PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 89. Number 4, December 1983

COBORDISM AND THE NONFINITE HOMOTOPY TYPE OF SOME DIFFEOMORPHISM GROUPS DANIEL S. CHESS Abstract. Unoriented cobordism, a geometric construction, and a theorem of Browder on finite //-spaces are used to give new examples of manifolds whose diffeomorphism groups have identity component of nonfinite homotopy type.

The nature of the group Diff(M) of smooth diffeomorphisms of a smooth manifold is of considerable current interest. It is known in many cases, and expected

for most M, that Diff0(M), the identity component of Diff(M) under the C°° topology, is not of finite homotopy type [1]. Our purpose is to give a simple construction of examples of this phenomenon. The existence of these examples follows directly from a result in the theory of //-spaces, a geometrical construction, and a calculation in the unoriented cobordism ring. Fact 1. Let X be an arcwise connected //-space. Then if X is of finite homotopy

type, m2(X)= 0[2J. Let 61* denote the unoriented cobordism ring. An element [M] of 61* is said to fiber over the «-sphere S" if and only if there is a representative M of [M] and a smooth fiber bundle p: M -» S". Denote by 61" the subset of elements of 61* which

fiber over S". Clearly 61" is an ideal of 61*. Fact 2. 6T+I c 61". The proof is by the following construction

[M] E 6l"+l is represented by F^M^S"+X.

due to H. Winkelnkemper.

Suppose

As 0 = [F X S"+x] E 6l"+1 by con-

sidering M + F X Sn+ ' (disjoint union), M is represented by

FX {-1,1} -^ M+ FX Sn+X -» S"+x, where

FX {-1,1} = 9(FX[-1,1]). Now

M + FX S"+x = FX {-1,1} X D"+] U FX {-1,1} X /)"+'

(FX {-1,1} XS",g), where g: (S", *) - (Diff(,F), id) is a smooth map and g : F X {-1, 1} X 5"' - F X {-1,1} X S" is given by g(x, -1, s) = (g(s)(x), -1, s) and g(x, I, s) = (x, 1, s). Received by the editors September 13, 1982 and, in revised form, May 2, 1983. 1980 Mathematics Subject Classification. Primary 57R50, 57R75. ©1983 American Mathematical

Society

0002-9939/83 $1.00 + $.25 per page

743

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

744

D. S. CHESS

Extend g to g on F X ([-!,- 4] U [4,1]) X S" by g(x, -t2, s) = (g(s)(x), and g(x, t2, s) = (x, t2, s). The manifold

-t2, s)

N = FX [-1, 1] X D"+i U FX [-1, 1] X D"+l (FX([-l,-2]u[\,l])XS",g) whose boundary is the disjoint union of M U F X S"+x and a fiber bundle over S"

with fiber

FX[-4,|]

UFX[-U]

= FX Sl

(3(FX[-M]),id) is the required cobordism.

We now describe the relation between the clutching functions for the boundary of N. Denote by S2v(Diff(F)) the appropriately topologized space of smooth maps of (5"', * ) to (Diff(F), id). Then there is an obvious map

e:Qs(Diff(F))

-Diff(FX

Sx)

given by e(l)(x, t) = (l(t)(x), t). Denoting by i2(Diff(F)) the loop space of Diff0(F), /: S2s(Diff(F)) -» i2(Diff(F)) is a homotopy equivalence. Regarding 77,(Diff0(F)) as 77,_,Qs(Diff(F)), we have the map è:77,(Diff0(F))

-^77,_,(njDiff(F))-77,_lD1ff(FX

Sx).

Given a smooth fiber bundle F -> M -* S" + ', n > 1, determined by a clutching class

g G 77„(Diff()(F)), the bundle over S" constructed above has fiber F X S] and clutching class ë(g) G ir„_,Diff(FX S]). Observation 3. Let 0 ¥= [M] G 613and let F -» M -» S3 be a representative of M with clutching class g. Then 0 =£ [g] G 772Diff0(F). Thus in order to show the existence of manifolds F with Diff0(F) of nonfinite homotopy type, it suffices to show 0 ¥= 61' for some i > 3. As 61* is a polynomial ring over Z2, 2:61* -» 61*; 2:íhi4 is an injective homomorphism. Let A', C 61* be the kernel of x¡ 61* -» Z2, where x is the mod 2 Euler characteristic.

