Nonabelian Algebraic Topology
Ronald Brown*
July 15, 2004
UWB Math Preprint 04.15
Abstract
This is an extended account of a short presentation with this title giv*
*en at the Min-
neapolis IMA Workshop on `n-categories: foundations and applications', Ju*
*ne 7-18, 2004,
organised by John Baez and Peter May.
Introduction
This talk gave a sketch of a book with the title Nonabelian algebraic topology *
*being written
under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker an*
*d Rafael
Sivera (Valencia) [6]. The aim is to give in one place a full account of work b*
*y R. Brown and
P.J. Higgins since the 1970s which defines and applies crossed complexes and cu*
*bical higher
homotopy groupoids. This leads to a distinctive account of that part of algebra*
*ic topology which
lies between homology theory and homotopy theory, and in which the fundamental *
*group and
its actions plays an essential role. The reason for an account at this Workshop*
* on n-categories
is that the higher homotopy groupoids defined are cubical forms of strict multi*
*ple categories.
Main applications are to higher dimensional nonabelian methods for local-to*
*-global prob-
lems, as exemplified by van Kampen type theorems. The potential wider implicati*
*ons of the
existence of such methods is one of the motivations of this programme.
The aim is to proceed through the steps
` ' ` '
(geometry)____//_underlying____//_(algebra)_//_(algorithms)_//_ computer *
* .
processes implementation
___________________________________
*I am grateful to the IMA for support at this Workshop, and to the Leverhulm*
*e Trust for general support.
1
Nonabelian Algebraic Topology *
* 2
The ability to do specific calculations, if necessary using computers, is seen *
*as a kind of test
of the theory, and one which also leads to seeking of new results; such calcula*
*tions seem at
this stage of the subject to require strict algebraic models of homotopy types.*
* We obtain
some nonabelian calculations, and it is this methodology which we term nonabeli*
*an algebraic
topology. It is fortunate that higher categorical structures, and in particular*
* higher groupoid
structures, do give nonabelian algebraic models of homotopy types which allow s*
*ome explicit
calculation. They have also led to new algebraic constructions, such as a nona*
*belian tensor
product of groups, of Lie algebras, and of other algebraic structures, with rel*
*ations to homology
of these structures (see the references in [3]).
1 Background
Topologists of the early 20th century dreamed of a generalisation to higher dim*
*ensions of the
nonabelian fundamental group, for applications to problems in geometry and anal*
*ysis for which
group theory had been successful. The dream seemed to be shattered by the disc*
*overy that
~Cech's apparently natural 1932 generalisation of the fundamental group, the hi*
*gher homotopy
groups ßn(X, x), were abelian in dimensions > 2. So `higher dimensional groups'*
* seemed to be
just abelian groups, and the dream seemed to be a mirage.
Despite this, the relative homotopy groups ßn(X, A, x) were found to be in *
*general non-
abelian for n = 2, and as a result J.H.C. Whitehead in the 1940s introduced the*
* term `crossed
module' for the properties of the boundary map
@ : ß2(X, A, x) ! ß1(A, x)
and the action of ß1(A, x) on ß2(X, A, x). In investigating `adding relations t*
*o homotopy groups'
he proved the subtle result (call it Theorem W) that if X is obtained from A by*
* adding 2-cells,
then ß2(X, A, x) is the free crossed ß1(A, x)-module on the characteristic maps*
* of the 2-cells.
The proof used transversality and knot theory.
A potentially new approach to homotopy theory derived from the expositions *
*in Brown's
1968 book [2] and Higgins' 1971 book [11], which in effect suggested that most *
*of 1-dimensional
homotopy theory can be better expressed in terms of groupoids rather than group*
*s. This
led to a search for the uses of groupoids in higher homotopy theory, and in par*
*ticular for
higher homotopy groupoids. The basic intuitive concept was generalising from th*
*e usual partial
compositions of homotopy classes of paths to partial compositions of homotopy c*
*lasses (of some
form) of cubes. One aim was a higher dimensional version of the van Kampen theo*
*rem for the
fundamental group. A search for such constructs proved abortive for some years *
*from 1966.
