The handle decompositions of 1-connected smooth or piecewise linear 5-manifolds are considered. The basic notions and constructions necessary for proving further results are introduced

Shkol’nikov, Yu.A.

Ukrains’kyi Matematychnyi Zhurnal

We prove the existence of the exact handle decomposition of a simply-connected smooth or PL 5-manifold with a standard simply-connected boundary of signature 0, the triviality of a 5-dimensional h-cobordism with ends of such type and the uniqueness, up to diffeomorphism (PL-isomorphism), of a smooth (or PL) h-cobordism between the given simply-connected 4-manifold and the corresponding standard manifold

Shkol`nikov, Yu.A.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Handle decompositions of simply-connected five-manifolds. III

Handle decompositions of simply-connected five-manifolds. I

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

Shkol'nikov, Yu.A.

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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Handle decompositions of simply-connected five-manifolds. II

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Handle decompositions of simply-connected five-manifolds. I

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)