Composition constructions on diffeomorphisms of SpSq

Pacific Journal of Mathematics

COMPOSITION CONSTRUCTIONS ON DIFFEOMORPHISMS OF S p × Sq R EINHARD S CHULTZ

Vol. 42, No. 3

March 1972

PACIFIC JOURNAL OF MATHEMATICS Vol. 42, No. 3, 1972

COMPOSITION CONSTRUCTIONS ON DIFFEOMORPHISMS OF Sp x S9 REINHARD SCHULTZ

It is shown that a map from Sp+r to Sp in the image of the J-homomorphism sends particular types of diffeomorphisms of S* X S« into diffeomorphisms of Sp+r X S«. This is applied to the problem of determining the diffeomorphism type of an exotic (p + q + r + l)-sphere obtained by attaching Dp+r+1 x Sq to Sp+r x Dq+1 via a diffeomorphism of Sp+r X Sq. The first section of this paper deals with the generalities of the construction. In the next two sections we exploit these ideas in a study of the plumbing pairing (compare [4]), and we show that the pairing vanishes in an infinite class of cases (3.2) and some selected low dimensions of other types (3.4). The last section of this paper uses the basic construction to investigate the existence of smooth semifree circle actions on homotopy nine spheres [5]. In particular, it is shown that if Σ9 does not bound a spin manifold, then it has no semifree circle action with 5-dimensional fixed point set. See [23] for further nonexistence theorems concerning semifree circle actions on homotopy spheres and some geometric applications. 1* Constructions for compositions* Let a e πp+r(Sp) and β e πp(SOq+1) be given. Then it is well known that β induces a diffeomorphism of Sp x Sq and the composition β aeπp+r(SOq+d induces a diffeomorphism of Sp+r x Sg. If a is in the image of / : πr(SOp) —• πp+r(Sp) we shall give a geometric procedure for passing from the diffeomorphism induced by β to that induced by β a. In all our p applications a will be the nontrivial homotopy class in πp+1(S ). PROPOSITION 1.1. Let X be an H-space, let jeπr(SOp), and let β 6 πp(X), where p Ξ> 2. If h is the diffeomorphism of Sr x Sp induced by 7, the map π: Sr x Sp —> Sp is projection, and q: Sr x Sp -— Sp+r is the collapsing map, then the following formula holds:

h*π*β = π*β-q*(β-J(y)) . p

r

The dot represents multiplication in [S x S , X]. The above result was proved for X a double loop space in [21, Appendix]. Proof. Without loss of generality the diffeomorphism h maps r+i *x D to itself by the identity (* is the basepoint). Thus if 739

740

REINHARD SCHULTZ p

K = S

r

x S

r+1

U * x D

,

then Λ extends to a homeomorphim of K. It is easy to see that K has the homotopy type of Sp V Sp+r, since the map (π, g):K->Sp x S p + r p p+r deforms to a map into S V S which induces isomorphisms in homology (the maps π and q are extensions of the maps on Sp x Sr to K). An explicit homotopy inverse to the above map may be constructed as follows: Define g: Sp V Sp+r —»K so that g\Sr is inclusion into * x Sr and g maps gP + r

=

2Jp

χ

gr y g,-i

χ

2Jr+l

by sending (x, y) e Dp x Z) r + 1 to (p(x), y) e K Q Sp x D r + 1 , where p maps D p to S p by collapsing the boundary. We then compute g*h*π*β from its restrictions to Sp+r and Sr; an argument like that of [21, Appendix] shows that h*π*β has the desired form in [K, X]. The proposition follows upon restriciting to Sp x Sr. 1.2. Leέ f be a diffeomorphism of Sp x £ r representing 7 αwd Ze£ g be a diffeomorphism of Sp x S g represented by β e πp(SOg) (notation as in l l ) . T%eπ the commutator diffeomorphism [lq x / , g x l r ] o^ Sp x Sq x Sr represents the homotopy class q*βJ(j) e [Sr x Sp, SOq+1]. COROLLARY