However in 1974 we observed that Theorem W gave a universal property for ho*
*motopy
in dimension 2, which was suggestive. It also seemed that if the putative highe*
*r dimensional
Nonabelian Algebraic Topology *
* 3
groupoid theory was to be seen as a success it should at least recover Theorem *
*W. But Theorem
W is about relative homotopy groups! It therefore seemed a good idea for us to *
*look at the
relative situation in dimension 2, that is, to start from a based pair X* = (X,*
* A, C), where C
is a set of base points, and define a homotopy double groupoid æ(X*) using maps*
* of squares.
The simplest, and most symmetric, possibility seemed to be to consider R2(X*) a*
*s the set of
maps f : I2 ! X which take the edges of I2 to the subspace A and take the verti*
*ces to the set
C, and then to form æ2(X*) as a quotient by homotopy rel vertices of I2 and thr*
*ough elements
of R2. Amazingly, this gave a double groupoid, whose 1-dimensional part was the*
* fundamental
groupoid ß1(A, C) on the set C. The proof is not entirely trivial, and uses a f*
*illing technique.
Previous to this, work with Chris Spencer in 1971-2 had investigated the no*
*tion of double
groupoid, shown a clear relation to crossed modules, and introduced the notion *
*of connection
on a double groupoid so as to define a notion of commutative cube in a double g*
*roupoid,
generalising the notion of commutative square in a groupoid. Once the construct*
*ion of æ2(X*)
had been given, a proof of a 2-dimensional van Kampen theorem came fairly quick*
*ly. The
paper on this was submitted in 1975, and appeared in 1978 [5] (after a third re*
*feree's report,
the reports of the first two referees being withheld). The equivalence between *
*double groupoids
with connection and crossed modules established by Brown and Spencer [7], and s*
*o relating the
double groupoid æ2(X*) to the second relative homotopy groups, enabled new calc*
*ulations of
some nonabelian second relative homotopy groups in terms of the associated cros*
*sed module,
and so enabled calculation of some homotopy 2-types.
Thus there is considerable evidence that the nonabelian fundamental group i*
*s naturally
generalised to various forms of crossed modules or double groupoids, and that t*
*hese structures
enable new understanding and calculations in dimension 2. The important point *
*is that in
dimension 2 we can easily define homotopical functors, that is, functors define*
*d in terms of ho-
motopy classes of maps, in a manner analogous to that of the fundamental group *
*and groupoid,
and then prove, but not so easily, theorems which enable us to calculate direct*
*ly and exactly
with these functors without using homological techniques.
As an example, we give the following problem. Given a morphism ' : P ! Q o*
*f groups,
calculate the homotopy 2-type of the mapping cone X of the induced map of class*
*ifying spaces
B' : BP ! BQ. This 2-type can be described completely in terms of the crossed *
*module
@ : '*(P ) ! Q `induced' from the identity crossed module 1 : P ! P by the morp*
*hism ' [8].
There are some general results on calculating induced crossed modules, whil*
*e for some
calculations, we have had to use a computer. The following table, taken from [8*
*], contains the
results of computer calculations, using the package XMOD [1] in the symbolic co*
*mputation
system GAP [9]. The examples are for Q = S4 and various subgroups P of Q. The c*
*omputer
has of course full information on the morphisms @ : '*(P ) ! S4 in terms of gen*
*erators of the
groups in the table. The third column gives ß2 = Ker @ = ß2(X). The numbers 128*
*.2322 and
96.67 refer to groups in the GAP4 table of groups.
Nonabelian Algebraic Topology *
* 4
_________________________________
| | P | ' P | ß | |
|_|_____________|__*______|__2__|_|_
| | <(1, 2)> GL(|2, 3) |C | |
| | | | 2 | |
| | S G|L(2, 3) |C | |
| | 3 | | 2 | |
| | <(1, 2), (3, 4)>S|C |C | |
| | |42 | 2 | |
| | D | S C | C | |
| | 8 | 4 2 | 2 | |
| | C C |SL(2, 3) |C | |
| | 3 3| | 6 | |
| | <(1, 2)(3, 4)>12|8.2322C |C3 | |
| | | 4| 2 | |
| | C |96.67 |C | |
|_|_______4______|_________|_4__|_|
In fact the pairs of equal groups in the second column give pairs of isomorphic*
* crossed modules,
though the proof of this needs some calculation.
