Handle decompositions of simply-connected five-manifolds. II
Handle decompositions of simply connected smooth or piecewise linear five-manifolds are considered. The basic notions and constructions necessary for proving further results are introduced.
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| author | Shkol’nikov, Yu. A. Школьников, Ю. А. |
| author_facet | Shkol’nikov, Yu. A. Школьников, Ю. А. |
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| description | Handle decompositions of simply connected smooth or piecewise linear five-manifolds are considered. The basic notions and constructions necessary for proving further results are introduced. |
| first_indexed | 2026-03-24T03:24:16Z |
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UDC 515.162
Yu. A. Shkol'nikov, cand. phys.-math. sci. (Inst . Math. Acad. Sci. Ukraine, Kiev)
HANDLE DECOMPOSITIONS OF SIMPLY-CONNECTED
FIVE-MANIFOLDS. II
P03KJIA,U HA pyqKH O,UH03B'H3HHX Il'HTHBHMIPHIIX
MHOrOBil,UIB. II
The handle decompositions of simply-connected smooth or piecewise-linear five-manifolds are con
sidered. The basic notions and constructions necessary for proving fu rther results are introduced.
PoJr m111aE:TbCJI po3KJ1a,ll Ha p)"lKH 0/lHOJB' Jl3HHX r JlaIIK HX a6o KYCKOBO-JliHiRHHX II' .>IT HBHMipHHX
MHOl"OBHAiB, HaseneHi OCHOBHi llOHJirr.ll j KOIICTPYKL1ii, Heo6xin11i l!Jlll onep)KaHH.ll nonaJll,UJHX pe-
3YJl bTaTiB.
The main result of th is paper is Theorem 3 asserting that the D. Barden ·s handle de
composition of a closed I-connected srriooth or PL 5-manifold is geometrically diag
onal. It is obtained as a consequence of Theorem 2 apparently describing the con
struction of the C. T. C. Wall's diffeomorphisms for each of I-connected 4-manifolds
s2 X S2 # s2 X s2 and s2 ~ s2 # s2 X s2. The basic notions and tools necessary to
prove these theorems were presented in [ l].
4. D. Barden's constructions. As was proved by D. Barden in [2J, any closed I
connected 5-manifold is diffeomorphic to the finite connected sum of 5-manifolds of
certain types. These manifolds are constructed as follows.
Consider standard 5-manifolds M = A * A and X =B * A-, where A and B are
the elementary 5-manifolds designed above. Let V be either M or X; then V ad
mits an exac t handle decomposi tion V = h0Uhf Uh}. which induces the canonical
handle decomposition of av = h0U hf1 U hf2 U hi, U h'f2 Uh4 wi th the canonical basis
{ a1, b1, a2 , b2 } of Hi(aV). All cycles of this basis can be realized by 2-spheres em
bedded in av. The spheres a1• and ii2 arc determined by the cores of 5-dimensional
') ? - -
2-handles h1, and h2 of V. the spheres b1, and bi are the h-spheres of these
handles. The intersection form Q(aV) in the canonical basis is
(~ ~)a,(~ ~) if av=aM or G ~) a, G ~) if av=ax.
Consider the following nondegenerate matrices with integer coefficients:
[~
0 0
-:1 B(k) = [~
0 0
-2kJ l 0 0 k
A(k) =
k l'· 0 ' 2k 0 ' l
0 0 l 0 0 I
(1)
l -2k 2(1-2k) -4k 0
0 2k- l 2k k - l
C(k) =
2k 0 0 l - 2k
1-k -2(k-l) l-2k 0
For any integer k ~ 1, specify automorphisms fi... of the group 112(oM) and auto
morphisms gk• and hh of H2(aX) as follows: f .. {a 1, b 1, a2 , b2 } = {a 1, b 1, a2 ,
© YU. A. SHKOL'NIKOV, 1993
1282 ISSN 004/ -6053. Y,q, . Mam. >Kyp11., 1993. 111 . .J5. N' 9
HANDLE DECOMPOSITION~ OF SIMPLY-CONNECTED FIVE-MANIFOLDS 1283
I
b2} A(k), gko{a 1, b 1, a2, b2} = {a 1, b 1, a 2, b2}B(k), hh{a1, b 1, a 2, b2} = {a1, b1,
a2 , b2 } C(k).
