Cohomology of metacyclic groups
Johannes
Huebschmann
1-72
Abstract: Let ${\mathbf{e}}:1 \to N \to G \to K \to 1$ be an extension of a finite cyclic group $N$ by a finite cyclic group $ K$. Using homological perturbation theory, we introduce the beginning of a free resolution of the integers ${\mathbf{Z}}$ over the group ring ${\mathbf{Z}}G$ of $G$ in such a way that the resolution reflects the structure of $G$ as an extension of $N$ by $K$, and we use this resolution to compute the additive structure of the integral cohomology of $ G$ in many cases. We proceed by first establishing a number of special cases, thereafter constructing suitable cohomology classes thereby obtaining a lower bound, then computing characteristic classes introduced in an earlier paper, and, finally, exploiting these classes, obtaining upper bounds for the cohomology via the integral cohomology spectral sequence of the extension ${\mathbf{e}}$. The calculation is then completed by comparing the two bounds.
$L\sp p$ bounds for spectral multipliers on nilpotent groups
Michael
Christ
73-81
Abstract: A criterion is given for the ${L^p}$ boundedness of a class of spectral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theorem of Hörmander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.
Pair-dense relation algebras
Roger D.
Maddux
83-131
Abstract: The central result of this paper is that every pair-dense relation algebra is completely representable. A relation algebra is said to be pair-dense if every nonzero element below the identity contains a "pair". A pair is the relation algebraic analogue of a relation of the form $ \{ \langle {a,a} \rangle,\langle {b,b} \rangle \}$ (with $a= b$ allowed). In a simple pair-dense relation algebra, every pair is either a "point" (an algebraic analogue of $\{ \langle {a,a} \rangle \}$) or a "twin" (a pair which contains no point). In fact, every simple pair-dense relation algebra $ \mathfrak{A}$ is completely representable over a set $U$ iff $ \vert U\vert= \kappa + 2\lambda$, where $\kappa$ is the number of points of $\mathfrak{A}$ and $\lambda$ is the number of twins of $\mathfrak{A}$. A relation algebra is said to be point-dense if every nonzero element below the identity contains a point. In a point-dense relation algebra every pair is a point, so a simple point-dense relation algebra $ \mathfrak{A}$ is completely representable over $U$ iff $\vert U\vert= \kappa$, where $ \kappa$ is the number of points of $ \mathfrak{A}$. This last result actually holds for semiassociative relation algebras, a class of algebras strictly containing the class of relation algebras. It follows that the relation algebra of all binary relations on a set $ U$ may be characterized as a simple complete point-dense semiassociative relation algebra whose set of points has the same cardinality as $U$. Semiassociative relation algebras may not be associative, so the equation $(x;y);z= x;(y;z)$ may fail, but it does hold if any one of $x,y$, or $z$ is $1$. In fact, any rearrangement of parentheses is possible in a term of the form ${x_0}; \ldots;{x_{\alpha - 1}}$, in case one of the
An infinite family in ${}\sb 2\pi\sp {\rm s}\sb *$ at Adams filtration seven
Wên Hsiung
Lin
133-149
Abstract: We prove the family $ \{ h_i^2{h_3}{d_1}\}$ in $ \operatorname{Ext}_A^{7,\ast}({\mathbb{Z}_2},{\mathbb{Z}_2})$ detects homotopy elements in the $2$-adic stable homotopy of spheres $_2\pi_{\ast}^S$ where $A$ is the $\bmod \;2$ Steenrod algebra.
Waldhausen's classification theorem for finitely uniformizable $3$-orbifolds
Yoshihiro
Takeuchi
151-200
Abstract: We define a map between two orbifolds. With respect to this map, we generalize $3$-manifold theory to $3$-orbifolds. As the main goal, we generalize the Waldhausen's classification theorem of Haken $ 3$-manifolds to finitely uniformizable $3$-orbifolds. For applications of the developed theory, we introduce an invariant for links and tangles by using the orbifold fundamental group. With the invariant, we classify a class of links and show the untangling theorem.
The associated Askey-Wilson polynomials
Mourad E. H.
Ismail;
Mizan
Rahman
201-237
Abstract: We derive some contiguous relations for very well-poised $_8{\phi _7}$ series and use them to construct two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials. We then use these solutions to find explicit representations of two families of associated Askey-Wilson polynomials. We identify the corresponding continued fractions as quotients of two very well-poised $_8{\phi _7}$ series and find the weight functions.
Twisted calibrations
Timothy A.
Murdoch
239-257
Abstract: The methods of calibrated geometry are extended to include nonorientable submanifolds which can be oriented by some real Euclidean line bundle. Specifically, if there exists a line bundle-valued differential form of comass one which restricts to a submanifold to be a density, then the submanifold satisfies a minimizing property. The results are applied to show that the cone on the Veronese surface minimizes among a general class of comparison $3$-folds.
