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The Bing staircase construction for Hilbert cube manifolds
M. Handel/ Hilbert d&be Manifolds
30
non-sii:o,lt x1!.;.ed mapping cyunder theorem for embeddings between Q-manifolds, and a ‘%~b,I theorem” for Keller cubes (i.e. if two &eller cubes intersect in cube &X!I *,weirunion is a Hilbert cube). J. Quinn and R. Wong s how ~\a+:%;A:? : fact follows immediately from the partial sum theorem. Gc;“s~x&:ing a finite dimensional result of (essentially) Bin we 64rove: +itit for certain decompositions 9 of Q, all of elemea t s ii-e homeomorphic to Ik for some fixed Q/ $9 x I$?-=::3 (see Theorem 2.1 and its Addendum). coroiA~~B~i:n’~ i6,the West Mapping Cylinder lheorem [12] w estat9Ei~hr:d
of the Mapping Cylinder Theorem has ele~~:x~szy requiring only an understanding of A>ll-known a anQ AI~~v:;I~~~ l:3,4], and two results from infinite dimensi Isotcpp~ t !T i ension) Theorem and the Collaring Theore of CZx:~:;ra-3t:~ ‘s notes [5]. :~;;dg’~&.~,)l$$. ; ne proof actually gives the stronger result (Corollary (-&z%__& 3 :J A c QI and f: A + Q2 is any map, then the stabi sgxw~( Ctl, ...I &) x Q is homeomorphic to 0. This was pre $2FL~ie4~ pfld&r , “s”[ 131, but it required a separate arg ament unrel h”,:.r~~; .I,;3i;;t :vlinder Theorem. This
pry
iif
X C: CJ IPIP>cell-like set, which now follows from Edwards
3&. Such a statement would be immediz
tSG paper to get around the problem.
31
icking out the reIevant &taik from [4] n 2 contains statemrxts of ction 4 contains a p3-oofof application of Bryant’s Reduction corollaries, the first two of which
the the 13, are
modelled on th: Mikbert cube Q, of closed intervak I = [0, 11.Vk represent Q as nd let Q(n) := In X In+l X In+2X . The metric l
l
l
Q, is a Q-manifold. is a Z-set in W if A is contained in a en A will be collared in W I semi-cc)ntinimwi.e. ; the w:X+X/9 is closzii. When
e will only list the v Yl(a --f(a),
i CA} for the
32
Ad.Han&l~HilkrtCbbe Manifolds
I 2.1.. C&eariz Q-manifold factor am! a closed embeddhg f : Y x I --) thequoti~ent map, Then w X ido : W X Q-, moiplaism.
Theorem 2.1 holds with I replaced by Ik for a
2.2. The finite dimensional analogue of Thear Dby replacing W with a finite dimension ‘, was proved (essentially) by Bryant [3,4]. .3. (West Mapping Cylinder Theo Q-man~old ftzctors.llzen a!x ido : M(f) x Q + Y x kt A, X, and Y be Q-man Then XqY is a QX u Y/(a -f(a); a E A). The followiq corollary was observed by
2.5 Let X b an AR (i.e. a containedin 4 Q-manifold
contains the heart tif the argument way.
Q &
33
ti at TV decompositoin
{f(y x I) x q; y E Y, q E Qj c W x Q 0 3 a homeomcxphism H : 3 x Q + such that both (w. q) and
r now we wil! assume that I has the standard inate direction of Q. Thus
8X d is a Z-set. The proposition (with S’ = 0) folfows by ion) Theorem [S, Section 41 to the natural [c;e- E, b -t el) x Q1 x 0 for sufficiently small r&y, ( t ) holds for soprae6’ > O., repeatedly we construct the desired homesmorphism
tain &(O C& e 6,) and a homeomorci1 that:
h’ satisfies Pqwsition 3.1(3) with (a, & y) rep1 Observe th2.tfor SE [& S,-11,1 z 2, . . . , tt - 1 (1) .r’lf(y )a !)xqlxs =identity, j>i-1. Also,
h i-2 - hi-'(f(yx
I)xqlxs)
and thus (2) #,‘I@- . k $-I(f (y x I) x ql x si = identity, j e summarized as follows:
amei%n of g: Cektain tapering and support restrictions are left out easy for tl’j :,: reader to fill in. I a :Y I + I x I be a homlesmorphiem with motion c tireEy in the second I Let g1: I/. ‘4 direction s .t(::hthat I g*(’ j\- i/n,
l]X[O, 1-(i+l)/(n+l)])=[l-i/G
l]X[+,&] 9ori==1,2 ,...,
n-2.
