Equivariant concordance of invariant knots
Neal W.
Stoltzfus
1-45
Abstract: The classification of equivariant concordance classes of high-dimensional codimension two knots invariant under a cyclic action, T, of order m has previously been reported on by Cappell and Shaneson [CS2]. They give an algebraic solution in terms of their algebraic k-theoretic $\Gamma$-groups. This work gives an alternative description by generalizing the well-known Seifert linking forms of knot theory to the equivariant case. This allows explicit algorithmic computations by means of the procedures and invariants of algebraic number theory (see the subsequent work [St], particularly Theorem 6.13). Following Levine [L3], we define bilinear forms on the middle-dimensional homology of an equivariant Seifert surface $ {B_i}(x,y) = L(x,{i_ + }(T_{\ast}^iy))$, for $i = 1, \cdots ,m$. Our first result (2.5) is that an invariant knot is equivariantly concordant to an invariant trivial knot if and only if there is a subspace of half the rank on which the ${B_i}$ vanish simultaneously. We then introduce the concepts of equivariant isometric structure and algebraic concordance which mirror the preceding geometric ideas. The resulting equivalence classes form a group under direct sum which has infinitely many elements of each of the possible orders (two, four and infinite), at least for odd periods. The central computation (3.4) gives an isomorphism of the equivariant concordance group with the subgroup of the algebraic knot concordance group whose Alexander polynomial, $ \Delta$, satisfies the classical relation $\left\vert {\prod\nolimits_{i = \,1}^m {\Delta \left( {{\lambda ^i}} \right)} } \right\vert\, = \,1\,$, where $\lambda$ is a primitive mth root of unity. This condition assures that the m-fold cover of the knot complement is also a homology circle, permitting the geometric realization of each equivariant isometric structure. Finally, we make an explicit computation of the Browder-Livesay desuspension invariant for knots invariant under an involution and also elucidate the connection of our methods with the results of [CS2] by explicitly describing a homomorphism from the group of equivariant isometric structures to the appropriate $\Gamma$-group.
Partially conservative extensions of arithmetic
D.
Guaspari
47-68
Abstract: Let T be a consistent r.e. extension of Peano arithmetic; $\Sigma _n^0$, $\Pi _n^0$ the usual quantifier-block classification of formulas of the language of arithmetic (bounded quantifiers counting ``for free"); and $ \Gamma$, $\Gamma '$ variables through the set of all classes $ \Sigma _n^0$, $ \Pi _n^0$. The principal concern of this paper is the question: When can we find an independent sentence $\phi \, \in \,\Gamma$ which is $\Gamma '$-conservative in the following sense: Any sentence $\chi$ in $\Gamma '$ which is provable from $T + \phi$ is already provable from T? (Additional embellishments: Ensure that $ \phi$ is not provably equivalent to a sentence in any class ``simpler'' than $ \Gamma$; that $ \phi$ is not conservative for classes ``more complicated'' than $ \Gamma '$.) The answer, roughly, is that one can find such a $\phi$, embellishments and all, unless $\Gamma$ and $\Gamma '$ are so related that such a $\phi$ obviously cannot exist. This theorem has applications to the theory of interpretations, since ``$\phi$ is $\Gamma$-conservative'' is closely related to the property ``$T + \phi$ is interpretable in T"-or to variants of it, depending on $\Gamma$. Finally, we provide simple model theoretic characterizations of $ \Gamma$-conservativeness. Most results extend straightforwardly if extra symbols are added to the language of arithmetic, and most have analogs in the Levy hierarchy of set theoretic formulas (T then being an extension of ZF).
Antiholomorphic involutions of analytic families of abelian varieties
Allan
Adler
69-94
Abstract: In this paper, we investigate antiholomorphic involutions of Kuga-Satake analytic families of polarized abelian varieties V. A complete set of invariants of the Aut(V)-conjugacy classes of antiholomorphic involutions of V is obtained. These invariants are expressed as cohomological invariants of the arithmetic data defining V. In the last section, the fibre varieties of Kuga-Satake type belonging to totally indefinite quaternion division algebras over totally real fields are investigated in more detail, and the cohomological invariants are related to results of Steve Kudla. The group of holomorphic sections of V is computed for this case. It is also shown that in general the fibre structure of V is intrinsic.
Some constructions of infinite M\"obius planes
Nicholas
Krier
95-115
Abstract: New infinite Möbius planes are constructed using transfinite induction. Any infinite affine plane A can be embedded in a Möbius plane M and the construction allows some groups of perspectivities of A to be extended to automorphism groups of M. Given $\left\{ {{A_\alpha },\,\alpha \, \in \,J} \right\}$, an infinite collection of s affine planes each of order s, there exist a Möbius plane M and a bijection b from J to the point set of M so that for each $\alpha \, \in \,J,\,{M_{b\left( \alpha \right)}}$ is isomorphic to $ {A_\alpha }$.