It is not claimed that these results are in themselves significant. But the*
*y do show that:
(i) there are feasible algorithms; (ii) the homotopy groups, even with their st*
*ructure as modules
over the fundamental group, represent but a pale shadow of the actual homotopy *
*type; (iii) this
homotopy type can sometimes be represented by convenient nonabelian structures;*
* (iv) we can
use strict structures for explicit calculations in homotopy theory; (v) the col*
*imit formulation
allows for some complete calculations, not just up to extension; (vi) there is *
*still the problem
of determining the triviality or not of the first k-invariant in H3(S4, ß2).
It may be argued that the use of crossed modules has not yet been extended *
*to the geometric
and analytic problems which we have described above as motivating the need for *
*a higher
dimensional version of the fundamental group and groupoid. This argument shows *
*that there
is work to be done! Indeed, there are few books on algebraic topology other th*
*an [2] which
mention even the fundamental groupoid on a set of base points.
Once the definitions and applications of the homotopy double groupoid of a *
*based pair of
spaces had been developed, it was easy to guess a formulation for a homotopy gr*
*oupoid and
GvKT in all dimensions, namely replace:
the based pair by a filtered space;
the crossed module functor by a crossed complex functor;
and the homotopy double groupoid by a corresponding higher homotopy groupoid.
However the proofs of the natural conjectures proved not easy, requiring develo*
*pments in both
the algebra of higher dimensional groupoids and in homotopy theory.
2 Main results
Major features of the work over the years with Philip J. Higgins and others can*
* be summarised
in the properties of the following diagram of categories and functors:
Nonabelian Algebraic Topology *
* 5
Diagram 2.1
filtered spacesH
rr88r dHHHdHH
æ rrrrrrrrr HHHHHH
rrrrr HHHHHH
rrrrrr HHHHH
xxrrr|r|rOrU*r B HHHH$$HH
cubical fl //crossedH
!-groupoids ______________________complexesoo_
with connections ~
Let us first say something about the categories FTop of filtered spaces, an*
*d !-Gpds of cubical
!-groupoids with connections. A filtered space consists of a (compactly generat*
*ed) space X and
an increasing sequence of subspaces
X* : X0 X1 X2 . . .Xn . . .X.
With the obvious morphisms, this yields the category FTop.
One example of a filtered space is a manifold equipped with a Morse functio*
*n. This example
is studied using crossed complex techniques in [12, Chapter VI] (free crossed c*
*omplexes are there
called homotopy systems).
Another, and more standard, example of a filtered space is a CW -complex wi*
*th its skeletal
filtration. In particular, the n-cube In, the product of n-copies of the unit i*
*nterval I = [0, 1],
with its standard cell structure, becomes a filtered space In*, and so for a fi*
*ltered space X*
we can define its filtered cubical singular complex R2(X*) which in dimension n*
* is simply
FTop (In*, X*).
Thus R2(X*) is an example of a cubical set: it has for n > 1 and i = 1, . .*
*,.n, face maps
@i from dimension n to dimension (n - 1), and in the reverse direction degenera*
*cy maps "i, all
satisfying a standard set of rules. We also need in the same direction as the "*
*i an additional
degeneracy structure, called connections, i, i = 1, . .,.n, which are defined *
*by the monoid
structures max , min on I. There are also in dimension n and for i = 1, . .,.n *
*partially defined
compositions Oigiven by the usual gluing a Oib of singular n-cubes a, b such th*
*at @+ia = @-ib.
There is an equivalence relation on R2(X*)n given by homotopy through fil*
*tered maps rel
vertices of In. This gives a quotient map
p : R2(X*) ! æ2(X*) = (R2(X*)= ). *
* (*)
It is easily seen that æ2(X*) inherits the structure of cubical set with connec*
*tions. A major
theorem is that the compositions Oi are also inherited, so that æ2(X*) becomes *
*a cubical !-
groupoid with connections. The proof uses techniques of collapsing on subcompl*
*exes of the
n-cube. A development of these techniques to chains of partial boxes proves th*
*e surprising
result, essential for the theory, that the quotient map p of (*) is a cubical K*
*an fibration. This
Nonabelian Algebraic Topology *
* 6
can be seen as a rectification result, by which we mean a result using deformat*
*ions to replace
equations up to homotopy by strict equations.