i b i entifying the first
Then gj %1duces82:f(‘YxI)X~1XI--*f(YXI)XQIXI [second] I 1;\il:::torin I x I with the A factor in Y X 1 [Q, X I]. j Finally g; t ruduces g by extending g2 ambiently to W = Qi x B’using a standard tapering arg l,rfnent. 1 By .a con;rektely symmetric operation centering on W y Q, x 1 (rather than W x Q1 x 0) md pushing from 0 towards 1 (rather than from 1 towards 0) we can corresponding to Fig( 3. Fig. 3 is obtained construct hL: WxQ,xI+ WXQ+I
I
Fig. 3.
I I I 1
by turning Fig. 1:‘:upside down and is interpreted as Fig. 2 was except for the obvious changes (points to the right of or below the heavy broken line/s are left fixed, points to the left are pi,;:;hed close to the shaded region). Specifically, h)Nsatisfies: (1)hsupp Ilk :‘.:UZ PJ&(f(Yx[O, ilnl)~ = hope: ition 3.2(2) with hb rep (3)k For s E [iJ”:n + 1), (i -t l)/( M)’ WtWXQt * hhl f (y x [(i - 1)/n, m3
)’ ‘T%WxC?* ..‘zk+i~f~Y x1)x
-4.
M. Han&l/Hilbert
C%be M~nifoldr
37
2/(n + en &) holds for i = 1. and hh have disjoint supports &e can compose them to get x Q1 x I. Fig. 4 corresponds to PIIIThus H satisfies: x Q1 X I either H(w, ql, t) = (w, ql, t) or 3y c Y such that both (w, (I*) and If s E [O,
Since
I
Fig. 4.
(HZ) Vs E I, 3 _tn interval [i/n, (i + 5)/n] c 2, i = 0,
. ..,
n - 5 such that Vy E Y,
41EQl WWxQ*Wf(y x 0 x 41x s)c NI1If(y x b/n, (i + 5iid x (II). l
Theinterval [i/ ft, (i +5)/n] is determined by looking ai I x s in Fig, 4 atad taking re it first intersects thPyleft hand shaded region to where it last and shaded region. and (HZ) correspo:ld to (i) and (ii) of Secc:jo,n3 (wrth 2r replacing E) we make the following adjutments. First, we choose E sufficiently large so that diam N&(f(y x [i/n, (i -t-5)/t:]) X ql)< E, Vy E Y, 41 E 611, i = 0, . . . 9 n - 5. Second, we could have arranged that diam (41 X 1) < E has b2en done. roof
is is a standard argument, e.g. QUS 8: Y-+(0,00), E
ha
) al-l
Me hfandel’.WdbertCl& Manif&fs
38
I Let Y s ri”_‘uD where C = lJZ_l Ci and D = ui”_ 1&; each IZiand D’ is compact and each IIfg‘ Y has a neighborhood in Y which intersects at most one Ci and at most one B’.L y infinitely zT,dqy simultaneous applications of tlhe compact case, we ,JJ& that (1) and (2) hold with fi’ replaced by hl and cauI finci h, : xQ+WXJ~ I ; V, E) rep1 ii 1:d b’r (C9 e/J). choose ;!r: Y -*(O, OO), S c 613, such ::lrat if c Nz&)(~(Y x 0 x q) and if diam B c Z!Cy), then diam hi(B) < ~(y)/3. Thus Y; (3) ’ (4) &an:; \I:il1(lV&)(f(y X I) X q))l< E(Y), Vy E CT By infim lit?y many simultaneous applications of the compact case, we ea h2 : W x Q ‘-: 8’ x Q such that (1) and (2) hold with ;H replaced by h2 and ( Y, E) replaced bql la,It 8). i H = hr h, satisfies (1) and (2) for (Y, E) as required. l
The pro& :Ilplitsinto the oDmpact and non-compact cases. We will only prove the compact d’~~!i :::, jeaving it to the reader to deduce the non-compact case as we did in I 9re proof c:I’ Theorem 2.1. 4 We appQ Pxyant’s Reduction [4, Section 21 to ourlsituation. Roughly, this states that to shyi.AKcopies of I”; one can apply the k = 1 Base k times to shrink the first coordinate d.1~ ection, the second coordinate direction’, etc., and then compose these homeomcxrp $1ms to shrink all direction.; at once. For the k = cam? case, we need only shrink the /LT-t IV coordinates for som.e large N be&se the remaining directions t are very :9?lr’;ufzl jrc The details follow. Coadi&:~rzf H2) (Section 3) implies that we could bdd a third condition to those already ix;;,;.--ledon the homeomorphism H produced by the proof of Theorem 2.1. (iii) FTWCE~ zh q E Q, 3 E I such that Vy E Y, h( f(yix I) x q) c Ns( f(y, t) X q). This prs~%1~c in a position to apply Bryant’s Reduction argument in a completely s~aigh~~~~l;rl.v~i,. z-dmanner. For finite I G k this producp a homeomorphism h : ‘!! x 0 -, W x LQs,:&isfying:
lI
0) bfdth) q) E VTx Q, either Iz(w, q) = I[w, q) or ai 1E Y such that both (w, q) and I k( w, q) E .h;tIj”(y x P) x q).