Fact 4. 2(KX) C 6i4 [3.7.3]. Hence 0 ¥=614 C 6l3. The generators of ^[A,] represented by elements of 614 which are given in §7 of [3] are all determined by S3 = Sp(l) actions on manifolds with evidently nontrivial rational pontryagin classes so that these examples differ from those of [1]. It is worthwhile to consider the examples of [3] more explicitly. Let F be R, C or H, let c7(F) be the group of unit norm elements of F, and let S(kF) denote the unit

sphere in kF. G(F)k+x denotes the (Â:+ l)-fold direct product of G(F), and 2A(F) denotes the A:-folddirect product of S(2F). We define a G(F)*+I action on 2*(F) X S((n + 1)F) by (?,,...,tk+x)((qx,px),...,

(qk,

pk),(px,...,pn+x))

= {{q^\P\txx),...,{qJt]x,tJ-Xp]t-x),...,{qkt-kx,tk_xpkt-kx))

{hP\tk\\,P2tk\\T--,Pn+\tk\\)

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

COBORDISM AND HOMOTOPY TYPE OF DIFFEOMORPHISM GROUPS

745

Denote the quotient manifold of this principal action by V(n, k; F). A point in the p(n + k) manifold V(n,k;F) (p = 1,2,4) is denoted [(qx, px),... ,(qk, pk), (px.#,+ ,)]• The map V(n, k;F) - FF(1) = Sx, S2, S4 given by [(qx,px),...,(qk,pk);

(/>,,..

.,/>„+,)]

-*[(px,qxj\

is a fiber map with fiber V(n, k — 1; F) and structure group (7(F) where the action

of G(F) on V(n, k — 1; F) is given by t[(qx,

px),...,

(qk-l,pk-i),(pl,...,Pn+x)]

= [{q\,tp\)A

COBORDISM AND THE NONFINITE HOMOTOPY TYPE OF SOME DIFFEOMORPHISM GROUPS DANIEL S. CHESS Abstract. Unoriented cobordism, a geometric construction, and a theorem of Browder on finite //-spaces are used to give new examples of manifolds whose diffeomorphism groups have identity component of nonfinite homotopy type.

The nature of the group Diff(M) of smooth diffeomorphisms of a smooth manifold is of considerable current interest. It is known in many cases, and expected

for most M, that Diff0(M), the identity component of Diff(M) under the C°° topology, is not of finite homotopy type [1]. Our purpose is to give a simple construction of examples of this phenomenon. The existence of these examples follows directly from a result in the theory of //-spaces, a geometrical construction, and a calculation in the unoriented cobordism ring. Fact 1. Let X be an arcwise connected //-space. Then if X is of finite homotopy

type, m2(X)= 0[2J. Let 61* denote the unoriented cobordism ring. An element [M] of 61* is said to fiber over the «-sphere S" if and only if there is a representative M of [M] and a smooth fiber bundle p: M -» S". Denote by 61" the subset of elements of 61* which

fiber over S". Clearly 61" is an ideal of 61*. Fact 2. 6T+I c 61". The proof is by the following construction

[M] E 6l"+l is represented by F^M^S"+X.

due to H. Winkelnkemper.

Suppose

As 0 = [F X S"+x] E 6l"+1 by con-

sidering M + F X Sn+ ' (disjoint union), M is represented by

FX {-1,1} -^ M+ FX Sn+X -» S"+x, where

FX {-1,1} = 9(FX[-1,1]). Now

M + FX S"+x = FX {-1,1} X D"+] U FX {-1,1} X /)"+'

(FX {-1,1} XS",g), where g: (S", *) - (Diff(,F), id) is a smooth map and g : F X {-1, 1} X 5"' - F X {-1,1} X S" is given by g(x, -1, s) = (g(s)(x), -1, s) and g(x, I, s) = (x, 1, s). Received by the editors September 13, 1982 and, in revised form, May 2, 1983. 1980 Mathematics Subject Classification. Primary 57R50, 57R75. ©1983 American Mathematical