An advantage of the functor æ and the category !-Gpds is that in this conte*
*xt we can prove
a Generalised van Kampen Theorem (GvKT), that the functor æ preserves certain c*
*olimits.
This yields precise calculations in a way not obtainable with exact or spectral*
* sequences.
The proof is `elementary' in the sense that it does not involve homology or*
* simplicial ap-
proximation. However the proof is elaborate. It depends heavily on the algebrai*
*c equivalence
of the categories !-Gpds and Crsof crossed complexes, by functors fl, ~ in the *
*Main Diagram.
This equivalence enables the notion of commutative cube in an !-groupoid, and t*
*he proof that
any composition of commutative cubes is commutative. It is also shown that flæ *
*is naturally
equivalent to a functor defined in a classical manner using the fundamental g*
*roupoid and
relative homotopy groups, as detailed below.
In detail we set (X*)1 = ß1(X1, X0), the fundamental groupoid of X1 on the*
* set of
base points X0, and for n > 2 we let (X*)n be the family of relative homotopy *
*groups
ßn(Xn, Xn-1, x), x 2 X0. Setting C0 = X0 and Cn = (X*)n we find that C obtain*
*s the
structure of crossed complex,
Diagram 2.2
. ._.__//_Cnffin//_Cn-1_//_._._.//_C2ffi2//_C1
t|| |t| |t| s||t||
fflffl|fflffl| |fflfflffflffl|flffl|
C0 C0 C0 C0
where the structure and axioms are those universally satisfied by this example *
* (X*).
If X* is the skeletal filtration of a reduced CW -complex X, then (X*) sho*
*uld be regarded
as a powerful replacement for the cellular chains of the universal cover of X, *
*with operators from
the fundamental group of X. However the use of crossed complexes rather than t*
*he cellular
chains allows for better realisation properties. Also the functor allows for *
*many base points,
so that groupoids are used in an essential way.
The GvKT for æ immediately transfers to a GvKT for this classical functor *
*. We again
emphasise that the proof of the GvKT for is elaborate, and tightly knit, but *
*elementary,
in that it requires no background in homology, or simplicial approximation. It*
* is a direct
generalisation of the proof of the van Kampen Theorem for the fundamental group*
*oid. Indeed
it was the intuitions that such a generalisation should exist that gave in 1966*
* the impetus to
the present theory.
With this GvKT we deduce in the first instance:
o the usual vKT for the fundamental groupoid on a set of base points;
Nonabelian Algebraic Topology *
* 7
o the Brouwer degree theorem (Sn is (n - 1)-connected and ßnSn = Z);
o the relative Hurewicz theorem;
o Whitehead's theorem that ß2(A [ {e2~}~, A, x) is a free crossed ß1(A, x)-*
*module;
o a more general excision result on ßn(A [ B, A, x) as an induced module (c*
*rossed module
if n = 2) when (A, A \ B) is (n - 1)-connected;
o computations of ßn(A [ B, A \ B, x) when (A, A \ B), (B, A \ B) are (n - *
*1)-connected.
Whitehead's theorem, and the last two results for n = 2, are nonabelian results*
*, and hence not
obtainable easily, if at all, by homological methods.
The assumptions required of the reader are quite small, just some familiari*
*ty with CW -
complexes. This contrasts with some expositions of basic homotopy theory, where*
* the proof of
say the relative Hurewicz theorem requires knowledge of singular homology theor*
*y. Of course
it is surprising to get this last theorem without homology, but this is because*
* it is seen as a
statement on the morphism of relative homotopy groups
ßn(X, A, x) ! ßn(X [ CA, CA, x) ~=ßn(X [ CA, x)
and is obtained, like our proof of Whitehead's theorem , as a special case of a*
*n excision result.
The reason for this success is that we use algebraic structures which model the*
* geometry and
underlying processes more closely than those in common use.
The cubical classifying space B2C construction for a crossed complex C is g*
*iven by the
geometric realisation of a cubical nerve N2 C defined by
(N2 C)n = Crs( (In*), C).
