A decidable fragment of the elementary theory of the lattice of recursively enumerable sets
M.
Lerman;
R. I.
Soare
1-37
Abstract: A natural class of sentences about the lattice of recursively enumerable sets modulo finite sets is shown to be decidable. This class properly contains the class of sentences previously shown to be decidable by Lachlan. New structure results about the lattice of recursively enumerable sets are proved which play an important role in the decision procedure.
Milnor's $\bar \mu $-invariants and Massey products
Richard
Porter
39-71
Abstract: The main result of this paper gives an interpretation of Milnor's $\overline \mu $-invariants of a link in terms of Massey products in the complement of the link. The approach presented here can be used to give topological proofs of results about the $\overline \mu $-invariants obtained by Milnor using different methods.
Crossed extensions
John G.
Ratcliffe
73-89
Abstract: We develop a natural five term exact sequence relating the second and third cohomology of groups. We show that this sequence is the proper framework for the problem of realizing an abstract kernel. As an application, we give an interpretation of the third cohomology of a group in terms of crossed sequences.
Twisted free tensor products
Elyahu
Katz
91-103
Abstract: A twisted free tensor product of a differential algebra and a free differential algebra is introduced. This complex is proved to be chain homotopy equivalent to the complex associated with a twisted free product of a simplicial group and a free simplicial group. In this way we turn a geometric situation into an algebraic one, i.e. for the cofibration $Y \to Y\,{ \cup _g}\,CX \to \Sigma X$ we obtain a spectral sequence converging into $ H(\Omega (Y\,{ \cup _g}\,CX))$. The spectral sequence obtained in the above situation is similar to the one obtained by L. Smith for a cofibration. However, the one we obtain has more information in the sense that differentials can be traced, requires more lax connectivity conditions and does not need the ring of coefficients to be a field.
Rotundity in Lebesgue-Bochner function spaces
Mark A.
Smith;
Barry
Turett
105-118
Abstract: This paper concerns the isometric theory of the Lebesgue-Bochner function space ${L^p}(\mu ,\,X)$ where $1 < p < \infty$. Specifically, the question of whether a geometrical property lifts from X to ${L^p}\,(\mu ,\,X)$ is examined. Positive results are obtained for the properties local uniform rotundity, weak uniform rotundity, uniform rotundity in each direction, midpoint local uniform rotundity, and B-convexity. However, it is shown that the Radon-Riesz property does not lift from X to ${L^p}\,(\mu ,\,X)$. Consequently, Lebesgue-Bochner function spaces with the Radon-Riesz property are examined more closely.
$K$-theory of hyperplanes
Barry H.
Dayton;
Charles A.
Weibel
119-141
Abstract: Let R be the coordinate ring of a union of N hyperplanes in general position in $ {\textbf{A}}_k^{n + 1}$. Then $\displaystyle {K_i}(R)\, = \,{K_i}(k)\, \oplus \,\left( {\begin{array}{*{20}{c}} {N\, - \,1} {n\, + \,1} \end{array} } \right)\,{K_{n + i}}(k).$ This formula holds for ${K_0},\,{K_1},\,{K_i}\,(i < 0)$, and for the Karoubi-Villamayor groups $K{V_i}\,(i \in \,{\mathbf{Z}})$. For $ {K_2}$ there is an extra summand $\bar R/R$, where $\bar R$ is the normalization of R. For ${K_3}$ the above is a quotient of ${K_3}(R)$. In §4 we show that $ {K_1}$-regularity implies $ {K_0}$-regularity, answering a question of Bass. We also show that $ {K_i}$-regularity is equivalent to Laurent ${K_i}$-regularity for $i \leqslant 1$. The results of this section are independent of the rest of the paper.
$\sigma $-id\'eaux engendr\'es par des ensembles ferm\'es et th\'eor\`emes d'approximation
Alain
Louveau
143-169
Abstract: This paper is motivated by the study of *-games on $\omega$, and by a question of D. A. Martin on the strength of the hypothesis ${\text{AD}}_\omega ^{\ast}$ that every *-game on $ {\omega ^\omega }$ is determined. A general study of the $\sigma $-ideals of subsets of ${\omega ^\omega }$ generated by closed sets encompasses the *-games and the ``perfect set property". Using associated games, we extend for these ideals many properties known for countable sets, under various hypotheses of determinacy. Our methods thus apply also to other examples of regularity properties, such as those introduced by A. S. Kechris. Finally, a general theorem of approximation by analytic sets in Solovay's model is proved which, together with the preceding results, gives the solution of Martin's problem: ${\text{AD}}_\omega ^{\ast}$ is true in Solovay's model.
Affine extensions of a Bernoulli shift
J.