‘Xr)xq)cN,(f(yxsxr)x
M Handel/HilbertCube Manifohs
39
for some s E P. Far su
Il’,(f(Y x s x qk-‘jx 4) c
N*Jf(y
x s x r) x q)
(k(f(y x Ik) x q;:! c 2F, vq E Q. show that the decomposition satisfies the Bing Shrinking Critwion.
The proofs of Corollaries 2.3 and 2.4 are well-known (e.g, [l, CL151). ~M(j‘xid&Q (,fxid&C~ Q+ Y X Q), we may assume that X and Y are Q-manit’olds. Choose an embedding g : X + Q such that g(X) is a Z-set in Q. Y x Q by F(x) = (f(x), g(x)). Then F(X) is an embedded Z-set o d in Y X Q [S, Collaring Theorem]. This implies that M(F) = f
M(J3=~~xCO,11~ YxQ/((x,o)-(f(x),
M(f)=Xx[O,
g(x));xEx}
l]u Y/{(x,O)-f(x);xEX},
it is clear that M(.f) = M(F)/{y X Q; y E Y}. Apply the addendum with k = 00 to M(F)! to show that x 8. eomorphism can 6-approxifirate ar XidO It is an easy matter to check that the for any pre-assigned E.
40
/ i
[l] R.D. Anldexson and N. Kroonenberg, Open problems in infinitedim#nsionaltoplogy, Math, Centre T.bsc%s52 (1974) 141-175. 1 [2] R.H. Ping, The Cartesian product of a certain non-manifoldand a line is; 4, Ann. of Math,70(2) I (1959)>399-412, [3] J,L, Bryant, Euclidean space modulo a cell, Fund. Math. 63 (1968) 43-51. [43 J.L. &qunt, Euclidean n-space modulo an (n - l)-cell, Trans. Am. Ma/h. Sot. 179 (1973) 181192. , [5] T.A. Urupman, Notes on Hilbert cube manifolds, University of Kentuckk, Lexington, KY. [6] T.A, cilaprnan, Text of lectures given at the Greensborc Conference in Gctober 1975, to appear. [7] R.D. F&iarards,to appear. i [8] R.D. Edwards and L.C. Glaser, A method for shrinking decompositi$ns of certain manifolds, Trans. Am. Math. Sot. 165 (1972)45-S% (91 M, Ijandel, On certain sums of Hilbert cubes, General Topology and Appl. 9 (1) (1978) 19-28. [lo] RX. Miller, Mapping cylinder neighborhoods of some ANR’s, Ann. of M&h. 103 (1976) 417-427, [ 11) R. Sher, The union of two Hilbert cubes meeting in a Hilbert cube needi not be a Hilbert cube, to i __ aypEm. [12] J.E. West, Mapping cylinders CiXHilbert cube factors, General Topoiogy and Appl. 1 (1971) 13 1-125. [13] J.E. West, Sums of Hilbert cuL= factors, Pacific J. Math., 54 (1974) 2934304. [14) J.B. West, Mapping Hilbert cube manifolds to ANR’s, Am. of Math. 104 (1977) 1-18.