Normal two-dimensional elliptic singularities
Stephen Shing Toung
Yau
117-134
Abstract: Given a weighted dual graph such that the canonical cycle $ K'$ exists, is there a singularity corresponding to the given weighted dual graph and which has Gorenstein structure? This is one of the important problems in normal surface singularities. In this paper, we give a necessary and sufficient condition for the existence of Gorenstein structures for weakly elliptic singularities. A necessary and sufficient condition for the existence of maximally elliptic structure is also given. Hence, the above question is answered affirmatively for a special kind of singularities. We also develop a theory for those elliptic Gorenstein singularities with geometric genus equal to three.
Free states of the gauge invariant canonical anticommutation relations. II
B. M.
Baker
135-155
Abstract: A class of representations of the gauge invariant subalgebra of the canonical anticommutation relations (henceforth GICAR) is studied. These representations are induced by restricting the well-known pure, nongauge invariant generalized free states of the canonical anticommutation relations (henceforth CAR). Denoting a state of the CAR by $\omega$, and the unique generalized free state of the CAR such that $\omega \left( {a{{\left( f \right)}^{\ast}}a\left( g \right)} \right)\, = \,\left( {f,Tg} \right)$ and $\omega \left( {a\left( f \right)a\left( g \right)} \right)\, = \,\left( {Sf,g} \right)$ by ${\omega _{S,T}}$, it is shown that a pure, nongauge invariant state ${\omega _{S,T}}$ induces a factor representation of the GICAR if and only if $Tr\,T\left( {I - T} \right)\, = \,\infty$.
Generic sets and minimal $\alpha $-degrees
C. T.
Chong
157-169
Abstract: A non-$ \alpha$-recursive subset G of an admissible ordinal $\alpha$ is of minimal $\alpha $-degree if every set of strictly lower $\alpha$-degree than that of G is $ \alpha$-recursive. We give a characterization of regular sets of minimal $ \alpha$-degree below $ 0'$ via the notion of genericity. We then apply this to outline some 'minimum requirements' to be satisfied by any construction of a set of minimal $ \aleph _\omega ^L$-degree below $0'$.
Analytic extensions and selections
J.
Globevnik
171-177
Abstract: Let G be a closed subset of the closed unit disc in C, let F be a closed subset of the unit circle of measure 0 and let $\Phi$ map G into the class of all open subsets of a complex Banach space X. Under suitable additional assumptions on $\Phi$ we prove that given any continuous function $f:\,F \to X$ satisfying $f(z)\, \in \,{\text{closure(}}\Phi (z))\,(z\, \in \,F\, \cap \,G)$ there exists a continuous function f from the closed unit disc into X, analytic in the open unit disc, which extends f and satisfies $\tilde f(z)\, \in \,\Phi (z)\,(z\, \in \,G\, - \,F)$. This enables us to generalize and sharpen known dominated extension theorems for the disc algebra.
Rational inner functions on bounded symmetric domains
Adam
Korányi;
Stephen
Vági
179-193
Abstract: It is shown that the rational inner functions on any bounded symmetric domain are given by a generalized version of a formula found by Rudin and Stout in the case of the polydisc. In particular, it is shown that all rational inner functions are constant on symmetric domains which have no irreducible factor of tube type.
Local $H$-maps of classifying spaces
Timothy
Lance
195-215
Abstract: Let BU denote the localization at an odd prime p of the classifying space for stable complex bundles, and let $f:BU \to BU$ be an H-map with fiber F. In this paper the Hopf algebra $ {H^{\ast}}(F,\textbf{Z}/P)$ is computed for any such f. For certain H-maps f of geometric interest the p-local cohomology of F is given by means of the Bockstein spectral sequence. A direct description of ${H^ {\ast} }(F,{{\textbf{Z}}_{(P)}})$ is also given for an important special case. Applications to the classifying spaces of surgery will appear later.
Cell-like $0$-dimensional decompositions of $S\sp{3}$ are $4$-manifold factors
R. J.
Daverman;
W. H.
Row
217-236
Abstract: The main result is the title theorem asserting that if G is any upper semicontinuous decomposition of $ {S^3}$ into cell-like sets which is 0-dimensional, in the sense that the image of the nondegenerate elements in ${S^3}/G$ is 0-dimensional, then $G\, \times \,{S^1}$ is shrinkable, and $ \left( {{S^3}/G} \right)\, \times \,{S^1}$ is homeomorphic to ${S^3}\, \times \,{S^1}$.
