BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY
GREGORY LUPTON, N. CHRISTOPHER PHILLIPS, CLAUDE L. SCHOCHET,
AND SAMUEL B. SMITH
Abstract.Let A be a unital commutative Banach algebra with maximal
ideal space Max(A). We determine the rational H-type of GLn(A), the group
of invertible n x n matrices with coefficients in A, in terms of the rat*
*ional
cohomology of Max(A). We also address an old problem of J. L. Taylor. Let
Lcn(A) denote the space of "last columns" of GLn(A). We construct a natu*
*ral
isomorphism
~Hs(Max(A); Q) ~=ss2n-1-s(Lcn(A)) Q
for n > 1_2s+1 which shows that the rational cohomology groups of Max(A)*
* are
determined by a topological invariant associated to A. As part of our an*
*alysis,
we determine the rational H-type of certain gauge groups F(X, G) for G a*
* Lie
group or, more generally, a rational H-space.
1.Introduction
Let A be a unital commutative Banach algebra. By Max (A) we denote the
set of maximal ideals of the Banach algebra A topologized with the relative wea*
*k*-
topology. Let GLn (A) denote the group of invertible nxn matrices with coeffici*
*ents
in A. In this paper, we describe the rational homotopy of GL n(A) and also that*
* of
a second topological space associated to A.
The difficulty inherent in describing the ordinary homotopy theory of the gro*
*up
GL n(A) is apparent even in the simplest cases. For instance, when A = C we have
ssj(GL 2(C)) ~=ssj(S3)
for j > 1. Now ssj(S3) 6= 0 for infinitely many values of j, and these groups a*
*re
known only for j < 100 or so. Of course this particular problem becomes tractab*
*le
after rationalization, since J.-P. Serre showed that ssj(S3) is finite for j > *
*3.
In fact, substantially more can be said about GL n(C) rationally, using class*
*ical
results and methods. For observe that polar decomposition provides a canonical
deformation retraction of GLn(C) to the compact Lie group Un(C). By results goi*
*ng
back to H. Hopf, the rational cohomology algebra of Un(C) is an exterior algebra
generated in odd degrees 1, 3, . .,.2n - 1. From these facts, we deduce a ratio*
*nal
homotopy equivalence
GLn(C) 'Q S1 x S3 x . .x.S2n-1.
____________
Date: 11 September 2005.
2000 Mathematics Subject Classification. 46J05, 46L85, 55P62, 54C35, 55P15, *
*55P45.
Key words and phrases. commutative Banach algebra, maximal ideal space, gene*
*ral linear
group, space of last columns, rational homotopy theory, function space, rationa*
*l H-space, gauge
groups.
Research of the second author partially supported by NSF grant DMS 0302401.
1
2 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
(See Example 4.16 below.) This equivalence actually determines the structure of
GL n(C) as a rational H-space since a homotopy-associative H-space with only odd
rational homotopy groups is rationally homotopy-abelian. (See Corollary 4.27.)
Our first main result extends the above analysis to the general case. We dete*
*r-
mine the rational homotopy type of GL n(A) in terms of the rational ~Cech cohom*
*ol-
ogy groups of the space Max (A). In fact, we determine the full structure of GL*
* n(A)
as a rational H-space. To describe our results, we introduce some notation.
Given a space X with basepoint, we write XO for the path component of the
basepoint of X. Recall that if ss is an abelian group, then K(ss, n) is the cor*
*re-
sponding Eilenberg-Mac Lane space. (See Example 4.5.) If ss is countable, by a
result of J. Milnor, multiplication of loops gives K(ss, n) the structure of an*
* abelian
topological group, andQthis structure is unique up to homotopy. (Again, see Exa*
*m-
ple 4.5.) A product j 1K(ssj, j) of Eilenberg-Mac Lane spaces with its standa*
*rd
loop multiplication consequently also has the structure of an abelian topologic*
*al
group. (This multiplication is rarely unique, even up to homotopy. See Exam-
ple 4.6.)
Our first main result is the following.
Theorem 1. Suppose that A is a unital commutative Banach algebra with maximal
ideal space Max (A). Let
V~h,j= ~H2j-1-h(Max (A); Q).
Then there is a natural rational H-equivalence
Yn 2j-1Y
GL n(A)O 'Q K(V~h,j, h)
j=1 h=1
where the product of Eilenberg-Mac Lane spaces has the standard loop multiplica*
*tion.
Thus GL n(A)O has the rational H-type of an abelian topological group.
As a consequence of Theorem 1, we address an old question of J. L. Taylor
regarding the cohomology of the space Max(A). In [27], Taylor raised the questi*
*on of
characterizing topological invariants of Max(A) in terms of invariants of the a*
*lgebra.
Taylor showed that classical results give a way to characterize Hn(Max (A); Z) *
*for
n = 0, 1, 2. To be more specific:
o H0(Max (A); Z) consists of formal combinations of idempotents of A. Tayl*
*or
derives this from the Shilov Idempotent Theorem.
o H1(Max (A); Z) is the quotient of GL (A) by the image of the exponential
map exp:A ! GL (A). Taylor derives this from the Arens-Royden Theo-
rem. For A = C(X), the algebra of continuous complex valued functions
on a compact Hausdorff space, and when X is metrizable, K. Thomsen
observes (see the introduction to Section 4 of [30]) that this is a (193*
*4)
theorem of Bruschlinsky.
o H2(Max (A); Z) is the Picard group Pic(A) of A by a result of Forster. T*
*he
Picard group consists of isomorphism classes of finitely generated proje*
*ctive
A-modules P which are invertible in the sense that there exists some fin*
*itely
generated projective module Q such that P Q is a free module of rank 1
over A. For a different, and deeper, version of the Picard group, see [6*
*].
We note that these results are easy in the case A = C(X); the main point is t*
*hat
they may be extended to commutative Banach algebras.
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 3
In addition, we have the result of J. Dixmier and A. Douady [11]:
o H3(X; Z) is isomorphic to the Brauer group of algebra bundles over X with
fibre K, the compact operators on a separable, infinite-dimensional Hilb*
*ert
space.
As far as we know, there has been no further progress on describing Hn(Max (A*
*); Z)
in a concrete fashion. However, the recent interesting work of A. B. Thom [28] *
*does
shed additional light on the problem, albeit in a very abstract context.
As regards K-theory, we can add the following results, which were known to
Atiyah, Adams, Swan and others.
o K0(Max (A)) is isomorphic to K0(A), which is defined (for any ring A) as
the quotient of the free abelian group generated by all finitely generat*
*ed
projective left A-modules modulo the isomorphism and direct sum relation*
*s.
o K1(Max (A)) is isomorphic to GL 1(A)=GL 1 (A)O, the quotient of the infi-
nite general linear group of A by the connected component of the identit*
*y.
Now consider the natural action of GLn (A) on the right A-module An. The space
of "last columns" Lcn(A) is defined to be the orbit of the last standard basis *
*vector
en 2 An under the action of GL n(A). This is a familiar space, studied for exam*
*ple
in [23]. With our second main result, we provide an answer to Taylor's question*
* for
the rational (~Cech) cohomology groups of the space Max (A).
Theorem 2. Let A be a unital commutative Banach algebra. Then the rational
cohomology groups of Max (A) are determined by a topological invariant of A via*
* a
natural isomorphism
~Hs(Max (A); Q) ~=ss2n-1-s(Lcn(A)) Q
for n > 1_2s + 1.
The paper is organized as follows. In Section 2 we introduce the topological
spaces associated to a Banach algebra that we will need. In Section 3 we begin *
*our
examination of the homotopy groups of the general linear groups of the commuta-
tive Banach algebras. The group GL n(A) is known to be homotopy equivalent to
the function space F (Max (A), GLn(C)) by an important result of Davie [10]. We
reduce the problem of identifying Lcn(A) to one of studying the path component
of the constant map in the function space F (Max (A), S2n-1) (Theorem 3.6). Con-
sequently, we are concerned with studying function spaces of the form F (X, G),
for G a topological group or, more generally, of the rational homotopy type of a
topological group. In Section 4 we develop some ideas, mostly from rational hom*
*o-
topy theory, sufficient to understand completely the rational homotopy type of a
function space of this latter form when X and G are finite complexes. The resul*
*ts
of this section are developed in greater generality than strictly necessary, an*
*d some
are of independent interest for the homotopy of function spaces (e.g. Theorem 4*
*.10).
Finally, in Section 5, we prove Theorems 1 and 2 by applying the results develo*
*ped
in Section 4.
For any topological spaces X and Y we let F (X, Y ) denote the set of continu*
*ous
functions X ! Y with the compactly generated topology. (See Section IV.3 of [31*
*].)
This topology agrees with the compact-open topology on compact sets. Hence using
this topology rather than the compact-open topology has no effect on homotopy
and singular homology. If X and Y are based spaces then we let Fo(X, Y ) denote
the subspace of basepoint-preserving maps. We use [X, Y ] to denote the set of *
*based
4 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
homotopy classes of based maps from X to Y. When working with a topological
group we take the basepoint to be the identity of the group. Basepoints are alw*
*ays
assumed to be non-degenerate, that is, the inclusion of the basepoint into the *
*space
is a cofibration, or equivalently, the pair consisting of the space and its bas*
*epoint is
an NDR-pair. (See page 7 and Section I.4 of [31].) Observe that f :X ! Y induces
maps f*: F (W, X) ! F (W, Y ) and f* :F (Y, Z) ! F (X, Z) given by composition
and pre-composition with f, respectively. We use this notation also for the maps
induced by f on basepoint-preserving function spaces and on sets of homotopy
classes of maps.
A function space is typically not path-connected. We take the basepoint of
F (X, Y ) and Fo(X, Y ) to be the constant map sending X to the basepoint of
to Y. Thus F (X, Y )O denotes the space of maps from X to Y which are freely
homotopic to the constant map. Similarly, Fo(X, Y )O denotes the space of based,
null-homotopic maps. If G is a topological group then the function space F (X, *
*G)
is one also, with multiplication given by pointwise multiplication of maps. In *
*this
case, the basepoint is the constant map carrying X to the identity of G. For any
compact space X we let C(X) denote the C*-algebra of continuous complex-valued
functions on X with the operations given pointwise. We denote by Mn(A) the
algebra of n x n matrices with entries in A. If A is a Banach algebra, then Mn(*
*A)
is also a Banach algebra in an obvious way.