The crossed complex C has a filtration C* by successive truncations, and this g*
*ives rise to
the filtered space B(C) = B2C* for which there is a natural isomorphism B(C) ~*
*=C. The
fundamental groupoid and homotopy groups of B2C are just the fundamental groupo*
*id and
`homotopy groups' of C. In particular, if X* is the skeletal filtration of a CW*
* -complex X, and
C = (X*), then ß1(C) ~=ß1(X, X0) and for n > 2, ßn(C, x) ~=Hn(Xex), the homolo*
*gy of the
universal cover of X based at x.
The category FTop has a monoidal closed structure with tensor product given*
* by
[
(X* Y*)n = Xp x Yq.
p+q=n
It is also fairly easy to define a monoidal closed structure on the category !-*
*Gpds since this
category is founded in the structure on cubes. The equivalence of this category*
* with that of
Nonabelian Algebraic Topology *
* 8
crossed complexes, Crs, then yields a monoidal closed structure on the latter c*
*ategory, with
exponential law
Crs(A B, C) ~=Crs(A, CRS(B, C)).
The elements of CRS (B, C) are: in dimension 0 just morphisms B ! C; in dimensi*
*on 1, are
homotopies of morphisms; and in dimensions > 2 are forms of `higher homotopies'*
*. There is a
complicated formula for the tensor product A B, as generated in dimension n b*
*y elements
a b, a 2 Ap, b 2 Bq, p + q = n, with a set of relations and boundary formulae*
* related to the
cellular decomposition of a product of cells Ep x Eq, where Ep has for p > 1 ju*
*st 3 cells of
dimensions 0, p - 1, p respectively.
Using the filtered 1-cube I1*, and the values on this of æ and , we have u*
*nit interval objects
in each of the categories FTop, !-Gpds, Crs. This, with the tensor product, gi*
*ves a cylinder
object and so gives a homotopy theory on these categories.
Another difficult result is that the functor preserves certain tensor pro*
*ducts, for example
of CW -filtrations. This result is needed to deduce the homotopy addition lemma*
* for a simplex
by induction, and also to prove the homotopy classification results.
A major result is that there is a natural bijection of homotopy classes
[ (X*), C] ~=[X, B2C]
for any crossed complex C and CW -complex X, with its skeletal filtration. This*
* is a special
case of a weak equivalence
B2(CRS( (X*), C)) ! (B2C)X .
In fact the published version of this is for the simplicial classifying space B*
* C, and this has
given also an equivariant version of these results.
In this theory the homotopy addition lemma (HAL) for a simplex falls out by*
* a simple
inductive calculation, since the usual representation of the n-simplex n as a *
*cone on n-1
is exactly modelled in the crossed complex case, where the cone is defined usin*
*g the tensor
product. The explicit formulae for the tensor product, and so for any cone cons*
*truction, give
an easy calculation of the HAL formula.
There is an acyclic model theory for crossed complexes similar to that for *
*chain complexes.
So we obtain an Eilenberg-Zilber type theorem for crossed complexes. This has b*
*een developed
with explicit formulae by Tonks.
3 Conclusion
The book that is being written is divided into three parts, of which Part I is *
*now available on
the web [6]. Part I deals with the case of dimensions 1 and 2, and the proof an*
*d applications
Nonabelian Algebraic Topology *
* 9
of the GvKT in this dimension. The reasons for this are that the step from dim*
*ension 1 to
dimension 2 is a big one, and the reader needs to grasp the new ideas and techn*
*iques. Also a
gentle introduction is needed to calculating with the (nonabelian) crossed modu*
*les.
Part II should be regarded as a kind of handbook on crossed complexes. The *
*main properties
of crossed complexes and of the functor are stated. Applications and calculat*
*ions are deduced.
Thus crossed complexes are presented as a basic tool in algebraic topology.
The proofs in Part III of these basic properties of crossed complexes, such*
* as the monoidal
closed structure, and the GvKT, require however the use of the category of cubi*
*cal !-groupoids
with connection. The theory of these is developed in Part III.
Appendices give some basic theory needed, such as some category theory (adj*
*oint functors,
colimits, etc.).
A question asked at the Workshop was: Why concentrate on filtered spaces? T*
*he only an-
swer I can give is that they provide a workable basis for a theory of higher ho*
*motopy groupoids.