30
non-sii:o,lt x1!.;.ed mapping cyunder theorem for embeddings between Q-manifolds, and a ‘%~b,I theorem” for Keller cubes (i.e. if two &eller cubes intersect in cube &X!I *,weirunion is a Hilbert cube). J. Quinn and R. Wong s how ~\a+:%;A:? : fact follows immediately from the partial sum theorem. Gc;“s~x&:ing a finite dimensional result of (essentially) Bin we 64rove: +itit for certain decompositions 9 of Q, all of elemea t s ii-e homeomorphic to Ik for some fixed Q/ $9 x I$?-=::3 (see Theorem 2.1 and its Addendum). coroiA~~B~i:n’~ i6,the West Mapping Cylinder lheorem [12] w estat9Ei~hr:d
of the Mapping Cylinder Theorem has ele~~:x~szy requiring only an understanding of A>ll-known a anQ AI~~v:;I~~~ l:3,4], and two results from infinite dimensi Isotcpp~ t !T i ension) Theorem and the Collaring Theore of CZx:~:;ra-3t:~ ‘s notes [5]. :~;;dg’~&.~,)l$$. ; ne proof actually gives the stronger result (Corollary (-&z%__& 3 :J A c QI and f: A + Q2 is any map, then the stabi sgxw~( Ctl, ...I &) x Q is homeomorphic to 0. This was pre $2FL~ie4~ pfld&r , “s”[ 131, but it required a separate arg ament unrel h”,:.r~~; .I,;3i;;t :vlinder Theorem. This
pry
iif
X C: CJ IPIP>cell-like set, which now follows from Edwards
3&. Such a statement would be immediz
tSG paper to get around the problem.
31
icking out the reIevant &taik from [4] n 2 contains statemrxts of ction 4 contains a p3-oofof application of Bryant’s Reduction corollaries, the first two of which
the the 13, are
modelled on th: Mikbert cube Q, of closed intervak I = [0, 11.Vk represent Q as nd let Q(n) := In X In+l X In+2X . The metric l
l
l
Q, is a Q-manifold. is a Z-set in W if A is contained in a en A will be collared in W I semi-cc)ntinimwi.e. ; the w:X+X/9 is closzii. When
e will only list the v Yl(a --f(a),
i CA} for the
32
Ad.Han&l~HilkrtCbbe Manifolds
I 2.1.. C&eariz Q-manifold factor am! a closed embeddhg f : Y x I --) thequoti~ent map, Then w X ido : W X Q-, moiplaism.
Theorem 2.1 holds with I replaced by Ik for a
2.2. The finite dimensional analogue of Thear Dby replacing W with a finite dimension ‘, was proved (essentially) by Bryant [3,4]. .3. (West Mapping Cylinder Theo Q-man~old ftzctors.llzen a!x ido : M(f) x Q + Y x kt A, X, and Y be Q-man Then XqY is a QX u Y/(a -f(a); a E A). The followiq corollary was observed by
2.5 Let X b an AR (i.e. a containedin 4 Q-manifold
contains the heart tif the argument way.
Q &
33
ti at TV decompositoin
{f(y x I) x q; y E Y, q E Qj c W x Q 0 3 a homeomcxphism H : 3 x Q + such that both (w. q) and
r now we wil! assume that I has the standard inate direction of Q. Thus
8X d is a Z-set. The proposition (with S’ = 0) folfows by ion) Theorem [S, Section 41 to the natural [c;e- E, b -t el) x Q1 x 0 for sufficiently small r&y, ( t ) holds for soprae6’ > O., repeatedly we construct the desired homesmorphism
tain &(O C& e 6,) and a homeomorci1 that:
h’ satisfies Pqwsition 3.1(3) with (a, & y) rep1 Observe th2.tfor SE [& S,-11,1 z 2, . . . , tt - 1 (1) .r’lf(y )a !)xqlxs =identity, j>i-1. Also,
h i-2 - hi-'(f(yx
I)xqlxs)
and thus (2) #,‘I@- . k $-I(f (y x I) x ql x si = identity, j e summarized as follows:
amei%n of g: Cektain tapering and support restrictions are left out easy for tl’j :,: reader to fill in. I a :Y I + I x I be a homlesmorphiem with motion c tireEy in the second I Let g1: I/. ‘4 direction s .t(::hthat I g*(’ j\- i/n,
l]X[O, 1-(i+l)/(n+l)])=[l-i/G
l]X[+,&] 9ori==1,2 ,...,
n-2.