Acknowledgement. We would like to thank Daniel Isaksen and Jim Stasheff
for helpful discussions.
2.The Spaces
In this section, we let A be any Banach algebra. While we will be interested
later exclusively in the case A is commutative and unital, the results of this *
*section
can be stated in somewhat greater generality. We follow Corach and Larotonda
[7], [8] and Rieffel [23] in discussing the spaces associated to A that we will*
* need.
Following Section 3 of [30], we handle non-unital algebras by unitizing them and
making slight modifications of the definitions. In order to preserve functorial*
*ity, we
therefore unitize all algebras. Thus, the definitions of GL n(A) and Lcn(A) giv*
*en
below do not agree with those in the introduction. We will show in Proposition *
*2.11
below that, when A is unital, they are nevertheless equivalent.
Notation 2.1. If A is a Banach algebra, then we denote by A+ its unitization
A C, with the elements written a + ~ . 1 (or just a + ~) for a 2 A and ~ 2 C,*
* and
the multiplication obtained by applying the distributive law in the obvious way.
(See Definition 1 in Section 3 of [4], or the end of Section 1.2 of [20].)
The unitization is again a Banach algebra. If A is a C*-algebra, then there i*
*s a
unique choice of norm making A+ a C*-algebra (Theorem 2.1.6 of [20]).
Remark 2.2. When forming A+ , we add a new identity even if A is already unital.
In this case, A+ is isomorphic to the Banach algebra direct sum A C. Letting *
*1A
and 1A+ denote the identities of A and A+ , the isomorphism is given by
a + ~ . 1A+ 7! (a + ~ . 1A , ~).
This map is a homeomorphism and if A is a C*-algebra it is an isometry.
Definition 2.3. Let A be a Banach algebra. We set
GL n(A) = {a 2 Mn(A+ ): a is invertible and a - 1 2 Mn(A)},
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 5
with the relative topology from Mn(A+ ).
Note that Mn(A) is a subset of Mn(A+ ) in an obvious way, and that, in this
definition, 1 denotes the identity of Mn(A+ ).
Remark 2.4. Let A be a Banach algebra. For a = (aj,k) 2 GL n(A) and x =
(x1, . .,.xn) 2 (A+ )n, define ax to be the usual left action of a 2 Mn(A+ ), t*
*he
right A+ -module endomorphisms of (A+ )n when elements of (A+ )n are viewed as
column vectors. This defines a jointly continuous left action of GL n(A) on (A+*
* )n.
Definition 2.5. Let A be a Banach algebra. We define
Lgn(A) = {(x1, . .,.xn) 2 (A+ )n :x1, . .,.xn-1, xn - 1 2 A and
P n
there are y1, . .,.yn 2 A+ such that k=1ykxk}=.1
We define Lcn(A) to be the orbit of the last standard basis vector en = (0, . .*
*,.0, 1) 2
(A+ )n under the action of GL n(A) on (A+ )n of Remark 2.4. We let fl :GLn (A) !
Lcn(A) (or flA if it is necessary to specify the algebra) be fl(a) = a . en. Fi*
*nally, we
define TL n(A) GL n(A) to be {a 2 GLn(A): a . en = en}. All these spaces are
given the relative norm topology from (A+ )n or Mn(A+ ) as appropriate.
As we will see in Proposition 2.11, the definitions of GLn (A) and Lcn(A) sim*
*plify
to those given in the introduction when A is unital.
Remark 2.6. One immediately checks that TL n(A) is the subgroup of GL n(A)
consisting of matrices of the form
~ ~
x 0
c 1 ,
where x 2 GL n-1(A) and c is any row of elements of A of length n - 1.
Lemma 2.7. Let A be a Banach algebra.
(1) The closed vector subspace
Ln = {x = (x1, . .,.xn) 2 (A+ )n :x - en 2 An}
is invariant under the action of GL n(A).
(2) The space Lgn(A) is invariant under the action of GL n(A).
(3) We have Lcn(A) Lgn(A), and Lcn(A) is equal to the set of all n-tuples *
*of
elements of A+ which occur as the last column of an element of GL n(A).
Proof.We prove Part (1). Let a 2 GL n(A) and let x 2 Ln. Then a - 1 2 Mn(A),
from which it easily follows that (a - 1)x 2 An. Also x - en 2 An. So
ax - en = (a - 1)x + (x - en) 2 An.
For Part (2), let a 2 GL n(A) and let x 2 Lgn(A). Then a.x-en 2 An by Part (1*
*).
Choose y = (y1, . .,.yn) 2 (A+ )n such that y1x1 + . .+.ynxn = 1. Regarding y as
aProw vector, form the matrix product z = y . a-1 2 (A+ )n. Then one checks that
n
k=1zk(a . x)k = 1. So a . x 2 Lgn(A).
We prove Part (3). To identify Lcn(A) as the space of last columns, simply
observe that a . en = x if and only if (x1, . .,.xn) is the last column of a. T*
*he first
statement follows from this observation and (2).
6 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
The following theorem is primarily due to Corach and Larotonda (Theorem 1
of [7]) in the unital case, with parts due to Thomsen (Section 3 of [30]) in the
C*-algebra case. We will follow [30] for most of the proof, since it uses machi*
*nery
more familiar in Banach algebras.
Theorem 2.8. Let A be a Banach algebra. Then:
(1) For each x 2 Lgn(A), the map a 7! a.x is an open mapping (not surjective)
from GL n(A) to
Ln = {x = (x1, . .,.xn) 2 (A+ )n :x - en 2 An}.
(2) The map fl defines a principal locally trivial fibre bundle with structu*
*ral
group TL n(A).
(3) The sequence
TLn(A) -! GL n(A) -fl!Lcn(A)
is a Serre fibration, with base point preserving maps.
(4) The construction of the Serre fibration is natural in A, using the obvio*
*us
maps (described explicitly in the proof).
(5) The spaces GL n(A), TL n(A), and Lcn(A) are homeomorphic to open sub-
sets of Banach spaces, and have the homotopy type of CW complexes.
(6) We have Lgn(A)O = Lcn(A)O, and the map fl restricts to an open map of
GLn(A)O onto Lcn(A)O.
Proof.For C*-algebras (not necessarily unital), Part (1) is a slight strengthen*
*ing
of Lemma 3.3 of [30]. We follow the proof of Theorem 8.3 of [22]. Let a 2 GL n(*
*A)
and let x 2 Lgn(A). Set y = a . x, and let " > 0. We find ffi > 0 such that whe*
*never
z 2 Ln and kzk - ykk < ffi for 1 k n, then there is b 2 GL n(A) such that
kb - ak < " and b . x = z. This will clearly suffice to prove Part (1).
Now, because y 2 Lgn(A) (Lemma 2.7(2)), there are r1, . .,.rn 2 A such that
r1y1 + . .+.rnyn = 1. Set
M = max(kr1k, . .,.krnk),
and choose ffi > 0 so small that
ffin2M < 1 and (1 - ffin2M)-1 < _"_kak.
Let (z1, . .,.zn) 2 An satisfy kzk - ykk < ffi for all k. Let c be the matrix w*
*ith (j, k)
entry equal to (zj- yj)rk. Then c 2 Mn(A) because zj- yj 2 A for all j. Also,
cj,1x1 + . .+.cj,nxn = (zj- yj)r1x1 + . .+.(zj- yj)rnxn = zj- yj.
Therefore (1 + c)y = z, whence b = (1 + c)a satisfies bx = z. Finally,
Xn
kck k(zj- yj)rkk < ffin2M.
j,k=1
Therefore 1 + c is invertible in Mn(A), with inverse
(1 + c)-1 = 1 - c + c2 - c3 + . ...
In particular, (1 + c)-1 - 1 2 Mn(A), so c 2 GL n(A), and
k(1 + c)-1k (1 - kck)-1 < (1 - ffin2M)-1 < _"_kak.
So b 2 GL n(A) and kb - ak < ". This proves Part (1).
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 7
For C*-algebras, Parts (2) and (3) are in Corollary 3.5 of [30] and the discu*
*s-
sion after Lemma 3.7 of [30] (except for the part about the maps preserving the
basepoints, which is immediate from our conventions on basepoints), and Part (6)
follows from Lemma 3.7 of [30]. For Banach algebras, the only required change is
in the definition of Lgn(A); see Definition 2.5.
The second statement in Part (5) for Lcn(A) is Lemma 3.6 of [30]. For Banach
algebras, we first recall that, by Corollary 5.5 in Chapter 4 of [17], every op*
*en
subset of a Banach space has the homotopy type of a CW complex. So it is enough
to prove the first statement. For Lcn(A), Part (1) implies that Lcn(A) is an op*
*en
subset of the Banach space Ln there. Also, GL n(A) is homeomorphic to the open
subset of An2 consisting of those (aj,k)nj,k=12 An2 such that
2 3
a1,1+ 1 a1,2 . . . a1,n
66 a2,1 a2,2+ 1 . . . a2,n 7
64 .. . . . 77
. .. .. .. 5
an,1 an,2 . . .an,n+ 1
is invertible in A+ . Similarly, TL n(A) is homeomorphic to an open subset of
An(n-1).
Finally, we consider Part (4). Let A and B be Banach algebras, and let ': A !*
* B
be a continuous homomorphism (not necessarily unital even if A and B are both
unital). Then there is a continuous unital homomorphism '+ :A+ ! B+ given by
'+ (a + ~ . 1A ) = '(a) + ~ . 1B . One checks immediately that the correspondin*
*g map
'+n:Mn(A+ ) ! Mn(B+ ) sends GL n(A) into GL n(B), and, using the description
given in Remark 2.6, that it sends TLn (A) into TLn (B). Similarly, the map
(x1, . .,.xn) 7! ('+ (x1), . .,.'+ (xn))
is easily seen to send Lgn(A) into Lgn(B) and to send (0, . .,.0, 1) to (0, . .*
*,.0, 1).
If a 2 GL n(A) and x = (x1, . .,.xn) 2 Lgn(A), then the image of a . x in Lgn(B*
*) is
equal to
'+n(a) . ('+ (x1), . .,.'+ (xn)).
It follows that Lcn(A) is sent to Lcn(B), and that we have a commutative diagram
TLn(A) ----! GL n(A) --flA--!Lcn(A)
? ? ?