Proof. By a direct computation it follows that the commutator diffeomorphism sends (x, y, z) e Sp x Sq x Sr into (a?, [goixj-'goπffa y)]z, y) , where βr0: S2' —> SO g + 1 is such that f(x, y) — (x, go(x)y) and the multiplication within the brackets comes from the group structure of SOq+1. By 1.1 we know that the homotopy class of gQπf = f*π*β is the product of the class of goπ, τr*/S with the class q*βJ(y), and the corollary follows from this. We shall need generalizations of 1.2 in §4. Suppose that h is a p q q diffeomorphism of S x S which is the identity near Ώ\ x S (compare p q [17, 2.3]). Then h is homotopic to a map sending (x, y) e S x S to p (x, hf(x)y), where h! is a continuous map from S into the space Gq+1 of self-maps of Sq with degree + 1. Let β be the homotopy class in πp(Gq+1) so obtained; it is immediate β is a pseudo-isotopy invariant of h and that the formula β(hjι^ = β(h^ + β(h2) holds. For any integer s satisfying l^s^p — 1, the diffeomorphism h induces a diffeoίs] p s s morphism h of S " x S x Sq such that hίs]\D x Sq = h\DpxSq (for some coordinate disk D in Sp~s x Ss) and h[s} = 1 off D x S. PROPOSITION

1.3.

Let h: Sp x Sq —>SP x Sq be a

diffeomorphism

COMPOSITION CONSTRUCTIONS OF DIFFEOMORPHISMS OF S*> x S« p

741

9

which is the identity near D + x S , and let f be a diffeomorphism of Sp x Sr induced by 7 e πp(SOr). Ifβe πp(Gq+1) is the class associated to h, p q r then the commutator diffeomorphism [lq x f,h x l r ] of S x S x S is βguαί £0 α diffeomorphism of the form kίP] for some k on Sp+r x S9 P r q which is the identity off D ^ x S . Furthermore, the homotopy class associated to k is the composition β J(y) e πp+r(Gq+ί). Proof. Pick a representative / of 7 which is the identity near r p D + x S . Then the commutator above is the identity near the complement of Dp x Dr x Sq, and hence does have the form kίP]. The ίP} homotopy class of k may be computed from that of k by a computation of the commutator similar to that of 1.2. Notice that — β is the homotopy class associated to h~\ Suppose that q = 3 and the above diffeomorphism h of Sp x S 3 is equivariant with respect to the free action of S1 on the second factor by scalar multiplication. Then a diffeomorphism h of Sp x S2 is induced on the quotient manifold, and a homotopy class β e πp(G3) is obtained. Given / and 7 as above, one may form the commutators [13 x /, h x l r ] and [12 x f, hx lr] and obtain diffeomorphisms k of Sp+r x S3 and I of Sp+r x S2; the diffeomorphism k is equivariant with respect to the action of Sι on S3, and ί is the diffeomorphism which it induces on p+r 2 S x S. PROPOSITION 1.4. In the above notation, the homotopy class in πp+r(Gz) determined by I is βj(j).

The proof is straightforward. 2* Plumbing formulas* Throughout this paper &(M) will denote the group of pseudoisotopy classes of orientation-preserving diffeomorphisms of the smooth closed manifold Mn. Recall that there is a canonical map σ: Γn+1-+ 2${M) (compare [21, 1. 7]), whose image we shall call Δ\ this map is 1 — 1 if M is a product of spheres [21, 1. 7]. Furthermore, Δ is contained in the center of &{M) (compare [17, p. 529]). The plumbing pairing σp,q:

πp(SOq)

x πq(SOp)

> Γ P+q+l

may be defined as follows. There are canonical maps of πp(SOq) and v 9 πq(SOp) into j2f{S x S ) as noted previously and we denote the homotopy and pseudo-isotopy classes by the same letter. Let / represent ueπp(SOq) and g represent veπq(SOp); without loss of generality / is the identity near Ό\ x S9 and g is the identity near Sp x Dη+. The commutator [u, v] is thus represented by a diffeomorphism [/, g] which

742

REINHARD SCHULTZ

is the identity off D^. x Dl, and hence [u, v] = σ{i) e Δ, for some p q 7 e Γp+q+1. Since σ is 1 — 1 for M = S x S , 7 is uniquely determined, and we define σ{u, v) to be this element 7. The bilinearity of a follows easily from the definition. REMARK. Often the definition of the plumbing pairing σp>q is extended to πp(SOq) x πq(SOp+ι). This is done by taking 7 e Γp+q+1to be defined by the attaching construction p+q+i

S

=

P+I

D

χ

S9\JiutvjSp

x Dq+1

.