i b i entifying the first
Then gj %1duces82:f(‘YxI)X~1XI--*f(YXI)XQIXI [second] I 1;\il:::torin I x I with the A factor in Y X 1 [Q, X I]. j Finally g; t ruduces g by extending g2 ambiently to W = Qi x B’using a standard tapering arg l,rfnent. 1 By .a con;rektely symmetric operation centering on W y Q, x 1 (rather than W x Q1 x 0) md pushing from 0 towards 1 (rather than from 1 towards 0) we can corresponding to Fig( 3. Fig. 3 is obtained construct hL: WxQ,xI+ WXQ+I
I
Fig. 3.
I I I 1
by turning Fig. 1:‘:upside down and is interpreted as Fig. 2 was except for the obvious changes (points to the right of or below the heavy broken line/s are left fixed, points to the left are pi,;:;hed close to the shaded region). Specifically, h)Nsatisfies: (1)hsupp Ilk :‘.:UZ PJ&(f(Yx[O, ilnl)~ = hope: ition 3.2(2) with hb rep (3)k For s E [iJ”:n + 1), (i -t l)/( M)’ WtWXQt * hhl f (y x [(i - 1)/n, m3
)’ ‘T%WxC?* ..‘zk+i~f~Y x1)x
-4.
M. Han&l/Hilbert
C%be M~nifoldr
37
2/(n + en &) holds for i = 1. and hh have disjoint supports &e can compose them to get x Q1 x I. Fig. 4 corresponds to PIIIThus H satisfies: x Q1 X I either H(w, ql, t) = (w, ql, t) or 3y c Y such that both (w, (I*) and If s E [O,
Since
I
Fig. 4.
(HZ) Vs E I, 3 _tn interval [i/n, (i + 5)/n] c 2, i = 0,
. ..,
n - 5 such that Vy E Y,
41EQl WWxQ*Wf(y x 0 x 41x s)c NI1If(y x b/n, (i + 5iid x (II). l
Theinterval [i/ ft, (i +5)/n] is determined by looking ai I x s in Fig, 4 atad taking re it first intersects thPyleft hand shaded region to where it last and shaded region. and (HZ) correspo:ld to (i) and (ii) of Secc:jo,n3 (wrth 2r replacing E) we make the following adjutments. First, we choose E sufficiently large so that diam N&(f(y x [i/n, (i -t-5)/t:]) X ql)< E, Vy E Y, 41 E 611, i = 0, . . . 9 n - 5. Second, we could have arranged that diam (41 X 1) < E has b2en done. roof
is is a standard argument, e.g. QUS 8: Y-+(0,00), E
ha
) al-l
Me hfandel’.WdbertCl& Manif&fs
38
I Let Y s ri”_‘uD where C = lJZ_l Ci and D = ui”_ 1&; each IZiand D’ is compact and each IIfg‘ Y has a neighborhood in Y which intersects at most one Ci and at most one B’.L y infinitely zT,dqy simultaneous applications of tlhe compact case, we ,JJ& that (1) and (2) hold with fi’ replaced by hl and cauI finci h, : xQ+WXJ~ I ; V, E) rep1 ii 1:d b’r (C9 e/J). choose ;!r: Y -*(O, OO), S c 613, such ::lrat if c Nz&)(~(Y x 0 x q) and if diam B c Z!Cy), then diam hi(B) < ~(y)/3. Thus Y; (3) ’ (4) &an:; \I:il1(lV&)(f(y X I) X q))l< E(Y), Vy E CT By infim lit?y many simultaneous applications of the compact case, we ea h2 : W x Q ‘-: 8’ x Q such that (1) and (2) hold with ;H replaced by h2 and ( Y, E) replaced bql la,It 8). i H = hr h, satisfies (1) and (2) for (Y, E) as required. l
The pro& :Ilplitsinto the oDmpact and non-compact cases. We will only prove the compact d’~~!i :::, jeaving it to the reader to deduce the non-compact case as we did in I 9re proof c:I’ Theorem 2.1. 4 We appQ Pxyant’s Reduction [4, Section 21 to ourlsituation. Roughly, this states that to shyi.AKcopies of I”; one can apply the k = 1 Base k times to shrink the first coordinate d.1~ ection, the second coordinate direction’, etc., and then compose these homeomcxrp $1ms to shrink all direction.; at once. For the k = cam? case, we need only shrink the /LT-t IV coordinates for som.e large N be&se the remaining directions t are very :9?lr’;ufzl jrc The details follow. Coadi&:~rzf H2) (Section 3) implies that we could bdd a third condition to those already ix;;,;.--ledon the homeomorphism H produced by the proof of Theorem 2.1. (iii) FTWCE~ zh q E Q, 3 E I such that Vy E Y, h( f(yix I) x q) c Ns( f(y, t) X q). This prs~%1~c in a position to apply Bryant’s Reduction argument in a completely s~aigh~~~~l;rl.v~i,. z-dmanner. For finite I G k this producp a homeomorphism h : ‘!! x 0 -, W x LQs,:&isfying:
lI
0) bfdth) q) E VTx Q, either Iz(w, q) = I[w, q) or ai 1E Y such that both (w, q) and I k( w, q) E .h;tIj”(y x P) x q).