One can easily calculate that all ii,. preserve the intersection form Q(oM), whereas
gh and h"- preserve Q(dX). By the Wall 's theorem (3), there exist diffeomorphisms
fk of oM, and gk and hk of ax, which induce the diffeomorphisms fko, gk•' and h"-
011 H 2(oM) and H2(oX). For k > l introduce closed I-connected 5-manifolds Mk=
= MU,.(-M), X[B(k)] = XU8 (-X), and X[C(k)] = XU1 (-X) for k ~ l. Introduce also
lk. .t l.4:
M = X = S5 M = S2 x S3 and X = BU (-B) where g = id Since ::-.B = S2 x } Q ' 00 ' 00 goo ' OQ • 0 ~
~ s2 =- O::JP 2 # (-O::JP 2), the H2(dB) admits also a basis {p, q} such that each of p
and q corresponds to the summand O::JP 2. One can easily specify the diffeomorphism
g_1 of dB, which induces the following automorphism g_ 1• of H 2 (dB): g_J.: {p, q}
➔ {p. -q}. In the canonical basis {a= p, b = p - q} of oB, the automorphis!Jl g_i.
is represented by the matrix C ~1} Put X_ 1 ==XU 8_1(-B). By definition , all 5-
manifolds constructed above admit exact handle decompositions.
The matrices B(k) and C(k) differ from those considered in [2] because instead of
the canonical basis for H 2(0B) and ax = oA # dB "' dB * dB as in [2], the
corresponding bases {p , q} and {p1, q1, p2 , q2 } are used. When fixing the canonical
basis, the matrices B(k) and C(k) change to ( l ).
Lemma 5 [2].
1) Hz(Mk) = ~k EB~. k* 1, oo;
2) H2(X_ 1) = ~2, H2(X~) = H2(M ~) = ~ ;
3) Hi(X[B(k)]) = ~2k EB ~ ik• H2(X[(k)]) = ~Zk-l EB ~ 4k_2 , 0 < k < oo.
Any I-connected closed 5-manifold W admits the linking form b(x, y) = x o ye
e (Q / ~ on tors (H2(W)). This is a nonsingular nondegenerate skew-symmetric inte
ger bilinear form. In [2], a b-basis {z1, z2, x 1, y1, • • • , x,,., y,,.} was constructed, i.e.,
the basis in which z I has an odd order cp, z2 has the order 2cp, and b(z 1, z2) = l / <p;
both x; and Y; have an odd order 0; and bt, y.)= 1 / e,.; on the other pairs (u, v) of
the basis elements except, possibly, (z2 , z2) and ~ ;, y ,), i = 1, .. . , m, the value of
b(u , v) is 0. The elements z1 or both z1 and z2 may be missed from the b-basis.
In this case, we include z1 and z2 into the basis assuming them to be equal to zero. A
basis of the entire H2(W) is called a b-basis if it contains a b-basis of tors (H2(W)).
It is shown in [2] that a b-basis may be chosen to be minimal, i.e., such that it contains
a minimal number of elements. Since for each x e tors (H2(W)), we have b(x, x) = 0
or b(x, x) = 1 / 2, the minimal b-basis of tors (H2(W)) may be modified so that b(x,
x) = 0 for each element X of the b-basis except, possibly, for one element. For any
x e tors (H2(W)) we have b(x, x) * 0 if w 2(x) "le O ([2]). If w2(e) * 0 for each
e e Fr (/12(W)), then we can modify also a basis of Fr (H2(W)) so that w2(e) = 0 for
each element e of the basis except, possibly, for one element.