(The diffeomorphism class of the above manifold is a pseudo-isotopy invariant; compare [17, §5].) The fact that this extends the previous definition follows from 2.1 below. If one of vx or v2 is not in the image of πq(SOp) in πq(SOp+1), the formula σ(u, vx + v2) = σ(u, vx) + σ(%, v2) may be false. 2.1. Lei (7(7) e J g ^ ( Λ P ) be given, and suppose Qn+1 is a closed smooth manifold which Mn separates into two components N and P. Then the identification manifold N\Jσ(r)P is diffeomorphic to LEMMA

The proof of this follows exactly the same pattern as the more specialized remark in [17, 2.3, p. 526]. We include another result on manifolds obtained by boundary identification which will be used in subsequent proofs. Let M, N, P be compact connected smooth manifolds with boundary, and let Ql9 Q2 be closed smooth manifolds such that dM = Qly dN = Qx (J Q2 (a disjoint union), dP = Q2, If h and k are diffeomorphisms of Q1 and Q2 respectively, then one can form the smooth manifold X(h,k) - M\JhN\JkQ

.

2.2. Suppose there is a diffeomorphism φ of N mapping Qx and Q2 into themselves such that φ \ Qί = h! and φ\Q2 = k\ Then X(h, k) is diffeomorphic to X(h'~ιh, kkf). LEMMA

Proof. Define the diffeomorphism Φ piece by piece. Let Φ\M{J P be the identity, and let Φ\N= q of Milnor-Munkres-Novikov [16, p. 583] has a similar description, although for computational purposes it is best described as the map τv q: πp(SOq) x πq(PD/O) —> πp+q(PD/O) for which τ(a, β) = β J(a). For our purposes it will be convenient to interpret these pairings in terms of pseudo-isotopy classes of diffeomorphisms of

COMPOSITION CONSTRUCTIONS ON DIFFEOMORPHISMS OF S? x S«

743

a product of three spheres; other elaborations are obviously possible. One may define pairings σ': πp(SOr+1)

x πq(SOp)

T : τΐp(SOr+1)

x Γq+ι

> Γp+q+1 > Γp+q+ι

via induced diffeomorphisms of Sp x Sq~r x Sr in much the same way that σ and r are defined. One uses the fact that there are embeddings Sp x Sq~r x Dr+1 g S p x Dq+1 £ £^+«+1 and glues together S p x S ? ~ r x Dr+1 and the closure of its complement in Sp+g+1 via the diffeomorphism of the product obtained. We remark that πg(SOp) maps into [S9~r x S% SOP] via maps fixed off Dg+~r x Ό\\ likewise, Γg+1 acts via diffeomorphisms fixed off Dq+r x Dr+. Levine has defined a 4-linear map δ which generalizes σ and τ to some extent (see [18, §7]; actually, δ is expressible in terms of a and r ) . The result stated below could also be formulated in terms of δ and a related pairing δf. REMARK.

PROPOSITION 2.3. Let i: SOr+1 —• SOq+1 be the inclusion. σ'(a, β) = σ{a, i*β) and τ'(β, a) = τ(i*β, a).

Then

Proof. Let Q1 = Spx Sq~r and let Q2 = Spx Sq. Let N=SpxN' be the cobordism between Qt and Q2 constructed from the above embeddings; i.e. JSP = Dq+1 - Int Sq~r x Dr+ί. Finally, let p ? r r+1 p+1 ? M = S x S " x D and let P = D x S. Then Sp+q+1 = M\JιN\JiD and MUi-ΛΓ = Sp x D ? + 1 . If we can extend the diffeomorphisms induced by σ, and r on Sp x S g and by 0"', r' on p q r r S x S ~ x S to diffeomorphisms of iV, then the result will follow by q 2.2. But let ψ: D x I—> N' be a proper embedding such that ( i ) ψ-\Sq) - Dq x 1 and f (Dξ. x l ) s Int Ό\ ( ϋ ) ψ-1(Sq~r x Sr) = D g x 0 and i/r(D*+ x θ ) g I n t D Γ r x I>+. p Then πg(SOp) and JΓ ? + 1 act on S x N' and i\Γ' via maps which are the identity outside the image of ψ and only depend on the Dq q coordinate. The map a e Γq+1 extends to S x N' via natural product extensions. On the other hand, there is an explicit embedding of N' in Rq+1 = Rr+1 x Rq~r given by the relation 2 2 2 2 iSΓ' = {(x,y)\ \x\ + \y\ ^l and (2|a?| - l) /4 + \y\ ^ 1/16}. (See the figure below.) This smooth manifold is invariant under the standard linear action of SOr+1 on Rq+ί, and the restricted action on the boundaries is again standard. In particular, the restriction to r q r r S x S ~ is merely the usual action on S crossed with the identity on