'+n?y ?y'+n ?y
TLn(B) ----! GL n(B) ----!flLcn(B),
B
which is what naturality means.
The spaces GL n(A) and Lcn(A) are, in general, not connected. When we focus
on homotopy theoretic properties of these spaces in subsequent sections, we will
restrict our attention to the connected components GL n(A)O and Lcn(A)O of the
basepoints. The following result justifies this restriction.
Proposition 2.9. Let A be a Banach algebra. The connected components of the
space GL n(A) are path connected and all homeomorphic. The same holds for
Lcn(A). The homeomorphisms can be chosen to be natural with respect to both
a single Banach algebra homomorphism and the map fl of the fibration of Theo-
rem 2.8(3).
8 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
Proof.The connected components of GL n(A) and Lcn(A) are open because these
spaces are homeomorphic to open subsets of Banach spaces, by Theorem 2.8(5).
Now note that connected open subsets of Banach spaces are path connected.
For the second part, let a 2 GL n(A). Then b 7! ab is a homeomorphism from
GL n(A) to itself which sends GL n(A)O to a connected, open, and closed subset *
*of
GL n(A), which must be the connected component of GL n(A) containing a. Simi-
larly, the map ben 7! aben, for b 2 GL n(A), is a homeomorphism from Lcn(A)O to
to the connected component of Lcn(A) containing aen. By definition, every eleme*
*nt
of Lcn(A) has the form ben for some b, so this argument applies to all componen*
*ts.
For naturality, if ': A ! B, and if a 2 GL n(A), we take the maps involving B
to be multiplication by '+n(a).
As a direct consequence of Theorem 2.8, we obtain a long exact homotopy se-
quence relating the homotopy groups of GL n(A) and Lcn(A). This sequence for
C*-algebras is due to Thomsen (Section 3 of [30]). Note that, in the statement
below, ss0(GL n-1(A)) and ss0(GL n(A)) are possibly nonabelian groups, and that
ss0(Lcn(A)) is just a set. That the sequence ends with 0 means that ss0(GL n(A)*
*) !
ss0(Lcn(A)) is surjective.
Theorem 2.10. For a Banach algebra A and for each n > 0 there is a long exact
sequence
. ._.________//_ssk(GL n-1(A))_//_ssk(GL n(A))_//_ssk(Lcn(A))BC_
________________________________________________
GFfflffl|
ssk-1(GL n-1(A))_____// . . ._______//ss1(GL n(A))__//_ss1(Lcn(A))BC_
________________________________________________
GFfflffl|
ss0(GL n-1(A))_____//ss0(GL n(A))____//ss0(Lcn(A))_______//0
which is natural in A.
Proof.The long exact sequence in homotopy for the Serre fibration of Theorem 2.*
*8(3)
gives the sequence of the theorem with ss*(TL n(A)) in place of ss*(GL n-1(A)).*
* (The
map ss0(GL n(A)) ! ss0(Lcn(A)) is surjective because GL n(A) ! Lcn(A) is surjec-
tive.) The map GL n-1(A) ! TLn(A) given by
~ ~
x 7! x0 01
is a deformation retraction (see the description of TLn(A) given in Remark 2.6)*
* and
hence induces an isomorphism on homotopy. This gives the sequence as stated.
Naturality is immediate from naturality of the fibration, Theorem 2.8(4).
We now turn to the simplifications in the unital case. Stated informally, if *
*A is
unital, then GL n(A) is the invertible group of Mn(A), the space Lcn(A) is the *
*set
of last columns of invertible elements of Mn(A), etc. The proof is immediate, a*
*nd
is omitted.
Proposition 2.11. Let A be a unital Banach algebra, and identify A+ with the
Banach algebra direct sum A C as in Remark 2.2. In the obvious way, further
identify Mn(A+ ) with Mn(A) Mn(C) and (A+ )n with An Cn. Then:
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 9
(1) The map a 7! (a, 1) defines an isomorphism of topological groups from the
group of invertible elements of Mn(A) to GL n(A).
(2) The map x 7! (x, (0, . .,.0, 1)) defines a homeomorphism from
P n
{(x1, . .,.xn) 2 An :There are y1, . .,.yn 2 A such that k=1ykxk}= 1
to Lgn(A).
These maps convert the obvious action of the invertible group of Mn(A) on Lgn(A)
into the restriction of the action defined in Remark 2.4 to GL n(A). In particu*
*lar,
they identify Lcn(A) with the orbit of (0, . .,.0, 1) under the action of the i*
*nvertible
group of Mn(A), and TL n(A) with the invertible elements of Mn(A) whose last
column is (0, . .,.0, 1).
These maps are natural for continuous unital homomorphisms of unital Banach
algebras.
3. Commutative Banach algebras
Suppose now that A is a commutative unital Banach algebra. In this case,
Max (A) is a non-empty compact Hausdorff space, and the Gelfand transform pro-
vides an algebra homomorphism
: A ! C(Max (A)).
See Section 17 of [4] or Section 1.3 of [20]. Recall that if A = C(X) then Max *
*(A) =
X and the Gelfand transform is an isomorphism (Theorem 2.1.10 of [20]). In
general, induces a continuous unital homomorphism
n :A Mn(C) -! Mn(C(Max (A))) = F (Max (A), Mn(C))
which restricts to a natural continuous homomorphism
n :GLn(A) -! GL n(C(Max (A))) = F (Max (A), GLn(C)).
Here we recall that the group operation in the space F (Max (A), GLn(C)) is poi*
*nt-
wise multiplication of functions.
Remark 3.1. For naturality statements, it is important to note that a unital
homomorphism ': A ! B of commutative unital Banach algebras induces, in a
functorial manner, a continuous function '*: Max(B) ! Max (A). If M B is a
maximal ideal, then '*(M) = '-1(M). If we think of the maximal ideal space as
consisting of all (continuous) unital homomorphisms to C, then '*(!) = ! O '.
It is then immediate that A 7! C(Max (A)) is a covariant functor. Thus it mak*
*es
sense to say that : A ! C(Max (A)) is natural. Similarly, A 7! ~H*(Max (A); Q)*
* is
a covariant functor.
The following important result of Davie bridges the Gelfand transform gap.
Theorem 3.2 (A. Davie). Let A be a unital commutative Banach algebra with
maximal ideal space Max (A). Then the natural homomorphism
n :GLn(A) -! F (Max (A), GLn(C)).
induced by the Gelfand transform is a homotopy equivalence.
Proof.This is a special case of Theorem 4.10 of [10].
10 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
We note in passing that Davie's work together with very elementary reason-
ing will substitute for the Shilov Idempotent Theorem and for the Arens-Royden
Theorem in the computation of H0(Max (A); Z) and H1(Max (A); Z) respectively.
For our purposes, the significance of this result is that it identifies GL n(A)*
*, up to
H-equivalence (as in Definition 4.2), as a particular gauge group. We devote the
remainder of this section to obtaining a related identification of Lcn(A)O.
Theorem 3.3. Let A be a unital commutative Banach algebra with maximal ideal
space Max (A). Then the Gelfand transform induces a natural map
L n :Lcn(A)O -! Lcn(C(Max (A))O
which is a homotopy equivalence.
Proof.Set X = Max (A). Apply Theorem 2.10 to A and C(X), and naturality in
Theorem 2.10 to the Gelfand transform, obtaining a commutative diagram with
long exact rows which ends with
ss0(GL n-1(A)) ----! ss0(GL n(A)) ----! ss0(Lcn(A)) ----! 0
?? ? ?
y ?y ?y
ss0(GL n-1(C(X)))----! ss0(GL n(C(X)))----! ss0(Lcn(C(X)))----! 0.
Theorem 3.2 and the Five Lemma imply that ssk(Lcn(A)) ! ssk(Lcn(C(X))) is
an isomorphism for all k 1, and a direct argument shows that ss0(Lcn(A)) !
ss0(Lcn(C(X))) is a bijection of sets. Thus, L n is a weak homotopy equivalence*
* (on
every connected component, by Proposition 2.9). Since these spaces are homotopy
equivalent to CW complexes (by Theorem 2.8(5)), it follows from Theorem V.3.5
of [31] that L n is a homotopy equivalence.
We now want to explicitly identify Lcn(C(X)).
Lemma 3.4. Let X be a compact Hausdorff space. Identify Lgn(C(X)) with a sub-
set of C(X)n as in Proposition 2.11(2), and further identify C(X)n with F (X, C*
*n)
by sending (f1, . .,.fn) to x 7! (f1(x), . .,.fn(x)) Then the image of Lgn(C(X)*
*) is
exactly F (X, Cn - {0}).
Proof.Let f 2 F (X, Cn), and write f(x) = (f1(x), . .,.fn(x)).
Suppose f 2 F (X, Cn - {0}). Set
_____
gk(x) = ________fk(x)_______|f. 2 2
1(x)| + . .+.|fn(x)|
(The denominator is never zero by hypothesis.) Then g1f1 + . .+.gnfn = 1, so
(f1, . .,.fn) 2 Lgn(C(X)).
Now suppose that there is x 2 X such that f(x) = 0. Then everything in the id*
*eal
generated by f1, . .,.fn must vanish at x, so (f1, . .,.fn) is not in Lgn(C(X)).
Corollary 3.5. Under the obvious map F (X, Cn) ! C(X)n, the image of the space
F (X, Cn - 0)Ois exactly Lcn(C(X))O.
Proof.Combine Lemma 3.4 and Theorem 2.8(6).
The following theorem gives the desired identification of the space Lcn(A).
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 11
Theorem 3.6. Let A be a commutative unital Banach algebra. Then there is a
natural homotopy equivalence
Lcn(A)O ' F (Max (A), S2n-1)O.
Proof.The Gelfand transform induces a natural homotopy equivalence
Lcn(A)O -'!Lcn(C(Max (A)))O
by Proposition 3.3. Corollary 3.5 produces a natural homeomorphism
Lc n(C(Max (A)))O F (Max (A), Cn - {0})O.
The deformation retraction
Cn - {0} -! S2n-1
given by r(v) = v=kvk induces a natural homotopy equivalence
F (Max (A), Cn - {0})O -'!F (Max (A), S2n-1)O.
and so the composite
(Lcn(A))O ! Lcn(C(Max (A)))O ! F (Max (A), Cn - {0})O ! F (Max (A), S2n-1)O
is the required natural homotopy equivalence.