Thus we have constructed the basis of H2(W), which we call the minimal w2 - b-
ISSN 004/-6053. YKp. Mam. JKYP"· • 1993. m. 45, N• 9
1284 YU. A. SHKOL'NIKOV
basis.
Theorem 1 (the Barden decomposition theorem, [2)). For a11y b-basis { z1, z2,
x 1 , y 1, ••• , Xm, Ym, 11, ••• , I,} of H 2(W), there exists a diffeomorphism 'fl of W
into the ma11ifold
V = Mt t # Mx y # ... # Mx y # Me # ... # Me , (2)
1' 2 1' l r' r 1 ,
where Mt t = X 1 if the order cp of z 1 is 1, i.e. z 1 = 0, and Mt z.. =
I' 2 - I' -,.
= X[C((cp -1) / 2)) if cp > 1; H . . = M8 . if b(y , y .)= 0 and Mx . Y· = X[B(0;f 2))
x,, Y, I ' I 1' I
if b(y;, Y;Fi' 0, where 0; is the order of X; and y;; Me; = M_ if w2(e;)= 0 and
Me,=X_ if w2(e;);t 0. For each pair (u, v) = (z 1, z2) or (x;,Y;). the diffeo
morphism 'I' i11duces the isomorphism between gp (u, v) a11d H 2(Mu ), which
' '
.preserves the linki11g numbers. For each generator e; of Fr(H2(W)), we have
H2(Me) = ~ and w2(Me) = 0 if! w2(e;)= 0.
It follows from Theorem 1 that any b-basis of H 2(W) determines a handle de
composition of W which contains one 0-handle, one J-handle, and a pair of 2-handle
and 3-handle for each element of this basis. The minimal w2 - b-basis determines an
exact handle decomposition of W which contains at most one summand of type X for
each of tors (H2(W)) and Fr (H2(G)), all other summand being of type M. Since the
basis is minimal, the handle decomposition is exact. In what follows, we will consider
only such decompositions and call them the Barden handle decompositions.
s. Diffeomorphisms of manifolds s2 ~ s2 # s2 x s2 and s2 x 5 2 # s2 x 52 • Let
V denote either s2 X s2 # S2 X S2 or ax= S2 ~ s2 # s2 X S2. V admits an induced
canonical handle decomposition with the canonical basis {a1, b 1, a 2, b 2} and 2-
spheres { 12 1, b 1, 12 2, b2}, which realize this basis. {a2, b2} will always be considered
as a canonical basis of the second summand, i.e., of S2 x S 2. We prove here the
theorem which provides a geometric description of the Wall's diffeomorphisms of v.
Theorem 2. For V = aM or V = ax, let <p. be an automorphism of H 2(\/),
which preserves the intersection form Q(V). Let C be a matrix, which represents
cp. in the cano11ical basis { a1, b 1, a 2, b2} of an i11duced ca11onical handle decom-
position of V. Then there exists a diffeomorphism cp of V, which induces the
automorphism cp. 011 H2(V) a11d maps each sphere of {12 1, b1, 12 2, b2} into the
corresponding sphere of { ii 1, b1, 12 2, b2 }C, where the addition operation means
the connected summi11g and the minus sign means the altering of the oriellfation.
Fix the above-mentioned induced canonical handle decomposition of V. By re-
arranging the handles, we can construct the proper handle decomposition of V. Let 1)
be the corresponding diffeomorphism of V. The a-spheres of the proper handle de-
composition V=h0 Uhf1Uhf2Uhf1Uhf2Uh4 arein ah0 andthecoresofthese 2-han
dles determine the 2-spheres { 12, b, i, y} = T\ { 12 1, b 1, 12 2, b 2} which realize the ba
sis { a, b, x, y} = Tl. { a 1, b 1, a 2, b2} with geometric intersections and { x, y } corres-
ponds to the second summand S2 x s2. Consider a new proper handle decomposition
0 -2 -2 -2 -2 4 -2 -2 - 2 -2 2 2 2 2 V= h Uh11Uh12 Uh21Uh22Uh, where {h11 , h12 , h21 , h22 } = {h11 , h12 , h5_1, h22 }C,
the addition operation means the handle summing, and the minus sign means the
ISSN 0041-6053. YKp. Mam. ;,.;yp11.., 1993, m . 45, N • 9
HANDLE DECOMPOSITIONS OF SIMPLY-CONNECfED FIVE-MANIFOLDS 1285
altering of orientation for the core of a handle. If we construct a diffeomorphism 8 of
V, such that 8(hj) = h;J, .;, j = 1, 2, and then tum back to the induced canonical
handle decomposition, we obtain the diffeomorphism cp = 11-19r1 we are searching for.