‘Xr)xq)cN,(f(yxsxr)x
M Handel/HilbertCube Manifohs
39
for some s E P. Far su
Il’,(f(Y x s x qk-‘jx 4) c
N*Jf(y
x s x r) x q)
(k(f(y x Ik) x q;:! c 2F, vq E Q. show that the decomposition satisfies the Bing Shrinking Critwion.
The proofs of Corollaries 2.3 and 2.4 are well-known (e.g, [l, CL151). ~M(j‘xid&Q (,fxid&C~ Q+ Y X Q), we may assume that X and Y are Q-manit’olds. Choose an embedding g : X + Q such that g(X) is a Z-set in Q. Y x Q by F(x) = (f(x), g(x)). Then F(X) is an embedded Z-set o d in Y X Q [S, Collaring Theorem]. This implies that M(F) = f
M(J3=~~xCO,11~ YxQ/((x,o)-(f(x),
M(f)=Xx[O,
g(x));xEx}
l]u Y/{(x,O)-f(x);xEX},
it is clear that M(.f) = M(F)/{y X Q; y E Y}. Apply the addendum with k = 00 to M(F)! to show that x 8. eomorphism can 6-approxifirate ar XidO It is an easy matter to check that the for any pre-assigned E.
40
/ i
[l] R.D. Anldexson and N. Kroonenberg, Open problems in infinitedim#nsionaltoplogy, Math, Centre T.bsc%s52 (1974) 141-175. 1 [2] R.H. Ping, The Cartesian product of a certain non-manifoldand a line is; 4, Ann. of Math,70(2) I (1959)>399-412, [3] J,L, Bryant, Euclidean space modulo a cell, Fund. Math. 63 (1968) 43-51. [43 J.L. &qunt, Euclidean n-space modulo an (n - l)-cell, Trans. Am. Ma/h. Sot. 179 (1973) 181192. , [5] T.A. Urupman, Notes on Hilbert cube manifolds, University of Kentuckk, Lexington, KY. [6] T.A, cilaprnan, Text of lectures given at the Greensborc Conference in Gctober 1975, to appear. [7] R.D. F&iarards,to appear. i [8] R.D. Edwards and L.C. Glaser, A method for shrinking decompositi$ns of certain manifolds, Trans. Am. Math. Sot. 165 (1972)45-S% (91 M, Ijandel, On certain sums of Hilbert cubes, General Topology and Appl. 9 (1) (1978) 19-28. [lo] RX. Miller, Mapping cylinder neighborhoods of some ANR’s, Ann. of M&h. 103 (1976) 417-427, [ 11) R. Sher, The union of two Hilbert cubes meeting in a Hilbert cube needi not be a Hilbert cube, to i __ aypEm. [12] J.E. West, Mapping cylinders CiXHilbert cube factors, General Topoiogy and Appl. 1 (1971) 13 1-125. [13] J.E. West, Sums of Hilbert cuL= factors, Pacific J. Math., 54 (1974) 2934304. [14) J.B. West, Mapping Hilbert cube manifolds to ANR’s, Am. of Math. 104 (1977) 1-18.