4.H-Spaces, Function Spaces, and Rational Homotopy
Davie's Theorem (Theorem 3.2) gives a multiplicative homotopy equivalence be-
tween GL n(A) and the function space F (Max (A), GLn(C)). Further, as mentioned
in the introduction, polar decomposition gives a multiplicative homotopy equiva-
lence Un(C) -'! GL n(C). In this section, we give a complete description of the
rationalization of the group F (X, G) for G a compact Lie group provided X is a
finite complex. In fact, as it is necessary for our analysis we work in somewh*
*at
greater generality, allowing G to be an H-space with various restrictions. We a*
*lso
analyze the rationalization of the basepoint-preserving function spaces Fo(X, G*
*).
Throughout this section, we thus assume all spaces come equipped with a fixed,
non-degenerate basepoint.
We let G denote a connected H-space (Section III.4 of [31]). That is, we assu*
*me
G comes equipped with a continuous map ~: G x G ! G, for which there is an
element e 2 G such that the restrictions of ~ to {e}xG ! G and to Gx{e} ! G are
both homotopic to the identity map of G. The map ~ is called the multiplication,
and we will assume_as for topological groups_that the so-called identity element
e is chosen as the basepoint of G. In what follows, we will write either the pa*
*ir
(G, ~) or just G depending on whether or not we are concerned with the particul*
*ar
multiplication for G or just the fact that G is an H-space.
While we mainly have in mind the case in which (G, ~) is a topological group,
our results in this section require only that ~ be homotopy-associative and have
a homotopy inverse: a map :G ! G such that ~ O ( x 1) O : G ! G and
~O(1x )O : G ! G are both null-homotopic. That is, we will require that (G, ~)
be group-like as defined before Theorem III.4.14 of [31]. We begin by recalling*
* some
basic facts about group-like spaces in ordinary homotopy theory.
We will often require that G be a CW complex. In this context, recall the
following result of I. James, Corollaries 1.2 and 1.3 of [15].
12 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
Proposition 4.1 (I. James). Let (G, ~) be a homotopy-associative H-space. For
any CW complex X the set [X, G] has the structure of a group. If G is a CW comp*
*lex
then (G, ~) is group-like.
Definition 4.2. If (G, ~) and (G0, ~0) are H-spaces, then we say a map F :G ! G0
is an H-map if it satisfies
~0O (F x F ) ~ F O ~: G x G ! G0,
where "~" denotes homotopy of maps. An H-map F :G ! G0 that is a (weak)
homotopy equivalence is called a (weak) H-equivalence. We say two H-spaces G
and G0are H-equivalent if there exists an H-equivalence F :G -'!G0.
Next we recall the definition of the homotopy nilpotency of a group-like space
as introduced by Berstein and Ganea [3]. Suppose (G, ~) is a group-like space. *
*Let
'2: G x G ! G
denote the commutator map
'2(g1, g2) = ~(~(g1, g2), ~( (g1), (g2))).
Extend this definition by letting '1: G ! G be the identity map and, for n > 1,
defining 'n :Gn ! G by the rule
'n = '2 O ('n-1 x '1).
The following corresponds to Definition 1.7 in [3].
Definition 4.3. Given a group-like space (G, ~), the homotopical nilpotency Hni*
*l(G)
of (G, ~) is the least integer n such that the map 'n+1 above is null-homotopic*
*. If
Hnil(G) = 1 we say (G, ~) is homotopy-abelian
Finally, recall that the homotopy groups of a homotopy-associative H-space
(G, ~) come equipped with the Samelson bracket. (See Section X.5 of [31].) This
is a bilinear pairing
< , >: ssp(G) x ssq(G) ! ssp+q(G).
The following result is essentially Theorem 4.6 of [3].
Proposition 4.4 (I. Berstein and T. Ganea). If (G, ~) is H-equivalent to a loop
space then Hnil(G) is an upper bound for the longest nonvanishing Samelson brac*
*ket
in ss*(G).
Proof.By Theorem 4.6 of [3], Hnil( X) is an upper bound for the length of
the longest nonvanishing Whitehead product in ss*(X). Equivalently, by Theorem
X.7.10 of [31], Hnil( X) is an upper bound for the length of the longest nonvan-
ishing Samelson bracket in ss*( X).
Example 4.5. Recall that an Eilenberg-Mac Lane space K(ss, n) is a CW com-
plex with only one non-zero homotopy group, namely ssn(K(ss, n)) = ss. (See Sec-
tion V.7 of [31] for the basic theory of Eilenberg-Mac Lane spaces.) If ss is a*
*belian
and n 1, then K(ss, n) is homotopy equivalent to the loop space K(ss, n + 1).
Thus the Eilenberg-Mac Lane spaces are H-spaces with multiplication given by the
usual product of loops. By Theorem V.7.13 of [31] this H-structure is unique up
to homotopy. If ss is a countable group then K(ss, n) is actually an abelian to*
*po-
logical group by the Corollary to Theorem 3 on page 360 of [18]. In particular,
Hnil(K(ss, n)) = 1.
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 13
Example 4.6. More generally, letQss1, ss2, . .b.e any sequence of countable abe*
*lian
groups. Then the product space j 1K(ssj, j) also has the structure of an abel*
*ian
topological group via the product multiplication. We refer to this multiplicat*
*ion
as the standard multiplication for a product of Eilenberg-Mac Lane spaces. We
emphasize that a product of Eilenberg-Mac Lane spaces may admit many non-H-
equivalent H-structures. (For example, see Proposition Ia in [9].)
We now turn to the function space F (X, G). If (G, ~) is an H-space (respecti*
*vely,
a topological group) than F (X, G) inherits an H-space structure (respectively,*
* topo-
logical group structure) from that of G. Specifically, let
~b:F (X, G) x F (X, G) -! F (X, G)
be given by
b~(f1, f2)(x) = ~(f1(x), f2(x))
for x 2 X and f1, f2: X ! G. If the maps f1 and f2 preserve basepoints, then so
does b~(f1, f2). Hence Fo(X, G) inherits an H-space structure in the same way. *
*We
recall that, when X is the base space of a principal G-bundle, the group F (X, *
*G)
corresponds to the gauge group of the bundle and has been studied extensively.
(See, for example, [2].) We will assume from now on that the function spaces
F (X, G) and Fo(X, G) come equipped with the multiplication induced by G. We
next collect some basic facts regarding these H-spaces.
Proposition 4.7. Let X be a space and let G be a homotopy-associative H-space. *
*If
either X or G is a CW complex then the path components of F (X, G) are naturally
homotopy equivalent to each other and similarly for Fo(X, G).
Proof.In either case, Proposition 4.1 implies that the set [X, G] of based homo*
*topy
classes is a group. This result is an extension to H-spaces (or to group-like s*
*paces,
as appropriate) of the argument of Proposition 2.9.
Proposition 4.8. Let X be a compact space and let G be a homotopy-associative
CW H-space. Then F (X, G) and Fo(X, G) are H-equivalent to group-like CW com-
plexes.
Proof.By Theorem 3 of [19], the function spaces have the homotopy type of
CW complexes. The result follows from Proposition 4.1.
Proposition 4.9. Let X be a space and let G be a path-connected H-space. Then
there is a natural weak H-equivalence
F (X, G)O 'w G x Fo(X, G)O,
where the right-hand side has the obvious product H-structure. If X is a compact
metric space and G is a connected CW complex this is a natural H-equivalence.
Proof.By assumption the basepoint x0 2 X is non-degenerate, so that the inclusi*
*on
{x0} ! X is a cofibration. This implies that the induced map
F (X, Y ) ! F (x0, Y )
is a fibration for any space Y. (See Theorem I.7.8 of [31].) This map is just *
*the
evaluation map ! :F (X, Y ) ! Y defined by !(f) = f(x0). Taking Y = G and
restricting, we have a fibre sequence of path connected spaces
Fo(X, G)O -i!F (X, G)O -!!G.
14 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
This fibration has a section s, which assigns to g 2 G the constant function f(*
*x) = g.
Therefore the long exact homotopy sequence breaks into split short exact sequen*
*ces
s*
____________________________________*
*_____________________________________________________________________________*
*______________
--___________________________________*
*_____________________________
(4.1) 0_____//ssn(Fo(X, G)O)i*//_ssn(F (X,_G)O)!*//_ssn(G)//_0
for each n. Define an H-map ': G x Fo(X, G)O ! F (X, G)O by ' = b~O (s x i),
where b~is the multiplication on F (X, G)O described before Proposition 4.7. It
is straightforward to check that ' induces an isomorphism on homotopy groups.
Finally, if X and G satisfy the further hypotheses above then, by Proposition 4*
*.8,
the function spaces in question have the homotopy type of CW complexes. So a
weak homotopy equivalence is a homotopy equivalence, by Theorem V.3.5 of [31].
Regarding the homotopical nilpotency of F (X, G)O, we have the following resu*
*lt.
Theorem 4.10. Let X be a space and G a connected topological group, or, alter-
nately let X be a compact space and G a connected homotopy-associative CW H-
space. Then
Hnil(F (X, G)O) = Hnil(G).
Proof.With either hypothesis, F (X, G)O is group-like and so its homotopical ni*
*lpo-
tency is well-defined. (In the second case, see Proposition 4.8.) As in the p*
*roof
of Proposition 4.9, we have a section s: G ! F (X, G)O of the evaluation fibrat*
*ion
! :F (X, G)O ! G. It is easy to see that both ! and s are H-maps_recall that the
multiplication on F (X, G)O is induced from that of G. Let
b'n:F (X, G)nO! F (X, G)O
denote the commutator map for F (X, G)O. Then we have
b'nO sn = s O 'n :Gn ! F (X, G)O.
The inequality Hnil(F (X, G)O) Hnil(G) follows. On the other hand, suppose
Hnil(G) = n so that 'n+1: Gn+1 ! G is null-homotopic. We can identify
F (X, G)n+1O~=F (X, Gn+1)O.
The map b'n+1is then adjoint, via the exponential law, to the composition
'n+1
X x F (X, Gn+1)O__"_//_Gn+1____//G
where "(x, f) = f(x) is the generalized evaluation map. Since this composition *
*is
null-homotopic, b'n+1is null-homotopic.