Thus, our nearest aim is to construct a diffeomorphism 8.
The 2-ha,ndles of the proper handle decomposition of V are glued along a framed
link in S3 = 'iJhO of type (3) for V = 'iJM or type (4) for V = 'iJX.
tdtd (3)
ffitd (4)
Since any two links of type 3 are ambiently isotopic in S3 and the same holds also
for any two links of type 4, Theorem 2 will be proved if we show that the link for atta-
ching 2-handles 'ii;J, i, j = 1, 2, to S3 = oh0 is the same as that for attaching h.J.
Denote this property by r. The property r is equivalent to all mutual intersection in
dices of {a, fj, .i, ji }C being geometric (algebraic indices of {a, fj, x, ji}C are
equal to those of {a, b, x, y} because C preserves the intersection form).
Let Y be an arbitrary closed I-connected 4-manifold with the indefinite intersec-
tion form. Consider V = Y # S2 x S 2. In the proof of the Wall's Theorem (3), all the
generators of the group of automorphisms of H 2(V) preserving the intersection form
are presented. Let {x, y} be a canonical basis of H 2(S2 x S 2) and z be an arbitrary
element of H 2(Y). Consider the following automorphisms of H 2(V):
Ei: z ➔ z-(z · OO)y
x➔x-Ny+ro
y➔ y
E~: z ➔ z-(z·ro)y
x➔ x
y➔ y-Nx+ro.
where ro is the element of H2(Y) such that oo · ro = 2N e ~ - For ro e H2(V) such
that lro · rol = 1, if it exists, consider the automorphism S( w)
2
z ➔ z---(z·ro)co, x ➔ x, y ➔ y.
(J.)·(O
Consider also the following automorphisms
ISSN 0041-6053. YKp . MCllll.)KJpll., 1993, 111. 45, N"9
1286
z➔ z;
x➔ -x
y➔ -y
YU. A. SHKOL'NIKOV
x➔ y
y➔ x.
As was shown by Wall (3], in the case where Q(V) is even. the group of auto
morphisms of H2(V) preserving the intersection form Q(V) admits the following ge
nerators:
1) Ei, E~ for all Ol e H2(Y) with even w · w;
2) R0 , R1, R2.
In the case where Q(V) is odd, the generators are the same as specified in 1) and 2)
and also S(u) for a fixed u e H 2(V) such that lu · ul = 1 By applying this result to
V = ax with the.basis {a, b, x, y}, we obtain E~. 13 : a ➔ a - (2a + l3)y, b ➔ b-2ay,
x ➔ x - 2a(a + l3)y + 2aa + l3b , y ➔ y for any w = 2aa + l3b e H 2(S2 ~ S 2). E~.!3
can be obtained as a result of permuting x and y in E~. !3. Fixing u = a, we obtain
S(u) = 8-l EB E.
For V = aM, we have
for any w = aa + l3b , ·because W· w is always even, and E~.!3 as a result permuting x
and y in E&,!3.
It sufficies to pr9ve property r only for these generators, since the property is ob
vious for R0 , R 1, R2.