744

REINHARD SCHULTZ

FIGURE 1 q

r

S ~ . There are a homotopy class in πp(SOr+1) which induces a diffeomorphism of Sp x N' with the right restrictions to the boundaries, and this gives the desired extension of the map β. Since the pairings σ, τ and σ', τf are commutators in the restrictions of extendible maps, the diffeomorphisms associated to them are extendible, and the argument is completed. The Milnor-Munkres-Novikov pairing may be generalized to products of three spheres in still another manner. Let p, q, r be three positive integers, and let βeπp+q(SOr), cte Γp+q+r+1. Represent β by a class in [Sp x S9, SOr] which is constant off Ό\ x Ώ\ and induces a diffeomorphism on Sp x Sr x Sr. Represent a by a diffeomorphism of S9 x Sr which is the identity off Dl x DL. Then the commutator [a, β] is the identity off Ό\ x D\ x D+, and hence it determines an element τq(β,

a) of

Γp+q+r+ι.

THEOREM

2.4.

The pairing zq\ 7Γp+q(oUr)

x Γq+r+ι

> Γp+q+r+1

COMPOSITION CONSTRUCTIONS ON DIFFEOMORPHISMS OF S? X S*

745

is trivial. n

Proof. Let T be the ^-dimensional torus. Then a pairing τq may be formed using maps which are fixed off products involving (D\Y, (D+Y, and D+, and an argument similar to 2.3 shows that T

A = Γp+q+r+1 [20, p. 227]. Is the associated map into πp(Dq+r) trivial? The above information has been assembled to prove the following factorization formula relating the plumbing pairing σ and the MίlnorMunkres-Novikov pairing τ . THEOREM2.5. Let 7 e πr(SOq)(r ^p-2),ae If β is in the image of πp{SOp-r), then

πp(SOg), andβe

7r,p+(r+i(7f σPtq(a, β)) = σp+r,q(aJ(y),

πg(SOp).

β) .

Most of the interesting applications of the above formula occur when r — 1. The condition on β may be weakened to assuming that the Samelson product of 7 and β in SOp+ι vanishes. Proof. This basically reduces to an exercise in the algebra of the group £%r = &(SP x S9 x Sr). The homotopy groups involved in the statement 2.5 all map into £%r canonically, and we identify a given element with its image in ^ . Also, there is the composite map p: Γp+q+1 -Ϊ-* &(SP x Sq) —

&(S* x Sq x Sr) ,

the last map being cartesian product of a diffeomorphism with the identity on Sr. By definition of the Milnor-Munkres-Novikov pairing, we have the equation: *(α, 0)τ(7, σ(a, β)) = 7σ(α, ^ T ^ 1 . Now β and 7 are both in the image of [Sr x Sq, SOP+1] in &(SP x Sq x Sr). Since β comes from SOp^r and 7 comes from SOr+ί (a pair of commuting subgroups up to homotopy), these elements commute in the above homotopy group. Hence we may continue to alter the last expression in the equation:

However, by 1.2, 7αrΓ"1 = cca0 — S induces an epimorphism from τr4(G3) to πϋ(G3) by results of Toda [25, 5.1, 5.3, 5.8, 5.9,