We now consider the H-space F (X, G)O in the context of rational homotopy
theory. To do so, however, we will need to make some further restrictions on X
and G. In particular, for the remainder of this section, all spaces (except fun*
*ction
spaces) will be taken to be CW complexes.
We recall the basic facts about the rationalization of groups and spaces as d*
*evel-
oped in [14]. A connected CW complex X is nilpotent (Definition II.2.1 of [14])*
* if
ss1(X) is a nilpotent group and ss1(X) acts nilpotently (Section I.4 of [14]) o*
*n each
ssj(X) for j 2. A nilpotent space X is a rational space if its homotopy groups
ssj(X) are Q-vector spaces for each j 1. (This is the case P = ? of Definition
II.3.1 of [14].) If ss is a nilpotent group, then we denote by ssQ its rational*
*ization
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 15
in the sense of Definition I.1.1 of [14] (the case P = ?). If ss is abelian, su*
*ch as for
ss = ssj(X) with j 2 and also for ss1 of an H-space, then ssQ is just ss Q.*
* If X
is a nilpotent CW complex, then a function f :X ! Y is a rationalization if Y is
rational and if f induces an isomorphism
~=
f]:ssj(X)Q -! ssj(Y )
for each j 1. This is not the same statement as Definition II.3.1 of [14], bu*
*t is
equivalent by Theorem 3B in Section II.3 of [14].
With this terminology, we have the following theorem.
Theorem 4.11 (Hilton, Mislin, and Roitberg). Every nilpotent CW complex X
has a rationalization e: X ! XQ where XQ is a CW complex. The space XQ is
unique up to homotopy equivalence.
Proof.Existence is Theorem 3B in Section II.3 of [14]. For uniqueness, suppose
f :X ! Y is some other rationalization. Following Definition II.3.1 of [14], f
induces a bijection
f* :[Y, XQ] ! [X, XQ]
Thus there exists h: Y ! XQ such that hOf ' e. Using Theorem 3B in Section II.3
of [14], we see that h induces an isomorphism of rational homotopy groups since*
* f
and e do. Since Y and XQ are rational spaces, h is a weak equivalence. Finally,
since Y and XQ are CW complexes, Theorem V.3.5 of [31] implies that h is a
homotopy equivalence.
We use the notation X 'Q Y to mean that X and Y are rationally equivalent
spaces, that is, that XQ and YQ are homotopy equivalent. We give some examples
of rationalization maps which will figure prominently below.
Example 4.12. Consider an odd-dimensional sphere S2n-1. The groups ssj(S2n-1)
are finite except in the single degree j = 2n - 1. (See Theorem 9.7.7 of [25].)*
* So
the rationalization of S2n-1 is a map
e: S2n-1 ! K(Q, 2n - 1)
corresponding to a nontrivial class in H2n-1(S2n-1; Q). Thus, we have
S2n-1 'Q K(Z, 2n - 1) 'Q K(Q, 2n - 1),
and so, by Example 4.5, an odd-dimensional sphere has the rational homotopy type
of an abelian topological group. We note in passing that even-dimensional spher*
*es
are slightly more complicated, having two nontrivial rational homotopy groups.
There are many more spaces, like S2n-1, whose rationalizations are Eilenberg-
Mac Lane spaces or products of them.
Definition 4.13. Let X be a connected nilpotent CW complex. We say X is a
rational H-space if XQ is an H-space.
The following lemma and its corollary are well-known. (See, for instance, [24*
*].)
We include the statements and proofs for completeness.
Lemma 4.14. Let G be a connected CW H-space. Then G is nilpotent, and the
rationalization of G may be written in the form
Y
(4.2) e: G ! K(ssj(G) Q, j).
j 1
16 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
In particular, the rational homotopy groups of G correspond to a space of algeb*
*ra
generators of H*(G; Q). This rationalization is natural in the sense that a map*
* of
H-spaces f :G1 ! G2 gives rise to a homotopy-commutative diagram in which the
map on Eilenberg-Mac Lane spaces is induced by f] 1: ss*(G1) Q ! ss*(G2) Q.
Proof.The fundamental group of G is abelian and acts trivially, that is, G is a
simple space. (See Theorem 7.3.9, and the preceding discussion, in [25].) So G *
*is
nilpotent. By a classical theorem of H. Hopf (Corollary III.8.12 of [31]), the *
*rational
cohomology algebra H*(G; Q) is the tensor product of an exterior algebra on odd
generators with a polynomial algebra on even generators, that is, H*(G; Q) is a*
* free
commutative graded algebra. (See Example 6 in Section 3 of [13].) WriteQn1, n2,*
* . . .
for the degrees of these generators. We then obtain a map e: G ! jK(Q, nj)
such that, on rational cohomology, the image of the fundamental class of K(Q, n*
*j)
is the corresponding generator of H*(G; Q). Since H*(G; Q) is free over Q, the *
*map
e induces an isomorphism on rational cohomology. By Theorem 3B in Section II.3
of [14] again, e is a rational equivalence.
Thus we see that it is sufficient to determine the rational homotopy groups of
an H-space, or more generally of a rational H-space, in order to identify its r*
*ational
homotopy type.
Corollary 4.15. Let X be a rational H-space (Definition 4.13). Then there exists
a rational equivalence of the form
Y
X 'Q K(ssj(X) Q, j),
j 1
which is natural in the same sense as in Lemma 4.14.
Proof.Apply Lemma 4.14 to XQ, which is a CW complex by Theorem 4.11, and
is still connected, using the fact that the rationalization of a rational space*
* is a
homotopy equivalence.
Example 4.16. We specialize Lemma 4.14 to the Lie group Un(C). The coho-
mology H*(Un(C); Q) is the exterior algebra on one generator in each odd degree
from 1 through 2n - 1. (See page 412 of [5].) Thus, using Example 4.12,
Yn
Un(C) 'Q S1 x S3 x . .x.S2n-1 'Q K(Q, 2j - 1),
j=1
as mentioned in the introduction. Alternately, we may compute the rational ho-
motopy of Un(C) from the long exact sequences of the fibrations Uk-1(C) -!
Uk(C) -! S2k-1 for 2 k n (used in the analysis in Section IV.10 of [31]). In
particular, by IV.10.17 of [31], the map Un-1(C) ! Un(C) is an isomorphism on
ssk for k 2n - 3.
Remark 4.17. The preceding examples display various possible situations which
can occur in the context of H-spaces and rationalization.QAs we show below (Cor*
*ol-
laries 4.26 and 4.27), the rational equivalence Un(C) 'Q nj=1K(Q, 2j - 1) can
be taken to be multiplicative when the product of Eilenberg-Mac Lane spaces has
the standardQmultiplication. On the other hand, for a general H-space G, the map
G ! j 1K(ssj(G) Q, j) of Lemma 4.14 will rarely be be multiplicative, even
up to rational equivalence, if the product of Eilenberg-Mac Lane spaces has the
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 17
standard multiplication. (See Example 4.22.) Finally, the space S2n-1 is not an*
* H-
space for n 6= 1, 2, 4 (by J. F. Adams [1]) and so the question of the multipli*
*cativity
of S2n-1 ! K(Q, 2n - 1) in these cases is moot.
Turning to the rationalization of function spaces, the results we need depend*
* on
three basic developments. First, as has already been used, Theorem 3 of [19] sh*
*ows
that the path components of F (X, Y ) have the homotopy type of CW complexes
provided X is a compact metric space. (Recall we assume Y has the homotopy
type of a CW complex.) The second ingredient is Theorem II.3.11 of [14] and the
discussion that follows it. Nilpotence of the spaces is Corollary II.2.6 of [14*
*].
Theorem 4.18 (Hilton, Mislin, and Roitberg). Suppose that X is a finite CW com-
plex and Y is a nilpotent space with rationalization e: Y ! YQ. Then the compo-
nents of F (X, Y ) and Fo(X, Y ) are all nilpotent spaces and the induced maps
e*: Fo(X, Y )O -! Fo(X, YQ)O and e*: F (X, Y )O -! F (X, YQ)O
given by composition with e are rationalizations.
The third ingredient is an early result of R. Thom (Theorem 2 of [29]).
Theorem 4.19. (R. Thom) Let X be a Hausdorff topological space. Then there is
a natural isomorphism
ssj(F (X, K(ss, n))O) ~=Hn-j(X; ss)
for j 1.
If X is not a finite CW complex then one must specify which cohomology theory
is being used: Thom is using singular cohomology. This is not the best choice f*
*or
compact spaces; ~Cech theory is better, and in general the natural map from ~Ce*
*ch
theory to singular theory is neither injective nor surjective. We use Thom's re*
*sult
only for X a finite complex.
Combining the facts above, we obtain the following result.
Theorem 4.20. Let X be a finite CW complex and let G be a connected CW H-
space or, more generally, a rational H-space. Let Vj and eVjbe the rational vec*
*tor
spaces
M M
Vj = Hl-j(X; ssl(G) Q) and Vej= Hel-j(X; ssl(G) Q).
l j l j
Then there are natural rational equivalences
Y Y
F (X, G)O 'Q K(Vj, j) and Fo(X, G)O 'Q K(eVj, j).
j 1 j 1
Proof.Since G is a rational H-space, we apply Corollary 4.15 to write its ratio*
*nal-
ization as a product of Eilenberg-Mac Lane spaces
Y
GQ ' K(ssl(G) Q, l).
l 1
By Theorem 4.18, we have rational equivalences
F (X, G)O 'Q F (X, GQ)O and Fo(X, G)O 'Q Fo(X, GQ)O.
Now F (X, GQ)O and Fo(X, GQ)O are H-spaces. Thus, by Lemma 4.14, to prove the
theorem it suffices to compute the (rational) homotopy groups of these spaces.
18 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
Q Q
Using the standard homeomorphism of function spaces F (X, lYl) lF (X, Yl*
*),
we obtain i
Y j Y
F (X, GQ)O ' F X, l 1K(ssl(G) Q, l) F (X, K(ssl(G) Q,Ol)).
O l 1
Thom's result 4.19 now gives
l-j
ssj F X, K(ssl(G) Q, l) O~=H (X; ssl(G) Q).
Thus we see ssj(F (X, GQ)O) ~=Vj, as needed. Next, using this isomorphism and t*
*he
split short exact sequence (4.1) we compute ssj(Fo(X, GQ)O) ~=eVj, which comple*
*tes
the proof.