A more general theory, in some ways, is given by n-cubes of spaces, and the ass*
*ociated catn-
groups, but this does not have such an intuitive exposition as can be provided *
*for the filtered
case. Currently, there is in the absolute case no theory which works in dimensi*
*ons > 2, and
even the known dimension 2 case does not yet provide a range of explicit calcul*
*ations.
Peter May pointed out that, in contrast to stable homotopy theory, this wor*
*k brings the
fundamental group fully into the algebraic topology setting, and that such inpu*
*t is surely
needed for applications in algebraic geometry. Note that Grothendieck's increas*
*ingly influential
manuscript `Pursuing Stacks' [10] was intended as an account of nonabelian homo*
*logical algebra
(private communication). But that subject should in principle need `nonabelian*
* algebraic
topology' as a precursor.
The book makes no attempt to explain the work on catn-groups, which allows *
*for calculations
in homotopy theory which are more complex, and indeed more nonabelian, than tho*
*se given in
the planned book. That work thus continues the story of Nonabelian Algebraic To*
*pology, and
references to many authors and papers are given in [3].
The references which follow include some survey articles with wider referen*
*ces. We conclude
with a diagram of historical context for the current theory1.
References
[1] M. Alp and C. D. Wensley, XMod, Crossed modules and Cat1-groups: a GAP4 pac*
*kage,
(2004) (http://www.maths.bangor.ac.uk/chda/)
___________________________________
1I am grateful to Aaron Lauda for rendering this diagram in xypic.
Nonabelian Algebraic Topology *
* 10
[2] R. Brown, Elements of Modern Topology, McGraw Hill, Maidenhead, 1968. secon*
*d edition
as Topology: a geometric account of general topology, homotopy types, and t*
*he fundamental
groupoid, Ellis Horwood, Chichester (1988) 460 pp.
[3] R. Brown, `Higher dimensional group theory',
http://www.bangor.ac.uk/~mas010/hdaweb2.htm
[4] R. Brown, `Crossed complexes and homotopy groupoids as non commutative tool*
*s for higher
dimensional local-to-global problems', Proceedings of the Fields Institute *
*Workshop on Cat-
egorical Structures for Descent and Galois Theory, Hopf Algebras and Semiab*
*elian Cate-
gories, September 23-28, 2002, Contemp. Math. (2004). (to appear), UWB Math*
* Preprint
02.26.pdf (30 pp.)
[5] R. Brown and P. J. Higgins, On the connection between the second relative h*
*omotopy groups
of some related spaces, Proc.London Math. Soc., (3) 36 (1978) 193-212.
[6] R. Brown and R. Sivera, `Nonabelian algebraic topology', (in preparation) P*
*art I is down-
loadable from
(http://www.bangor.ac.uk/~mas010/nonab-a-t.html)
[7] R. Brown and C. B. Spencer, Double groupoids and crossed modules, Cahiers T*
*op. G'eom.
Diff., 17 (1976) 343-362.
[8] R. Brown and C. D. Wensley, `Computation and homotopical applications of in*
*duced crossed
modules', J. Symbolic Computation, 35 (2003) 59-72.
[9] The GAP Group, 2004, GAP - Groups, Algorithms, and Programming, version 4.4*
* , Tech-
nical report, (http://www.gap-system.org)
[10] A. Grothendieck, `Pursuing Stacks', 600p, 1983, distributed from Bangor. N*
*ow being edited
by G. Maltsiniotis for the SMF.
[11] P. J. Higgins, 1971, Categories and Groupoids, Van Nostrand, New York. Rep*
*rint Series,
Theory and Appl. Categories (to appear).
[12] V. Sharko, 1993, Functions on manifolds: algebraic and topological aspects*
*, number 131 in
Translations of Mathematical Monographs, American Mathematical Society.
Professor Emeritus R. Brown,
Department of Mathematics, University of Wales, Bangor, Dean St, Bangor,
Gwynedd LL57 1UT, United Kingdom
http://www.bangor.ac.uk/~mas010 email: r.brown@bangor.ac.uk
Nonabelian Algebraic Topology *
* 11
Some Context for Nonabelian Algebraic Topology
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