To prove the property r for 8-i EB£, consider 8-l in the basis {p, q} of
H 2(S2 ~ S2). This basis is realized by the embedded 2-spheres { p, q} and determined
by the handle decomposition with the 2-handles attached along the obvious framed
link. This link consists of two circles in S3 having framings 1 and - 1. The first sphere
corresponds top and the second to q. Since,bydefinition, 8-i (p)=p and 8_ 1(q)=
-q, the link is not changed and property r is obvious. Since the canonical basis { a,
b} of H 2(S2 ~ S2) is obtained from {p, q} with a= p and b = p - q, 8-i can be
performed with one Kirby move, hence, 8-l has property r in the canonical basis of
H2(S2~ S2). The same is, certainly, true for 8-l EBE in the canonical basis of ax.
If we prove the property r for E~.!3 and Ei, 13 • the proof of Theorem 2 will be
completed because R0, R 1, R2, and 8-i EB E are of order 2 and the diffeomorphisms
opposite to E~.!3 and ei,13 are the same as E~.!3 and E/;_,13 , but with different a
and [3 . Since x and y in e;, 13 and E~.!3 are symmetric, it suffices to prove
property r only for E~.!3 for V = ax or V = aM. For V = cJX, consider E~.!3 as
the product C2C 1, where the automorphisms C 1 and C2 act as follows
C1 : a➔ a'=a C2 : a' ➔a"=a'-(2a+l3)y'
b➔ b'=b b'➔b"=b' - 2ay'
x➔ x' =x-2a(a+l3)y+2aa+l3b x' ➔x"= x'
y➔y'=y y'➔ y"=y'.
ISSN 0041-6053 . YKp. -'tam. JKJP"·• 1993 , m . 45 , N" 9
HAND F. DECOMPOSITIONS OF SIMPLY-CONNECTED FIVE-MANIFOLDS 1287
Note that C I and C2 do not preserve the intersection form Q(oX). Having per
formed C 1 for a given proper handle decomposition h,1, i, j = l, 2 of ax, we obtain
a handle decomposition attached along the framed link on the left -hand side of the
picture.
In Fig. 1 we show the attaching circles of 2-handles. Near each circle, we show the
framing and the cycle in H 2(dX) determined by the core of the 2-handle attached to
this circle. Denote these circles by Ya'• Yb'• Yx• Yy'· Since yy' has a trivial framing and
links Yx• geometrically one time, we can apply the Kirby moves [4] to free Yx of Ya'
and Yb'· It readily follows from the definition of the Kirby move that the composition
of the Kirby moves we have just performed determines the automorphism C2 of
H 2(oX) applied to the link on the left-hand side. Thus, after performing C I and C 2,
we have a framed link on the right -hand side of the picture with all linking numbers
being geometric . The property f' for V = cJX is proved.
Fig. I
Fig. 2
ISSN 004 / -6053 . Y..:p . M (Jll l. »:Yf'II .. 1993.111. ./5 . N" 9
1288 YU. A. SHKOL'NIKOV
For V = dM, we have E&.!3 = C2C1 with
C1: a➔ a'= a
b➔ b'=b
x➔ x' =x - al3y +aa +J3b
y➔ y'=y
C2 : a'➔a"=a' - J3y'
b'➔ b" =b' -ay'
x' ➔x"=x'
y' ➔ y" =y'.
The applica.tion of C2 of H 2(cJM) to the link on the left-hand side of Fig. 2 is
equivalent to performing a series of Kirby moves with it to obtain a link with geome-
tric linking numbers on the right-hand side. This proves property r for E~.!3 and
completes the proof of Theorem 2.
6. Applications to the Barden handle decomposition. Here we use Theorem 2 to
prove the following theorem .
Theorem 3. All incidence indices of 3-handles and 2-handles in the Barden
handle decomposition of a closed I-connected 5-manifold are geometrically dia
gonal .
It suffices to prove this theorem for M.,,, X .,, , X_1, M1" X[B(k)] and X[C(k)].
For M.,,, X.,,, and X_1, the theorem is obvious. To prove it for other manifolds,
consider an exact handle decomposition of the standard 5-manifold W = M or W = X.