The ingredients of Theorem 4.20 have been combined before in the same way, for
specific X and G. (See, for instance, the computations in Section 2 of [2].) Wh*
*ile
Theorem 4.20 determines the full rational homotopy type of the function spaces *
*of
interest to us, these function spaces are H-spaces and so it is natural to ask *
*whether
we can determine their structure as such, at least up to rational H-equivalence*
*. We
take up this question for the remainder of the section.
Following the discussion before Theorem II.1.8 of [14], if (G, ~) is an H-spa*
*ce
then the rationalization GQ admits a multiplication ~Q making the rationalizati*
*on
map e: G ! GQ an H-map. Moreover, this H-space structure on GQ is uniquely
determined by that of G in the sense that if (GeQ, e~Q) represents a second rat*
*ional-
ization of (G, ~) then (GQ, ~Q) and (GeQ, e~Q) are H-equivalent. It is easy to *
*see that
if (G, ~) is group-like, then so is (GQ, ~Q).
Definition 4.21. We say two CW H-spaces (G, ~) and (G0, ~0) are rationally H-
equivalent if their rationalizations (GQ, ~Q) and (G0Q, ~0Q) are H-equivalent.
Example 4.22. The rationalization K(Z, n) ! K(Q, n) is clearly a rational H-
equivalence. On the other hand, the rationalization S2 ! K(Q, 1) x K(Q, 2) is
not a rational H-equivalence if both sides have the standard loop multiplicatio*
*ns.
To see that these spaces are not H-equivalent, we argue as follows. Let ' 2 ss2*
*(S2)
be the homotopy class of the identity map. As is well-known, [', '] 2 ss3(S2) i*
*s of
infinite order (see Theorem XI.2.5 of [31]), yielding a non-zero Whitehead brac*
*ket
in ss*(S2Q). Under the identification of Whitehead brackets in ss*(S2Q) with Sa*
*melson
brackets in ss*( S2Q) (Theorem X.7.10 of [31]), there is a corresponding non-ze*
*ro
Samelson bracket in ss*( S2Q). By Proposition 4.4 we have Hnil( S2Q) 2. On the
other hand, K(Q, 1) x K(Q, 2) is homotopy-abelian.
Definition 4.23. Given a connected CW homotopy-associative H-space (G, ~),
the rational homotopical nilpotency HnilQ(G) is the homotopical nilpotency of t*
*he
rationalization (GQ, ~Q) of (G, ~), that is, HnilQ(G) = Hnil(GQ). We say (G, ~)*
* is
rationally homotopy-abelian if HnilQ(G) = 1.
The Samelson bracket in ss*(G) induces a bracket in the rationalization giving
ss*(G) Q the structure of graded Lie algebra as in the following definition. *
*(See
the beginning of Section 21 of [13].)
Definition 4.24. A positively graded vector space L over Q is a graded Lie alge*
*bra
if L comes equipped with a bilinear, degree zero pairing < , > satisfying
(1) Anti-symmetry: = -(-1)deg(ff) deg(fi)
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 19
(2) Jacobi identity: > = <, fl> + (-1)deg(ff) deg(fi)>.
By Theorems X.5.1 and X.5.4 of [31], after tensoring with the rationals, ss*(*
*G) Q
with the induced bracket is a graded Lie algebra over Q.
The following result can be deduced in various ways; our proof is based on re*
*sults
of H. Scheerer in [24].
Theorem 4.25. Let (G, ~) be a connected CW homotopy-associative H-space. The
following are equivalent.
(1) (G, ~) has the rational H-type of an abelian topological group.
(2) (G, ~) is rationally homotopy-abelian.
(3) The Samelson Lie algebra ss*(G) Q, described above, is abelian, that i*
*s,
all brackets are zero.
(4) The rational equivalence
Y
e: G ! K(ssj(G) Q, j)
j 1
of Lemma 4.14 is a rational H-equivalence, where the product of Eilenber*
*g-
Mac Lane spaces has the standard multiplication.
Proof.The implication (1) =) (2) is immediate from definitions. The implica-
tion (4) =) (1) is the result of Milnor mentioned in Example 4.5 (Corollary to
Theorem 3 of [18]).
We prove (2) =) (3). Proposition 4.1 implies that (G, ~) is group-like, so
(GQ, ~Q) is group-like too. Corollary 2 in Section 0.1 of [24] therefore shows *
*that
(G, ~) is rationally H-equivalent to a loop space X with the usual multiplicat*
*ion
of loops. Now use Theorem 4.4.
Finally, for (3) =) (4) use Corollary 3 of [24], which gives a bijection betw*
*een
the set of H-maps between two rational group-like spaces and the space of homom*
*or-
phismsQbetween their rational SamelsonQalgebras. The rational Samelson algebra *
*of
j 1K(ssj(G), j) is abelian because j 1K(ssj(G), j) is rationally abelian. S*
*o if
ss*(G) Q is abelian then e trivially induces an isomorphism of rational Samel*
*son
algebras and is thus a rational H-equivalence.
The following useful consequences are well-known.
Corollary 4.26. The standard loop multiplication on a product of Eilenberg Mac
Lane spaces is the unique group-like, homotopy-abelian H-structure up to ration*
*al
H-equivalence.
Proof.Use (2) () (4) in Theorem 4.25.
Corollary 4.27. Let G be a connected CW homotopy-associative H-space. Suppose
H*(G; Q) is finite-dimensional. Then G is rationally homotopy-abelian.
Proof.Referring to the proof of Lemma 4.14, we see that finite-dimensionality of
H*(G; Q) implies that H*(G; Q) is an exterior algebra on odd-degree generators.
By that same result, we conclude that ss*(G) Q is zero in even degrees. Thus,
for degree reasons, the rational Samelson bracket on ss*(G) Q is trivial. Th*
*is
is Condition (3) of Theorem 4.25, so we conclude that G is rationally homotopy-
abelian.
20 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
Finally, we identify the H-spaces F (X, G)O and Fo(X, G)O, for X and G finite
complexes, up to rational H-equivalence. Note that a continuous map f :X !
Y induces H-maps f* :F (Y, G)O ! F (X, G)O and f* :Fo(Y, G)O ! Fo(X, G)O.
Similarly, an H-map h: G ! H induces H-maps h*: F (X, G)O ! F (X, H)O and
h*: Fo(X, G)O ! Fo(X, H)O.
Theorem 4.28. Let X be a finite CW complex. Let G be a connected CW homotopy-
associative H-space with H*(G; Q) finite-dimensional. Then F (X, G)O and Fo(X, *
*G)O
are homotopy-abelian after rationalization. Consequently, the rational equivale*
*nces
of Theorem 4.20 are actually rational H-equivalences where the products of Eile*
*nberg-
Mac Lane spaces have the standard multiplication. Moreover, these rational H-
equivalences are natural with respect to maps f :X ! Y and H-maps h: G ! H.
Proof.Corollary 4.27 implies that G is rationally homotopy-abelian. Thus, by
Theorem 4.10, F (X, GQ)O is rationally homotopy-abelian. Using Proposition 4.9
and the fact that the rational Samelson algebra of an H-product is the product *
*of
the Samelson algebras (see Example 21.4 of [13]), we see that the rational Same*
*lson
algebra of Fo(X, G)O is also abelian. Thus Fo(X, G)O is rationally homotopy-abe*
*lian
by Theorem 4.25. The naturality assertions now follow from the naturality of the
rational homotopy equivalence of a rational H-space with the appropriate product
of Eilenberg-Mac Lane spaces as in Corollary 4.15.
5. Conclusion: Passage to Limits
We now prove the main results of the paper. First, using our results above we
give a preliminary version of Theorem 1 in the special case the maximal ideal s*
*pace
Max (A) of A happens to be a finite complex.
Theorem 5.1. Suppose that A is a commutative unital Banach algebra and Max(A)
is a finite CW complex. Let Vh,j= H2j-1-h(Max (A); Q). Then there is a rational
H-equivalence
Yn 2j-1Y
GLn(A)O 'Q K(Vh,j, h),
j=1 h=1
where the product of Eilenberg-Mac Lane spaces has the standard loop multiplica*
*tion.
The equivalence is natural with respect to homomorphisms between commutative
unital Banach algebras with maximal ideal space a finite complex. Moreover, for
fixed A with Max (A) a finite complex the equivalence is natural in n in the fo*
*llowing
sense: For each n > 1, let i: GLn-1(A)O ! GL n(A)O denote the inclusion in the
upper left corner. Then, after the identifications
n-1M Mn
ssk(GL n-1(A)O) Q ~= Vk,j and ssk(GL n(A)O) Q ~= Vk,j
j=1 j=1
for each k 1 given above, the map
i] 1: ssk(GL n-1(A)O) Q ! ssk(GL n(A)O) Q
corresponds to the inclusion of vector spaces
n-1M Mn
Vk,j,! Vk,j.
j=1 j=1
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 21
Proof.By Theorem 3.2 we have an H-equivalence
GL n(A)O ' F (Max (A), GLn(C))O.
The rational homotopy groups of GL n(C) ' Un(C) occur in degrees 1, 3, . .,.2n -
1 by Example 4.16. Theorem 4.20 thus gives the needed rational equivalence.
Since H*(GL n(C), Q) is finite-dimensional, Theorem 4.28 implies this equivalen*
*ce
is actually a rational H-equivalence. Naturality with respect to Banach algebra
homomorphisms is a direct consequence of the naturality given in that theorem. *
*For
naturality with respect to n, we first need to know what the inclusion GLn-1(C)*
*O ,!
GL n(C)O does on the rational homotopy groups. For this, use the retractions to
the corresponding unitary groups and Example 4.16. Naturality with respect to n
now follows from the naturality in Theorem 4.28 and the fact that the inclusion
i: GLn-1(A)O ! GL n(A)O corresponds, via the H-equivalence of Theorem 3.2, to
the map F (Max (A), GLn-1(C)) ! F (Max (A), GLn(C))O induced by the inclusion
GL n-1(C)O ,! GL n(C)O.
We deduce the general case by considering limits of direct systems in our var*
*ious
settings. Recall that a unital homomorphism ': A ! B between unital Banach
algebras induces a group homomorphism GL n('): GLn(A) ! GL n(B). We first
consider various direct systems in the Banach algebra setting.
