.

On the structure of the first homology of the groupof equivariant diffeomorphisms of manifolds
with smooth torus actions
信州大学・理学部 阿部 孝順 (Kōjun Abe )Faculty of Science, Shinshu University§1.

Preliminaries

In the previous paper [AF5] we calculate the first homology of thegroup of equivariant diffeomorphisms of representation spaces of finitegroups, and apply to the case of the smooth orbifolds. In this talk weshall consider the case of smooth manifolds with smooth torus actions.
First we describe the previous results.
For a finite group G, let M be a smooth connected G-manifold. LetDG (M ) denote the group of G-equivariant smooth diffeomorphisms ofM which are G-isotopic to the identity through the isotopies with compact support. We recall the result the case where M is a finite dimensional G-module V . Let V G be the subspace of the fixed point set ofV . Let AutG (V ) denote the set of G-invariant automorphisms of V and
AutG (V )0 the identitiy component of AutG (V ). Then we have the following.
Theorem 1.1(1) If dim V G > 0, then DG (V ) is perfect.
(2) If dim V G = 0, then H1 (DG (V )) ∼= H1 (AutG (V )0 ).
We can decompose V = ⊕di=1 ki Vi , where Vi runs over the inequivalent
irreducible representation space of G and ki is a positive integer.
EndG (Vi ): the set of G-invariant endmorphisms of Vi
Then dim EndG (Vi ) = 1, 2 or 4.

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Corollary 1.2dz}|2{H1 (DG (V )) ∼= R × U (1) × · · · × U (1),dwhere d2 is the number of Vi with dim EndG (Vi ) = 2.
If M is a smooth orbifolds, then p ∈ M is said to be an isolated
singular point of M if there exists a local chart (Ui , φi ) around p suchthat p̃ is an isolated fixed point of (Γi , Ũi ) with πi (p̃) = p Let φi : Ui →Ũi /Γi , πi : Ũi → Ui be the canonical projection. Then we have.
Theorem 1.3 If a smooth orbifold M has {p1 , ..., pk } as the isolated
singular point set. Let (Γi , Vpi ) (1 ≤ i ≤ k) be the representaion spaceassociated to the isolated singular points. ThenH1 (D(M )) ∼= H1 (AutΓp1 (Vp1 )0 ) × · · · × H1 (AutΓpk (Vpk )0 ).
§2.

Orbit preserving G-diffeomorphisms

Let M be a connected closed G-manifold and B be the orbit space.
Then the natural projection π : M → B induces the group homomorphism P : DG (M ) → D(B).
Let (H0 ) be the principal orbit type of M and let {(Hi )| 0 ≤ i ≤ `}be the other G-orbit types of M . Put
Mi = {p ∈ M | (Gp ) = (Hi )},Wi = N (Hi )/Hi ,Fi = MiHi ,Bi = Fi /Wi .

Then qi : Fi → Bi is the principal Wi -bundle and πi : Mi → Bi is theassociated fiber bundle with the fiber G/Hi . Thus
Mi ∼= G/Hi ×Wi Fi .
Let {Ui,α }(i,α)∈Λi be an open covering of Bi such that there exists a
local section σi,α on Ui,α of qi . Then we have the transition functions{ϕi,αβ } of the principal Wi -bundle qi given by
ϕi,αβ (b)σi,β (b) = σi,α (b) (b ∈ Ui,α ∩ Ui,β ).
We shall study the group KerP which coincides with the group of orbit
preserving equivariant diffeomorphisms of M .
2Let h ∈ KerP . Then h induces the bundle isomorphisms hi ofπi (0 ≤ i ≤ `). We have smooth maps
si,α : Ui,α → Wi ((i, α) ∈ Λi ) satisfyinghi (σi,α (b)) = si,α (b)σi,α (b) (b ∈ Ui,α ).
It follows that, for b ∈ Ui,α ∩ Ui,β , we havesi,α (b)ϕi,αβ (b) = ϕi,αβ (b)si,β (b).
Then hi corresponds to the collections Si (h) = {si,α }(i,α)∈Λi satisfying(3.1) si,α · ϕi,αβ = ϕi,αβ · si,β on Ui,α ∩ Ui,β .
Put S(h) = {Si (h)| 0 ≤ i ≤ `}.
Definition 2.1 A cocycle of an orbit preserving G-diffeomorphism of
M is a collection S = {Si | 1 ≤ i ≤ `} such that
(1) Si is the set of smooth functions {si,α }(i,α)∈Λi from Ui,α to Wi
satisfying the condition (3.1),(2) Let V be a slice at p ∈ M . Then ψV : V → G · V is a smooth
map. Here the map ψV is given by ψV (v) = si,α (π(v)) · v if π(v) ∈ Ui,α .
By definition S(h) is a cocycle of an orbit preserving G-diffeomorphismof M for each h ∈ Ker P . Let S = {Si | 1 ≤ i ≤ `} be a cocycle of an orbit preserving G-diffeomorphism of M . Then we have a map h : M → M
defined byh([gHi , x]) = [g · si,α (qi (x)), x]
for g ∈ G, x ∈ qi−1 (Ui,α ).
Lemma 2.2h ∈ Ker P .