COMPOSITION CONSTRUCTIONS ON DIFFEOMORPHISMS OF S* x S*

753

pp. 39, 40, 43, 44] and the natural splitting πt(Gz) ~ ^(F2) ζ&πi(SO3). Futhermore, since π4 = 0 the fiber homotopy equivalence determined 4 2 by any element of π4(G3) is homotopic to a diffeomorphism 7i of S x S which is fixed near D4 x S2([21, 2.2] again), and hence g is explicitly constructible from gt by Proposition 1.3. If τ\ is a diffeomorphism of S4 x S3 fixed near DL x S 3 which covers 7i, then Proposition 1.4 ι 4 allows one to construct 7 from 7i in the same fashion. Let h: S x S 1 4 —> S x S be a diffeomorphism induced by the nontrivial class in KiiSOi) = Z2, so that g may be written as the commutator [h x 13, 1

3

Since 7 is fixed off Dt x S , results of Levine [17, 2.3] imply 4 4 that 7i may be written as a product uv, where u extends to S x D and v is fixed near the complement of D\ x D\ (and hence corresponds to an element of Γ8). According to Proposition 2.3, [h x 1,1 x v] is pseudo-isotopic to τ{η, v) e Γ9. Since this element is central, it is easy to compute that [h x 1,1 x 7] = [h x 1,1 x u]τ(η, v) But [h x 1,1 x 7] extends to S 5 x D4 because u extends to S4 x D 4 . Hence Lemma 2.2 implies that S5 x D4UrD* x S* is diffeomorphic to Sδx

D4\Jτ{η,v)D*

x SZ =

S%.

But JSΪ bounds a spin manifold (e.g., see [1]); since we have shown that this sphere is the same as 9

4

9

4

F x D \J-9 S - f(F x Int D )

and the latter is congruent to Σ9 mod 6Spin10, the fact that Σ9 e b Spin10 is immediate. Let y1 be the map defined above with factorization uv. The factor v may be shown to be nontrivial in certain cases if and only if the exotic 8-sphere has a semifree circle action with 4-dimensional fixed point set. Since such an action on the exotic 8-sphere may be explicitly constructed using [8, Th. 3], the factor v can in fact be nontrivial. An explicit construction of the action will be given elsewhere. REMARK.

REFERENCES 1. D. W. Anderson, E. H. Brown and F. P. Peterson, SU-cobordism, KO-ckaracteristic numbers, and the Kervaire invariant, Ann. of Math., 8 3 (1966), 54-67. 2. P. L. Antonelli, On the stable diffeomorphism of homotopy spheres in the stable

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REINHARD SCHULTZ

range, n < 2p, Bull. Amer. Math. Soc, 7 5 (1969), 343-346. 3. M. Barratt, M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence II, Topology, 9 (1970), 309-316. 4. G. Bredon, A π*-module structure for Θ* and applications to transformation groups, Ann. of Math., 8 6 (1967), 434-448. 5. W. Browder, Surgery and the Theory of Differentiate Transformation Groups, Proceedings of the Conference on Transformation Groups, New Orleans, 1967,1-46. SpringerVerlag, New York, 1968. 6. G. Brumfiel, On the homotopy groups of BPL and PL\O, Ann. of Math., 88 (1968), 291-311; ibid, II, Topology, 8 (1969), 305-311. 1 7. , Differentiate S actions on homotopy spheres, mimeographed. 8. D. Frank, An invariant for almost closed manifolds, Bull. Amer. Math. Soc, 7 4 (1968), 562-567. 9. P. J. Hilton and J. H. C. Whitehead, Note on the Whitehead product, Ann. of Math., 5 8 (1953), 429-442. 10. M. Hirsch, Embeddings and compressions of polyhedra and smooth manifolds, Topology, 4 (1966), 361-369. 11. W.-C. Hsiang, J. Levine and R. Szczarba, On the normal bundle of a homotopy sphere in Euclidean space, Topology, 3 (1964), 173-181. 12. I. M. James and J. H. C. Whitehead, The homotopy theory of sphere bundles over spheres I, Proc. London Math. Soc, (3) 4 (1954), 196-218. 13. M. Kervaire, An interpretation of G. Whitehead's generalization of H. Hopfs invariant, Ann. of Math., 6 9 (1959), 345-365. 14. M. Kervaire and J. Milnor, Groups of homotopy spheres, Ann. of Math., 7 8 (1963), 514-537. 15. R. Lashof (ed.), Problems in differential and algebraic topology, Seattle Conference, 1963, Ann. of Math., 8 1 (1965), 565-591. 16. J. Levine, A classification of differentiate knots, Ann. of Math., 8 2 (1965), 15-50. 17. , Self-equivalences of Sn X Sk, Trans. Amer. Math. Soc, 143 (1969), 523-544. 18. , Inertia groups of manifolds and diffeomorphisms of spheres, Amer. J. Math., 92 (1970), 243-258. 19. M. Mahowald and M. Tangora, Some differentials in the Adams spectral sequence, Topology, 6 (1967), 349-369. 20. S. P. Novikov, Differentiate sphere bundles, Trans. Amer. Math. Soc, (2) 6 3 (1967), 217-243. 21. R. Schultz, On the inertia group of a product of spheres, Trans. Amer. Math. Soc, 156 (1971), 137-153. 22. , The nonexistence of free S1 actions on some homotopy spheres, Proc. Amer. Math. Soc, 2 7 (1971), 595-597. 23. , Semifree circle actions and the degree of symmetry of homotopy spheres, Amer. J. Math., 9 3 (1971), 829-839. 24. M. Tangora, verbal communication. 25. H. Toda, Composition methods in homotopy groups of spheres, Ann. of Math. Study No. 49. Princeton University Press, Princeton, 1962. 26. C. T. C. Wall, Classification of (n — l)-connected 2n-manifolds, Ann. of Math., 7 5 (1962), 163-179. 27. , Classification problems in differential topology VI. Classification of (8 -l)-connected (2s + l)-manifolds, Topology, 6 (1967), 273-296. Received December 21, 1970 and revised form June 21, 1971. The author was partially supported by NSF Grant GP-19530. PURDUE UNIVERSITY