Let be a directed set. Let (A~)~2 be a direct system of C*-algebras, index*
*ed
by with C*-algebra maps '~,~:A~ ! A~. The direct limit A = lim-!~A~ exists
in the category of C*-algebras. In fact, we have the following characterization.
Theorem 5.2. The direct limit A = lim-!~A~ of a system of C*-algebra maps
'~,~:A~ ! A~ is characterized by the existence of C*-algebra homomorphisms
'~,1: A~ ! A such that:
(1) '~,1 O '~,~= '~,1 whenever ~ ~.
(2) IfSa 2 A~ and '~,1(a) = 0 then lim~k'~,~(a)k = 0.
(3) ~2 '~,1(A~) is dense in A.
Proof.See the proof of Proposition 2.5.1 of [21] for the construction, and see *
*The-
orem 6.1.2 of [20] and the preceding discussion for the case = N. That these
properties characterize the direct limit for = N is implicit in the proof of *
*Theo-
rem 6.1.2(b) of [20], and the proof for a general index set is the same. (Murph*
*y has
a slightly different formulation of the second condition. The key fact in the p*
*roof is
that an injective *-algebra homomorphism of C*-algebras is isometric, even with*
*out
assuming continuity to begin with.)
S
We emphasize that ~2 '~,1(A~) is usually not the entire direct limit, merely
a dense subalgebra.
Given a direct system (A~)~2 of C*-algebras with maps '~,~:A~ ! A~, we
obtain direct systems of matrix and function spaces. Recall Mn(A) denotes the
space of n x n matrices with entries in A. Write e'~,~:Mn(A~) ! Mn(A~) for the
map induced by applying '~,~entry-wise. For function spaces in the C*-algebra
context, we introduce the following notation. We deviate from our usual notation
for function spaces to follow the standard notation C(Y, A) for C*-algebras.
Notation 5.3. For any C*-algebra A and any space Y let C(Y, A) denote the
C*-algebra of continuous functions b: Y ! A, with pointwise operations and the
supremum norm kak = supy2Yka(y)k. If moreover ': A ! B is a homomorphism of
22 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
C*-algebras, we write __':C(Y, A) ! C(Y, B) for the map obtained by composition
with '.
The following result shows the direct limits in both cases are as expected. T*
*he
result is a special case of the statement that direct limits commute with maxim*
*al
completed tensor products, but our proof avoids the technicalities of completed
tensor products.
Lemma 5.4. Let (A~)~2 be a direct system of C*-algebras, with maps '~,~:A~ !
A~, and let A = lim-!~A~. Let Y be a compact Hausdorff space. Then:
(1) For n 2 N there is a natural isomorphism lim-!~Mn(A~) ~=Mn(A).
(2) Then there is a natural isomorphism lim-!~C(Y, A~) ~=C(Y, A).
Proof.For (1), the maps are obtained by applying '~,1 to each entry. In the
characterization of the direct limit (Theorem 5.2), Parts (1) and (3) are immed*
*iate,
and Part (2) follows from the fact that if the entries of a net of matrices con*
*verge
to zero, then so do the matrices.
We prove (2). The maps C(Y, A~) ! C(Y, A~) and C(Y, A~) ! C(Y, A) are of
course __'~,~and __'~,1, as in Notation 5.3.
Part (1) of the characterization of the direct limit is immediate. For Part (*
*2),
let b 2 C(Y, A~) with __'~,1(b)(y) = 0. Because homomorphisms of C*-algebras are
norm decreasing, the functions f~(y) = k'~,~(b(y))k decrease pointwise to 0. By
Dini's Theorem (Problem E in Chapter 7 of [16]), the convergence is uniform, th*
*at
is, lim~k__'~,~(b)k = 0.
We finish by proving Part (3) of the characterization of the direct limit. L*
*et
b 2 C(Y, A), and let " > 0. Choose a finite partition of unity (f1, . .,.fn) su*
*ch that
whenever x, y 2 supp(fk) then kb(x) - b(y)k < 1_3". Choose yk 2 supp(fk). Choose
~k 2 and ck 2 A~k such that k'~k,1(ck) - b(yk)k < 1_3". Choose ~ 2 such that
~ ~k for all k, and letPak = '~k,~(ck) 2 A~. Then k'~,1(ak) - b(yk)k < 1_3".
Define a 2 A~ by a(y) = nk=1fk(y)ak. For y 2 Y we have kb(yk) - b(y)k < 1_3"
whenever fk(y) 6= 0, whence
Xn
k'~,1(a(y)) - b(y)k fk(y) kak - b(yk)k + kb(yk) - b(y)k
k=1
Xn i"j 2"
2 _ fk(y) = __.
k=1 3 3
So k__'~,1(a) - bk < ".
Now given a direct system (A~)~2 of unital C*-algebras with unital maps
'~,~:A~ ! A~, we obtain a direct system (GL n(A~))~2 of invertible groups
with structure maps GL n('~,~): GLn(A~) ! GL n(A~) The following result is im-
plicit in the usual direct proofs that K1 commutes with direct limits of C*-alg*
*ebras.
The proofs we know in the literature, however, instead use Bott periodicity and*
* the
result for K0.
Theorem 5.5. Let (A~)~2 be a direct system of unital C*-algebras, with unital
maps '~,~:A~ ! A~, and let A = lim-!~A~. Then the maps GLn ('~,1): GLn(A~) !
GL n(A) induce a natural isomorphism lim-!~ssk(GL n(A~)) ~=ssk(GL n(A)) for k *
* 0.
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 23
Proof.Following Notation 5.3, set B~ = C(Sk, Mn(A~)) and B = C(Sk, Mn(A)).
Then ssk(GL n(A~)) = ss0(GL 1(B~)) and ssk(GL n(A)) = ss0(GL 1(B)), and Lemma 5*
*.4
identifies B naturally with lim-!~B~. Therefore it suffices to prove the lemma *
*when
k = 0 and n = 1.
We must prove:
(1) Every path component of GL 1(A) contains some element '~,1(b) for some
~ and some b 2 GL 1(A~).
(2) If b 2 GL 1(A~) and '~,1(b) 2 GL 1(A)O, then there exists ~ ~ such that
'~,~(b) 2 GL 1(A~)O.
The key ingredient is that if s 2 A is invertible, then so is any t 2 A with
kt - sk < ks-1k-1. We use this to prove the following claim: if b 2 A~ and '~,1*
*(b)
is invertible, then there exists ~ ~ such that '~,~(b) is invertible. Let a =*
* '~,1(b).
Set "0 = 1=(3kak). Choose ~0 2 and c0 2 A~0 such that k'~0,1(c0) - a-1k < "0.
Then
ka'~0,1(c0) - 1k < "0kak 1_3.
Choose ~1 ~, ~0. By Part (2) of the characterization of the direct limit, the*
*re
exists ~0 ~0 such that
k'~,~0(b)'~0,~0(c0) - 1k < 2_3.
In particular, '~,~0(b)'~0,~0(c0) is invertible. Similarly, there is ~ ~0 suc*
*h that
'~0,~(c0)'~,~(b) is invertible. Then '~0,~(b) is invertible.SThe claim is prove*
*d.
To prove (1), let a 2 GL 1(A), and use density of ~2 '~,1(A~) in A to choose
~0 and b 2 A~0 such that k'~0,1(b) - ak < ka-1k-1. Then the straight line path
from '~0,1(b0) to a is in GL 1(A). By the claim, there is ~ ~0 such that b =
'~0,~(b0) is invertible.
To prove (2), let b 2 GL 1(A~) satisfy '~,1(b) 2 GL 1(A)O. Let t 7! a0(t) be
a continuous path from '~,1(b) to 1 in GL 1(A), defined for t 2 [1, 2]. Regard
t 7! a0(t) as an invertible element a 2 C([1, 2], A). By Lemma 5.4(2), we have
C([1, 2], A) = lim-!~C([1, 2], A~), so there is ~0 2 and c0 2 C([1, 2], A~0) *
*such that
k__'~0,1(c0)-a0k < ka-10k-1. Without loss of generality ~0 ~. Let a 2 C([0, 3*
*], A)
be the concatenation of the constant path '~,1(b) on [0, 1], the path a0 on [1,*
* 2],
and the constant path 1 on [2, 3]. Let c 2 C([0, 1], A~0) be the concatenation *
*of
the straight line path from '~,~0(b) to c0(1) on [0, 1], the path c0 on [1, 2],*
* and the
straight_line path from c0(2) to 1 on [2, 3]. Then also k__'~0,1(c) - ak < ka-1*
*k-1. So
'~0,1(c) is invertible. Applying the claim to the direct system (C([0, 3], A~))*
*~2
(using Lemma 5.4(2) again), we find ~ ~0 such that __'~0,~(c) is invertible. *
*Since
'~0,~(c(3)) = 1, we get '~,~(b) = '~0,~(c(0)) 2 GL 1(A~)O.
Finally, we briefly consider direct limits of graded Lie algebras over Q. Let*
* be
a directed set, for ~ 2 let L~ be a graded Lie algebra over Q as in Definitio*
*n 4.24,
and suppose given graded Lie algebra maps _~,~:L~ ! L~ for ~ ~ satisfying the
usual coherence conditions. Define the direct limit lim-!~L~ to be the graded s*
*pace
given, in each degree n > 0, as the algebraic direct limit of the L~ in degree *
*n with
the bracket induced by the maps _~,1: L~ ! lim-!~L~. It is direct to check L wi*
*th
the induced bracket satisfies the axioms in Definition 4.24. We need one easy r*
*esult
in this context.
Theorem 5.6. Let (L~)~2 be a direct system of graded Lie algebras over Q with
graded Lie algebra structure maps _~,~:L~ ! L~. The direct limit lim-!~L~ satis*
*fies:
24 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
(1) Given a graded Lie algebra L over Q and graded Lie algebra maps '~: L~ !
L satisfying '~ O _~,~= '~ there exists a unique graded Lie algebra map
': lim-!~L~ ! L satisfying ' O _~,1 = '~.
(2) If each L~ is abelian then so is lim-!~L~.
Proof.The proof of (1) is a direct consequence of the result for direct limits *
*of
groups. The proof of (2) is direct from the definition of the bracket in lim-!~*
*L~.