Let S(M ) be the set of all cocycles of an orbit preserving G-diffeomorphismof M .
Corollary 2.3 Let S : Ker P → S(M ) be a map which assigns eachh ∈ Ker P to S(h). Then S is bijective.
Remark 2.4 M. Davis [DA] introduced the G-normal system of smooth
G-manifolds to classify the set of G-manifolds and suggested that the orbit preserving G-diffeomorphisms are expressed by using this system. We
can express them more easy way by using the above cocycles.

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Let M (m, R) denote the set of all m×m-matrices. Let f : Rn \Rm →M (m, R) be a smooth map. Define a map fˆ : Rn → Rm by½
f (x, y)xx ∈ Rm , y ∈ Rn−m (y 6= 0)
ˆf (x, y) =0x ∈ Rm , y = 0.
Lemma 2.5 If fˆ is a smooth map, then f can be extended to a smoothmap F : Rn → M (m, R).
If Hi is a subgroup of Hj , let ri,j : G/Hi → G/Hj be the canonicalprojection.
Corollary 2.6 Let h ∈ KerP . Assume that Hi is a subgroup of Hjand Uj,β is contained in the closure Ui,α of Ui,α . Then si,α is extended toa map s̄i,α on Ui,α satisfyingri,j (s̄i,α (b)) = sj,β (b) for b ∈ Uj,β .
Example 2.7(1) Assume that q0 : F0 → B0 is a trivial W0 -bundle. It follows
from Corollary 2.6 that each h ∈ Ker P corresponds to a smooth map
s : B → W0 satisfying the following condition. If b ∈ Bi (1 ≤ i ≤ `),then−1s(b) ∈ r0i(Wi ) = (N (H0 ) ∩ N (Hi ))/H0 .
(2) If M is a (2n − 1)-dimensional homotopy sphere with a smoothO(n)-action with B = D2 . Then
H0 = O(n − 2), H1 = O(n − 1),

W0 = O(2), W1 = O(1).

Thus Ker P is one to one correspondence with the smooth maps froms : D2 → SO(2) such that s(∂D2 ) = 1.
Example 2.8 Let M be a 2n-dimensional torus manifold with thelocal standard action. Note that N (H)/H = T n /H ∼= H c for eachtoral subgroup H in T n , where H c is the complementary torus subgroupof H. Let F(M ) be the set of smooth maps s : B → T n such thats(π(p)) ∈ (Tpn )c for each p ∈ M . Then Ker P is isomorphic to F(M ) asa group.

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Let V be a slice at p ∈ M with π(p) ∈ Ui,α . Let PV : DG (G · V ) →D(G · V /G) be the natural group homomorphism. Note that dim Ui,α =dim Fi /Wi .
Proposition 2.9
If dim Ui,α > 0, thenKer PV ⊂ [Ker PV , DG (G · V )].
§3.