Constructing framed $4$-manifolds with given almost framed boundaries
Steve J.
Kaplan
237-263
Abstract: Two methods are presented for constructing framed 4-manifolds with given almost framed boundaries. The main tools are the ``moves'' of Kirby's calculus of framed links. A new description is given for the $\mu$-in-variant of a knot and this description is used to study almost framed 3-manifolds.
Morse and generic contact between foliations
Russell B.
Walker
265-281
Abstract: Motivated by the recent work of J. Franks and C. Robinson, the study of the contact between two foliations of equal codimension is begun. Two foliations generically contact each other in certain dimensional sub-manifold complexes. All but a finite number of these contact points are ``Morse". In a recent paper by the author, a complete large isotopy ``index of contact'' is specified for two foliations of ${T^2}$. If contact is restricted to index 0 ("domed contact"), some sharp conclusions are made as to the topology of the manifold and isotopy classes of the two foliations. It is hoped that this work will lead to the construction of new quasi-Anosov diffeomorphisms and possibly to a new Anosov diffeomorphism.
Selection theorems for $G\sb{\delta }$-valued multifunctions
S. M.
Srivastava
283-293
Abstract: In this paper we establish under suitable conditions the existence of measurable selectors for $ {G_\delta }$-valued multifunctions. In particular we prove that a measurable partition of a Polish space into $ {G_\delta }$ sets admits a Borel selector.
Analogs of Clifford's theorem for polycyclic-by-finite groups
Martin
Lorenz
295-317
Abstract: Let P be a primitive ideal in the group algebra $K[G]$ of the polycyclic group G and let N be a normal subgroup of G. We show that among the irreducible right $K[G]$-modules with annihilator P there exists at least one, V, such that the restricted $K[N]$-module ${V_N}$ is completely reducible, a sum of G-conjugate simple $K[N]$-submodules. Various stronger versions of this result are obtained. We also consider the action of G on the factor $ K[N]/P \cap K[N]$ and show that, in case K is uncountable, any ideal I of $K[N]$ satisfying ${ \cap _{g \in G}}{I^g}\, = \,P\, \cap \,K[N]$ is contained in a primitive ideal Q of $K[N]$ with ${ \cap _{g \in G}}{I^g}\, = \,P\, \cap \,K[N]$.
Critical mappings of Riemannian manifolds
David D.
Bleecker
319-338
Abstract: We consider maps, from one Riemannian manifold to another, which are critical for all invariantly defined functionals on the space of maps. There are many such critical mappings, perhaps too numerous to suitably classify, although a characterization of sorts is provided. They are proven to have constant rank, with the image being a homogeneous minimal submanifold of the target manifold. Critical maps need not be Riemannian submersions onto their images. Also, there are homogeneous spaces for which the identity map is not critical. Many open problems remain.
Isometries of $L\sb{p}$-spaces associated with semifinite von Neumann algebras
P. K.
Tam
339-354
Abstract: The paper determines the structure of (classes of) linear isometries between ${L_p}$-spaces associated with semifinite normal faithful traces on von Neumann algebras, generalizing works of M. Broise and B. Russo. Also established are some auxiliary results on ${L_p}$ norm inequalities which are of interest by themselves.
The Dirichlet norm and the norm of Szeg\H o type
Saburou
Saitoh
355-364
Abstract: Let S be a smoothly bounded region in the complex plane. Let $ g(z,t)$ denote the Green's function of S with pole at t. We show that $\displaystyle \iint_S {\vert f'(z){\vert^2}\,dx\,dy\, \leqslant \,\frac{1}{2}\i... ... {\frac{{\partial g(z,t)}} {{\partial {n_z}}}} \right)}^{ - 1}}\vert dz\vert} }$ holds for any analytic function $ f(z)$ on $S\, \cup \,\partial S$. This curious inequality is obtained as a special case of a much more general result.
A new characterization of amenable groups
Jon
Sherman
365-389
Abstract: Paradoxical sets, which are a natural generalization of the type of sets made famous as Hausdorff-Banach-Tarski paradoxes, are defined in terms of piecewise translations. Piecewise translations are the generalization to arbitrary discrete groups of the maps used in the Banach-Tarski paradoxes as congruences by finite decomposition. A subset of a group is defined to be large if finitely many translates of it can cover the group. The main result of this paper is that a group is amenable if and only if it does not contain a large paradoxical set.
Simple periodic orbits of mappings of the initial
Louis
Block
391-398
Abstract: Let f be a continuous map of a closed, bounded interval into itself. A criterion is given to determine whether or not f has a periodic point whose period is not a power of 2, which just depends on the periodic orbits of f whose period is a power of 2. Also, a lower bound for the topological entropy of f is obtained.