It induces the canonical handle decomposi tion of the standard 4-manifold aw with
the canonical basis {a1, b 1, a 2, b 2} realized by the 2-spheres {a 1, b1, a2 , b2}
embedded into aw (b 1 and b2 are the b-spheres of 5-dimensional 2-handles of W
). Each of closed 5-manifolds Mk, X[B(k)], and X[C(k)] can be obtained as a double
of M, X, and X, respective ly , along the corresponding boundary diffeomorphisms fie
gk and hk. By Statement 3, the homomorphism a3: C 3 ➔ C2 can be represented in
the canonical basis of the boundary by the matrix at =fib;)· b, for Mk, at =
g"(b;)·b1 for X[B(k)], and a,,= h/b,)·b, for X[C(k)]. It is easy to calculate these
matrices for M", X[B(k)], and X[C(k)] to obtain
G ~k} . (20k -~k} (2(1~2k) l-02k}
respectively. By Theorem 2, all the coefficients of these matrices are geometric. Thus,
Theorem 3 is proved.
This theorem can be applied also to construct round Morse functions (5). Combi
ning it with the technique of A. T. Fomenko and V. V. Sharko [6], we obtain the follo
wing theorem.
Theorem 4. Any closed I-connected 5-manifold admits an exact round Morse
function.
1. Shkol' nikov Yu . A. Handle decompositions of simply-connected five-manifolds . I // YKp. MaT.
)KypH. -1993. - 45, N" 8. - C . I 151 - 1156.
2. Barden D. Simply-connected five-manifolds// Ann. Math. - I 965. - 82. - P. 365-385.
3 . Wall C. T. C. Diffeomorphisms of 4-manifolds // J. London Math. Soc. - 1964. - 35 . - P.
131- 140.
4. Kirlry R. A calculus of framed links in S' II Invent. Math. - 1978. - 45 . - P. 35-56.
5 . Asimov D. Round handles and singular Morse - Smale flows// Ann . Math. - 1975. - 102, N" I. -
P. 41-54.
6. WapKo B. B. <l>yHKLIHH Ha M110roo6pa3Hl!X. - KHes: HayK . llYMKa, 1990. - 196 c.
Received 12.06.92
ISSN 0041-6053. YKp. Mam. JK_YpH. , 1993, m. 45, N• 9
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| id | umjimathkievua-article-5931 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:24:16Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/43/a2013b2375cab1deb0b16dab817c3943.pdf |
| spelling | umjimathkievua-article-59312020-03-19T09:21:13Z Handle decompositions of simply-connected five-manifolds. II Розклад на ручки однозв’язних п’ятивимірних многовидів. II Shkol’nikov, Yu. A. Школьников, Ю. А. Handle decompositions of simply connected smooth or piecewise linear five-manifolds are considered. The basic notions and constructions necessary for proving further results are introduced. Розглядається розклад на ручки однозв’язних гладких або кусково-лінійних п’ятивимірних многовидів. Наведені основні поняття і конструкції, необхідні для одержання подальших результатів. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5931 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1282–1288 Український математичний журнал; Том 45 № 9 (1993); 1282–1288 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/5931/8545 https://umj.imath.kiev.ua/index.php/umj/article/view/5931/8546 Copyright (c) 1993 Shkol’nikov Yu. A. |
| spellingShingle | Shkol’nikov, Yu. A. Школьников, Ю. А. Handle decompositions of simply-connected five-manifolds. II |
| title | Handle decompositions of simply-connected five-manifolds. II |
| title_alt | Розклад на ручки однозв’язних п’ятивимірних
многовидів. II |
| title_full | Handle decompositions of simply-connected five-manifolds. II |
| title_fullStr | Handle decompositions of simply-connected five-manifolds. II |
| title_full_unstemmed | Handle decompositions of simply-connected five-manifolds. II |
| title_short | Handle decompositions of simply-connected five-manifolds. II |
| title_sort | handle decompositions of simply-connected five-manifolds. ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5931 |
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