Society

0002-9939/83 $1.00 + $.25 per page

743

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

744

D. S. CHESS

Extend g to g on F X ([-!,- 4] U [4,1]) X S" by g(x, -t2, s) = (g(s)(x), and g(x, t2, s) = (x, t2, s). The manifold

-t2, s)

N = FX [-1, 1] X D"+i U FX [-1, 1] X D"+l (FX([-l,-2]u[\,l])XS",g) whose boundary is the disjoint union of M U F X S"+x and a fiber bundle over S"

with fiber

FX[-4,|]

UFX[-U]

= FX Sl

(3(FX[-M]),id) is the required cobordism.

We now describe the relation between the clutching functions for the boundary of N. Denote by S2v(Diff(F)) the appropriately topologized space of smooth maps of (5"', * ) to (Diff(F), id). Then there is an obvious map

e:Qs(Diff(F))

-Diff(FX

Sx)

given by e(l)(x, t) = (l(t)(x), t). Denoting by i2(Diff(F)) the loop space of Diff0(F), /: S2s(Diff(F)) -» i2(Diff(F)) is a homotopy equivalence. Regarding 77,(Diff0(F)) as 77,_,Qs(Diff(F)), we have the map è:77,(Diff0(F))

-^77,_,(njDiff(F))-77,_lD1ff(FX

Sx).

Given a smooth fiber bundle F -> M -* S" + ', n > 1, determined by a clutching class

g G 77„(Diff()(F)), the bundle over S" constructed above has fiber F X S] and clutching class ë(g) G ir„_,Diff(FX S]). Observation 3. Let 0 ¥= [M] G 613and let F -» M -» S3 be a representative of M with clutching class g. Then 0 =£ [g] G 772Diff0(F). Thus in order to show the existence of manifolds F with Diff0(F) of nonfinite homotopy type, it suffices to show 0 ¥= 61' for some i > 3. As 61* is a polynomial ring over Z2, 2:61* -» 61*; 2:íhi4 is an injective homomorphism. Let A', C 61* be the kernel of x¡ 61* -» Z2, where x is the mod 2 Euler characteristic.

Fact 4. 2(KX) C 6i4 [3.7.3]. Hence 0 ¥=614 C 6l3. The generators of ^[A,] represented by elements of 614 which are given in §7 of [3] are all determined by S3 = Sp(l) actions on manifolds with evidently nontrivial rational pontryagin classes so that these examples differ from those of [1]. It is worthwhile to consider the examples of [3] more explicitly. Let F be R, C or H, let c7(F) be the group of unit norm elements of F, and let S(kF) denote the unit

sphere in kF. G(F)k+x denotes the (Â:+ l)-fold direct product of G(F), and 2A(F) denotes the A:-folddirect product of S(2F). We define a G(F)*+I action on 2*(F) X S((n + 1)F) by (?,,...,tk+x)((qx,px),...,

(qk,

pk),(px,...,pn+x))

= {{q^\P\txx),...,{qJt]x,tJ-Xp]t-x),...,{qkt-kx,tk_xpkt-kx))

{hP\tk\\,P2tk\\T--,Pn+\tk\\)

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

COBORDISM AND HOMOTOPY TYPE OF DIFFEOMORPHISM GROUPS

745

Denote the quotient manifold of this principal action by V(n, k; F). A point in the p(n + k) manifold V(n,k;F) (p = 1,2,4) is denoted [(qx, px),... ,(qk, pk), (px.#,+ ,)]• The map V(n, k;F) - FF(1) = Sx, S2, S4 given by [(qx,px),...,(qk,pk);

(/>,,..

.,/>„+,)]

-*[(px,qxj\

is a fiber map with fiber V(n, k — 1; F) and structure group (7(F) where the action

of G(F) on V(n, k — 1; F) is given by t[(qx,

px),...,

(qk-l,pk-i),(pl,...,Pn+x)]

= [{q\,tp\)A