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Stanford University Stanford, California 94305

DUGUNDJI

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Pacific Journal of Mathematics Vol. 42, No. 3

March, 1972

Catherine Bandle, Extensions of an inequality by Pólya and Schiffer for vibrating membranes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. J. Bernau, Topologies on structure spaces of lattice groups . . . . . . . . . . . . . . Woodrow Wilson Bledsoe and Charles Edward Wilks, On Borel product measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eggert Briem and Murali Rao, Normpreserving extensions in subspaces of C(X ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alan Seymour Cover, Generalized continuation . . . . . . . . . . . . . . . . . . . . . . . . . . Larry Jean Cummings, Transformations of symmetric tensors . . . . . . . . . . . . . Peter Michael Curran, Cohomology of finitely presented groups . . . . . . . . . . . . James B. Derr and N. P. Mukherjee, Generalized quasicenter and hyperquasicenter of a finite group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Erik Maurice Ellentuck, Universal cosimple isols . . . . . . . . . . . . . . . . . . . . . . . . Benny Dan Evans, Boundary respecting maps of 3-mainfolds . . . . . . . . . . . . . . David F. Fraser, A probabilistic method for the rate of convergence to the Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Raymond Taylor Hoobler, Cohomology in the finite topology and Brauer groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Louis Roberts Hunt, Locally holomorphic sets and the Levi form . . . . . . . . . . B. T. Y. Kwee, On absolute de la Vallée Poussin summability . . . . . . . . . . . . . . Gérard Lallement, On nilpotency and residual finiteness in semigroups . . . . . George Edward Lang, Evaluation subgroups of factor spaces . . . . . . . . . . . . . . Andy R. Magid, A separably closed ring with nonzero torsion pic . . . . . . . . . Billy E. Rhoades, Commutants of some Hausdorff matrices . . . . . . . . . . . . . . . Maxwell Alexander Rosenlicht, Canonical forms for local derivations . . . . . Cedric Felix Schubert, On a conjecture of L. B. Page . . . . . . . . . . . . . . . . . . . . . Reinhard Schultz, Composition constructions on diffeomorphisms of S p × Sq . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. P. Singhal and H. M. (Hari Mohan) Srivastava, A class of bilateral generating functions for certain classical polynomials . . . . . . . . . . . . . . . Richard Alan Slocum, Using brick partitionings to establish conditions which insure that a Peano continuum is a 2-cell, a 2-sphere or an annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . James F. Smith, The p-classes of an H ∗ -algebra . . . . . . . . . . . . . . . . . . . . . . . . . Jack Williamson, Meromorphic functions with negative zeros and positive poles and a theorem of Teichmuller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . William Robin Zame, Algebras of analytic functions in the plane . . . . . . . . . .

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