We can now give the proof of our first main result.
Proof of Theorem 1.By Theorem 3.2, it suffices to prove the rational H-equivale*
*nce
for algebras of the form C(X), with X compact Hausdorff, and to prove naturality
for unital homomorphisms between such algebras, that is, for continuous maps (in
the opposite direction) of compact Hausdorff spaces.
So let X be compact Hausdorff. By Theorem 10.1 in Chapter X of [12], there
exists a directed set and an inverse system (X~)~2 of finite CW complexes and
continuous maps f~,~:X~ ! X~ for ~ ~ with X lim-~X~. We then obtain
a direct system of C*-algebras (C(X~))~2 with maps '~,~:C(X~) ! C(X~) in-
duced by the f~,~. The maps X ! X~ determine an isomorphism of C*-algebras
lim-!~C(X~) ! C(X) because Y 7! C(Y ) is a contravariant equivalence of cate-
gories from compact Hausdorff spaces to commutative unital C*-algebras. Applying
Theorem 5.5 and restricting to the identity components, we obtain isomorphisms
lim-!~ssk(GL n(C(X~))O) ~= ssk(GL n(C(X))O) for k > 0. Since the tensor product
commutes with direct limits, we obtain
lim-!ssk(GL n(C(X~))O) Q ~=ssk(GL n(C(X))O) Q.
~
Set Vk~,j= H2j-1-k(X~; Q) and let
Mn Mn
H(f~,~): Vk~,j! Vk~,j
j=1 j=1
be the map induced on rational cohomology groups by f~,~. Since each Max(C(X~))
X~ is a finite complex, Theorem 5.1 applies to give a commutative diagram
~= L n ~
ssk(GL n(C(X~))O) _Q_________//j=1 Vk,j
GL n('~,~)]|1| H(f~,~)||
fflffl| ~= L n fflffl|~
ssk(GL n(C(X~))O) _Q_________//j=1 Vk,j
for each k > 0. By the continuity of ~Cech cohomology (Theorem 3.1 in Chapter X
of [12]; see page 110 for the definition of GR ), lim-!~Vk~,j~=~H2j-1-k(X; Q) =*
* ~Vk,j.
We conclude n
M
ssk(GL n(C(X))O) Q ~= ~Vk,j.
j=1
Naturality is clear.
It remains to prove the rational H-equivalence and show that GL n(C(X))O is r*
*a-
tionally homotopy-abelian. Consider the direct system (ss*(GL n(C(X~))O) Q)~2
of graded Lie algebras over Q with structure maps
GL n('~,~) 1: ss*(GL n(C(X~))O) Q ! ss*(GL n(C(X~))O) Q.
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 25
By Theorems 5.5 and 5.6(1) there is a graded Lie algebra isomorphism
lim-!ss*(GL n(C(X~))O) Q ~=ss*(GL n(C(X))O) Q.
~
By Theorems 5.1 and 4.25(3), each ss*(GL n(C(X~))O) Q is abelian. Therefore
ss*(GL n(C(X))O) Q is abelian by Theorem 5.6(2). The space GL n(C(X)) is
homotopy equivalent to a CW complex (Theorem 2.8(5)). So we may apply (3) =)
(2) and (3) =) (4) of Theorem 4.25.
Corollary 5.7. Let A be a commutative unital Banach algebra and n > 1. Let
i: GLn-1(A)O ! GL n(A)O denote the inclusion. Let ~Vh,j= ~H2j-1-h(Max (A); Q).
Then, after the identifications
n-1M Mn
ssk(GL n-1(A)O) Q ~= ~Vk,jand ssk(GL n(A)O) Q ~= ~Vk,j
j=1 j=1
for each k 1 given by Theorem 1, the map
i] 1: ssk(GL n-1(A)O) Q ! ssk(GL n(A)O) Q
corresponds to the inclusion of vector spaces
n-1M Mn
V~k,j,! ~Vk,j.
j=1 j=1
Proof.As in the proof of Theorem 1 above, we may assume A = C(X) and write
ssk(GL n-1(A)O) Q = lim-!ssk(GL n-1(C(X~))O) Q
~
and
ssk(GL n(A)O) Q = lim-!ssk(GL n-1(C(X~))O) Q
~
where each X~ is a finite complex. Now use naturality with respect to n in Theo-
rem 5.1.
We can now deduce Theorem 2.
Proof of Theorem 2.Consider the long exact sequence of Theorem 2.10 as far as
the last abelian term ss1(GL n(A)). In the groups ssk(-) for k 1, we may repl*
*ace
each space by the component containing the basepoint, obtaining
. ._._________//_ssk(GL n-1(A)O)_//ssk(GL n(A)O)__//ssk(Lcn(A)O)BC_
___________________________________________________
GFfflffl|
ssk-1(GL n-1(A)O)______// . . .________//ss1(GL n(A)O)__//ss1(Lcn(A)O).
Tensor with Q. Applying Corollary 5.7, we compute
ssk(Lcn(A)O) Q ~=~H2n-1-k(Max (A); Q)
for k > 1, as needed.
26 LUPTON, PHILLIPS, SCHOCHET, AND SMITH
We conclude with some remarks and a result concerning the rational homotopy
type of the space of last columns for a commutative unital Banach algebra. By
Theorem 3.6, we have Lcn(A)O ' F (Max (A), S2n-1)O. Thus Lcn(A)O is a nilpotent
CW complex when Max (A) is a finite complex by Theorem 4.18. Using Exam-
ple 4.12 and Theorem 4.20, we conclude in this case that Lcn(A)O is a rational
H-space. When Max (A) is not a finite complex, there is no guarantee that Lcn(A*
*)O
is a nilpotent space (although we know of no counterexamples), and we cannot
discuss its rational homotopy type. However, it follows from Theorem 2.8(5) that
Lcn(A)O has the homotopy type of a CW complex and thus admits a universal
cover. Our last result describes the rational homotopy type of the universal co*
*ver
of Lcn(A)O in the general case of a commutative unital Banach algebra. We need
one final result from rational homotopy theory.
Lemma 5.8. Let p: E ! B be a map of nilpotent CW complexes inducing a
surjection on rational homotopy groups. If E is a rational H-space then B is one
also.
Proof.This result is well-known in rational homotopy theory and easy to prove
using minimal models. Since we have avoided their use thus far, we give a proof
which does not use minimal models. Our argument adapts one given in [24]; we
reproduce it here for the sake of completeness.
As usual, we assume that the basepoint of each CW H-space (G, ~) is the ident*
*ity.
In addition, following Theorem III.4.7 of [31], we assume that the identity eG *
*is
strict, in the senseLthat ~(x, eG ) = ~(eG , x) = x for all x 2 G.
Write ss*(BQ) = i2JViwith each Via 1-dimensional rational vector space con-
centrated in degree ni. For each i, pick a basis element ffiof Vi. Since (pQ)# *
*:ss*(EQ) !
ss*(BQ) is surjective, we may choose fii 2 ssni(EQ) such that (pQ)# (fii) = ffi*
* for
each i. If ni is odd, then K(Q, ni) ' SniQand so we may regard fii as a map
fli:K(Q, ni) ! EQ. Now suppose that ni is even. Then K(Q, ni) ' SniQ. For
any space X, let "X :X ! X be the adjoint of the suspension of the identity
map of X. This map is natural in X. Since EQ is an H-space, by Theorem 8.14
of [26] there is a retraction r : EQ ! EQ of "EQ. Using this retraction, we ha*
*ve
a commutative diagram
K(Q, ni) _'__//_ OSniQ__fii//_O EQ
"SniQ|| r||
| fflffl|
SniQ__fi___//_EQ.
i
Now set fli= r O fii:K(Q, ni) ! EQ. Q
Choose a total order on J. For each finite subset F J, set KF = i2FK(Q,*
* ni).
We define a map ffF :KF ! BQ as follows. Write F = {i1, . .,.ir} with i1 < . .<.
ir. Then, using . for the multiplication in EQ, set
ffF (x1, . .,.xr) = pQ (((fli1(x1) . fli2(x2)) ..fli3(x3))...f.lir(xr)).
The induced homomorphism on homotopy groupsQrestricts to the inclusion Vi !
ss*(BQ) on each summand of ss*(KF ) ~= i2FVi.
If F1 F2 J are finite sets, there is an obvious map kF1,F2:KF1 ! KF2,
obtained by viewing KF1 as the set of elements x 2 KF2 such that xiis the basep*
*oint
of K(Q, ni) for i 2 F2\F1. Since the basepoints are the identities and the iden*
*tities
BANACH ALGEBRAS AND RATIONAL HOMOTOPY THEORY 27
are strict, one checks that ffF2 O kF1,F2= ffF1 for F1 F2 J. Accordingly, we
obtain a map ff: lim-!FKJF! BQ, which moreover induces an isomorphism on
homotopy groups and hence is a homotopy equivalence. The H-space structures
on the Eilenberg-Mac Lane spaces give obvious H-space structures on the products
KF , and the maps kF1,F2respect these structures and the associated homotopies.
So lim-!FKJFis an H-space, and thus BQ is also an H-space.
Theorem 5.9. Let A be commutative unital Banach algebra. Let L^cn(A)Odenote
the universal cover of Lcn(A)O. If Max (A) is a finite complex then Lcn(A)O is a
rational H-space. In general, L^cn(A)Ois a rational H-space.
Proof.When Max (A) is a finite complex the result is a consequence of Theo-
rem 4.18, Example 4.12 and Theorem 4.20, as argued before the statement of
Lemma 5.8. For the general case, observe that, by Theorem 2.10 and Corollary 5.*
*7,
the map G^Ln(A)O ! L^cn(A)Oinduced on universal covers by the natural map
GL n(A)O ! Lcn(A)O is a surjection on rational homotopy groups. As G^Ln(A)Ois
a topological group, the result follows from Lemma 5.8.
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Department of Mathematics, Cleveland State University, Cleveland OH 44115
E-mail address: G.Lupton@csuohio.edu
Department of Mathematics, University of Oregon, Eugene OR 97403-1222
E-mail address: ncp@darkwing.uoregon.edu
Mathematics Department, Wayne State University, Detroit MI 48202
E-mail address: claude@math.wayne.edu
Department of Mathematics, Saint Joseph's University, Philadelphia PA 19131
E-mail address: smith@sju.edu