Feldman;
D. J.
Rudolph;
C. C.
Moore
171-191
Abstract: (a) For any automorphism $\phi$ of a compact metric group G, and any $a > 0$, we show the existence of a free finite measure-preserving (m.p.) action of the twisted product $Z{ \times ^\phi }\,G$ whose restriction to Z is Bernoulli with entropy $a\, + \,h(\phi )$, $h(\phi )$ being the entropy of $\phi$ on G with Haar measure. (b) A classification is given of all free finite m.p. actions of $ Z\, \times {\,^\phi }\,G$ such that the action of Z on the $ \sigma$-algebra of invariant sets of G is a Bernoulli action. (c) The classification of (b) is extended to ``quasifree'' actions: those for which the isotropy subgroups are in a single conjugacy class within G. An existence result like that of (a) holds in this case, provided certain necessary and sufficient algebraic conditions are satisfied; similarly, an isomorphism theorem for such actions holds, under certain necessary and sufficient conditions. (d) If G is a Lie group, then all actions of $Z\, \times {\,^\phi }\,G$ are quasifree; if G is also connected, then the second set of additional algebraic conditions alluded to in (c) is always satisfied, while the first will be satisfied only in an obvious case. (e) Examples are given where the isomorphism theorem fails: by violation of the algebraic conditions in the quasifree case, for other reasons in the non-quasifree case.
Homeomorphisms of $S\sp{3}$ leaving a Heegaard surface invariant
Jerome
Powell
193-216
Abstract: We find a finite set of generators for the group ${\mathcal{H} _g}$ of isotopy classes of orientation-preserving homeomorphisms of the 3-sphere $ {S^3}$ which leave a Heegaard surface T of genus g in $ {S^3}$ invariant. We also show that every element of the group ${\mathcal{H} _g}$ can be represented by a deformation of the surface T in ${S^3}$ of a very special type: during the deformation the surface T is the boundary of the regular neighborhood of a graph embedded in a fixed 2-sphere. The only exception occurs when a subset of the graph contained in a disc on the 2-sphere is ``flipped over."
Equivariant $G$-structure on versal deformations
Dock S.
Rim
217-226
Abstract: Let ${X_0}$ be an algebraic variety, and $(\chi ,\,\Sigma )$ its versal deformation. Now let G be an affine algebraic group acting algebraically on ${X_0}$. It gives rise to a definite linear G-action on the tangent space of $\Sigma$. In this paper we establish that if G is linearly reductive then there is an equivariant G-action on $ (\chi ,\Sigma )$ which induces given G-action on the special fibre $ {X_0}$ and its linear G-action on the tangent space of the formal moduli $ \Sigma$. Furthermore, such equivariant G-structure is shown to be unique up to noncanonical isomorphism.
Diophantine sets over algebraic integer rings. II
J.
Denef
227-236
Abstract: We prove that Z is diophantine over the ring of algebraic integers in any totally real number field or quadratic extension of a totally real number field.
Constructing Smale diffeomorphisms on compact surfaces
Steve
Batterson
237-245
Abstract: A necessary condition for an isotopy class on a compact surface to admit a Smale diffeomorphism whose dynamics are a specified set of subshifts of finite type is that the Euler characteristic of the manifold be equal to a sum and difference of certain numbers obtained from the matrices representing the subshifts. In this paper it is shown that this condition is sufficient up to a finite power of the subshifts.
Destructible and indestructible Blaschke products
H. Stephen
Morse
247-253
Abstract: A special case of destructibility for Blaschke products is introduced and studied. An example is given of a destructible Blaschke product which becomes indestructible when a single point is deleted from its zero-set.
A theorem on free envelopes
Chester C.
John
255-259
Abstract: The free envelope of a finite commutative semigroup was defined by Grillet [Trans. Amer. Math. Soc. 149 (1970), 665-682] to be a finitely generated free commutative semigroup $F(S)$ with identity and a homomorphism $\alpha :\,S\, \to \,F(S)$ endowed with certain properties. Grillet raised the following question: does $\alpha (S)$ always generate a pure subgroup of the free Abelian group with the same basis as $ F(S)$? We prove this is indeed the case. It follows as a result of two lemmas. Lemma 1: Given a full rank proper subgroup H of a finitely generated free Abelian group F and a basis X of F there exists a surjective homomorphism $f:\,F\, \to \,{\textbf{Z}}$ such that f is positive on X and ${f_{\left\vert H \right.}}$ is not surjective. Lemma 2: A finitely generated totally cancellative reduced subsemigroup of a finitely generated free Abelian group F is contained in the positive cone of some basis of F. The following duality theorem is also proved. Let ${S^{\ast}}\, \cong \,\operatorname{Hom} (S,\,N)$ where N is the nonnegative integers under addition. Then $S\, \cong \,{S^{{\ast}{\ast}}}$ if and only if S is isomorphic to a unitary subsemigroup of a finitely generated free commutative semigroup with identity.
On positive contractions in $L\sp{p}$-spaces
H. H.
Schaefer
261-268
Abstract: Let T denote a positive contraction $ (T\, \geqslant \,0,\,\left\Vert T \right\Vert\, \leqslant \,1)$ on a space ${L^p}(\mu )\,(1\, < \,p\, < \, + \,\infty )$. A primitive nth root of unity $\varepsilon$ is in the point spectrum $P\sigma (T)$ iff it is in $\varepsilon$ is in both $ P\sigma (T)$ and ${L^q}({p^{ - \,1}}\, + \,{q^{ - \,1}}\, = \,1)$ which are in canonical duality and on which T (resp., $T'$) acts as an isometry. If, in addition, T is quasi-compact then the spectral projection associated with the unimodular spectrum of T (resp., $T'$) is a positive contraction onto a Riesz subspace of ${L^p}$ (resp., ${L^q}$) on which T (resp., $ T'$) acts as an isometry.