalence of the red structures is required for
the proof of the BrownHiggins GVKT.
Towards Non Commutative Algebraic Topology: UCL May 7, 2003 *
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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.
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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 *
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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* = ncube 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 *
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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 *
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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 *
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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 *
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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 *
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Down to earth and explain crossed modules
JHC Whitehead in 193950 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 *
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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 *
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Theorem (Mac LaneWhitehead, 1950 Crossed modules
classify all connected weak based homotopy 2types.
Crossed modules as candidates for 2dimensional
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
2dimensional 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 *
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Induced crossed modules (BrownHiggins, 1978).
f : P ! Q a group morphism.
8 9
>< M ______//_f* (M ) >
  =
crossed ~   crossed
 @
P module >: fflffl fflffl> Qmodule
____f_____// ;
P Q
f* : crossed P modules ! crossed Qmodules
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 *
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Example 1) Let f : P ! Q be a morphism of groups,
inducing a cofibration sequence
BP ! BQ ! C (Bf ).
Algebraic description of the 2type 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 2type 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
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0
2) ~ = 1 : F (R) ! F (R), ! : F (R) ! Q defined by
! : R ! Q. Then
0
@ : C (!) = !* (F (R)) ! Q
is the free crossed Qmodule 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 2cells.
This is important for relating combinatorial group
theory and 2dimensional topology. (Identities amon
relations. )
Towards Non Commutative Algebraic Topology: UCL May 7, 2003 *
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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 kinvariant in
3
H (Coker @, ker @).
Geometric significance of the table?
Towards Non Commutative Algebraic Topology: UCL May 7, 2003 *
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Conclusion:
Key inputs: VKT for the
fundamental groupoid ß1 (X, X0 ) on a set X0 of base
points (RB: 1967).
CLAIM: all of 1dimensional 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 *
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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!