Appliction to torus actions

M : 2n-dimensional torus manifold with local standard action
Let p be a fixed point of M . Let AutT n (Tp (M )) denote the set of T n equivariant linear automorphisms of Tp (M ). We have a group homomorphism Φp : DG (M ) → AutT n (Tp (M ))0 assighning each h ∈ DG (M ) tothe differential dhp at p. Set the homomorphism
YAutT n (Tp (M ))0 .
Φ = {Φp } : DG (M ) →p∈F (M )Here F (M ) is the fixed point set of M .
Since Tp (M ) is the standard representation space of T n , AutT n (Tp (M ))0
is isomorphic to (C∗ )n . Define the group homomorphismYΘ = (P, Φ) : DT n (M ) → D(M/T n ) ×(C∗ )n .
p∈F (M )Since M/T n has a structure of an orbifold and is locally diffeomorphic
to R̃n /Z2 n around the isolated singular point of M/T n , where R̃ is thenon-trivial 1-dimensional Z2 -module. By Corollary 1.2 we have.
Corollary 3.1 If M has m fixed points, then H1 (D(M/T n )) ∼= Rmn .
Proposition 3.2
DT n (M ).

Ker Θ is contained in the commutator subgroup ofThere exists the following group exact sequence.
ιΘ∗
Ker Θ/[Ker Θ, DT n (M )] →H1 (DT n (M )) →∗ H1 (DT n (M )×Y(C∗ )n ) → 1.

p∈FnBy Proposition 3.2 ι∗ = 0 and Θ∗ is isomorphic. From Corollary 3.1 wehave.
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Theorem 3.3 Let M be a 2n-dimensional torus manifold with local
standard action. If M has m fixed points, thenH1 (DT n (M ) ∼= (R × C∗ )mn .
In order to prove Proposition 3.2, we need the following lemmas.
Lemma 3.4 (Fragmentation lemma)
Let M be a smooth G-manifold and {Vi | 1 ≤ i ≤ n} be a G-invariant
finite open covering of M . Let N be an open neighborhood of the identity
in DG (M ). Then there exists an open neighborhood N0 ⊂ N of theidentity with the following properties: For any f ∈ N0 , there exist fi ∈N, 1 ≤ i ≤ n, such thata) fi is G-isotopic to the identity through an equivariant C ∞ isotopy
whose support is contained in Vi , andb) f = fn ◦ fn−1 ◦ · · · ◦ f1 .
Theorem 3.5 (Bierstone [BI1], Schwarz [SC2])
Let N be a smooth G-manifold. Then each smooth isotopy on N/Gwith compact support lifts to a smooth G-equivariant isotopy on N .
Proof of Proposition 3.2 :Combining Fragmentation lemma and Theorem 3.5, the proof of Proposition 3.2 is reduced to the case M = T n · V , where V is a slice of a pointp ∈ M . Then M = T n · V ∼= T n ×Hp V . Let PV : T n · V → V /Hp be thenatural projection. Then it is enough to prove thatKer Θ ⊂ [Ker PV , DT n (T n · V )].
By Proposition 2.9, if p is not a isolated fixed point of T n , then
Proposition 3.2 is valid. Assume that p is a isolated fixed point of M .
Let T n · V = V and Θ be the compsositionΘ = (PV , Φp ) : DT n (V ) → D(V /T n )×AutT n (Tp (M ))0 ∼= D(V /T n )×(C∗ )n .
Let h ∈ Ker Θ. Then h is an orbit preserving equivariant diffeomorphism of V with compact support and dhp = 0. From the linearlization
theorem by Sternberg we can prove that h ∈ [Ker Φp , DT n (V )] by using
the contraction map.

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§4.

S 1 -action on 3-manifolds

Let M be a smooth closed 3-manifold with a smooth U (1)-action.
Let n1 and n2 = m − n1 be the numbers of the exceptional orbits U (1) · pwith U (1)p = Z2 and U (1)p = Zk (k ≥ 3), respectively. The we have thefollowing.
Proposition 4.1n +nn1}|2{z}| 2 { z∼H1 (D(M/U (1))) = R × · · · × R × U (1) × · · · × U (1) .

Theorem 4.2
n +nn +2n11}| 2{z}| 2 { zH1 (DU (1) (M )) ∼= R × · · · × R × U (1) × · · · × U (1) .

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