Grothendieck's existence theorem in analytic geometry and related results
Siegmund
Kosarew
259-306
Abstract: We state and prove several kinds of analytification theorems of formal objects (such as coherent sheaves and formal complex spaces) which are in the spirit of Grothendieck's algebraization theorem in [EGA, III]. The formulation of the results was derived from deformation theory and especially M. Artin's work on representability of functors. The methods of proof depend heavily on a deeper study of cotangent complexes and resolvants. As applications one can deduce the convergence of formal versal deformations in diverse situations.
On topological classification of function spaces $C\sb p(X)$ of low Borel complexity
T.
Dobrowolski;
W.
Marciszewski;
J.
Mogilski
307-324
Abstract: We prove that if $ X$ is a countable nondiscrete completely regular space such that the function space $ {C_p}(X)$ is an absolute $ {F_{\sigma \,\delta }}$-set, then ${C_p}(X)$ is homeomorphic to ${\sigma ^\infty }$, where $\sigma = \{ ({x_i}) \in {{\mathbf{R}}^\infty }:{x_i}= 0$ for all but finitely many $i\}$. As an application we answer in the negative some problems of A. V. Arhangel'skii by giving examples of countable completely regular spaces $ X$ and $Y$ such that $X$ fails to be a ${b_R}$-space and a $k$-space (and hence $X$ is not a $ {k_\omega }$-space and not a sequential space) and $Y$ fails to be an $ {\aleph _0}$-space while the function spaces ${C_p}(X)$ and ${C_p}(Y)$ are homeomorphic to ${C_p}(\mathfrak{X})$ for the compact metric space $\mathfrak{X}= \{ 0\} \cup \{ {n^{ - 1}}:n= 1,2, \ldots \}$.
A representation theorem for cyclic analytic two-isometries
Stefan
Richter
325-349
Abstract: A bounded linear operator $T$ on a complex separable Hilbert space $\mathcal{H}$ is called a $2$-isometry if ${T^{\ast 2}}{T^2} - 2{T^{ \ast }}T + I = 0$. We say that $T$ is analytic if ${ \cap _{n> 0}}\,{T^n}\,\mathcal{H}= (0)$. In this paper we show that every cyclic analytic $ 2$-isometry can be represented as multiplication by $z$ on a Dirichlet-type space $D(\mu)$. Here $\mu$ denotes a finite positive Borel measure on the unit circle. For two measures $\mu$ and $\nu$ the $2$-isometries obtained as multiplication by $ z$ on $D(\mu)$ and $D(\nu)$ are unitarily equivalent if and only if $ \mu = \nu$. We also investigate similarity and quasisimilarity of these $ 2$-isometries, and we apply our results to the invariant subspaces of the Dirichlet shift.
Homogeneous distributions on the Heisenberg group and representations of ${\rm SU}(2,1)$
R. C.
Fabec
351-391
Abstract: A 'Fourier' transform of tempered distributions on the Heisenberg group is defined to analyze homogeneous distributions relative the group of dilations $(z,t) \mapsto (r\,z,{r^2}t)$, $r \in {\mathbf{R}}$. An inversion formula is derived for the abelian central Fourier transform of the distribution. These formulas are applied to the family of homogeneous distributions defining the intertwining operators for the group $ {\text{SU}}(2,1)$. Explicit unitary structures are determined on subquotient representations and their spectral decompositions on the minimal parabolic subgroup are obtained.
Sharp square-function inequalities for conditionally symmetric martingales
Gang
Wang
393-419
Abstract: Let $f$ be a conditionally symmetric martingale taking values in a Hilbert space $\mathbb{H}$ and let $S(f)$ be its square function. If ${\nu _p}$ is the smallest positive zero of the confluent hypergeometric function and $ {\mu _p}$ is the largest positive zero of the parabolic cylinder function of parameter $p$, then the following inequalities are sharp: $\displaystyle \Vert f \Vert _{p} \leq \nu_{p}\Vert S(f)\Vert _{p}$ if$\displaystyle \;0 < p \leq 2,$ $\displaystyle \Vert f \Vert _{p} \leq \mu_{p} \Vert S(f)\Vert _{p}$ if$\displaystyle \;p \geq 3,$ $\displaystyle \nu_{p}\Vert S(f)\Vert _{p}\; \leq\; \Vert f\Vert _p$ if$\displaystyle \; p \geq 2.$ Moreover, the constants $\nu_p$ and $\mu_p$ for the cases mentioned above are also best possible for the Marcinkiewicz-Paley inequalities for Haar functions.
Szeg\H o's theorem on a bidisc
Takahiko
Nakazi
421-432
Abstract: G. Szegö showed that $\displaystyle \inf \;\int_0^{2\pi } {\vert 1 - f{\vert^2}w\,d\theta /2\pi = \exp \;\int_0^{2\pi } {\log \,w\,d\theta /2\pi } }$ where $ f$ ranges over analytic polynomials with mean value zeros. We study extensions of the Szegö's theorem on the disc to the bidisc. We show that the quantity is a mixed form of an arithmetic mean and a geometric one of $w$ in some special cases.
$2$-to-$1$ maps on hereditarily indecomposable continua
Jo
Heath
433-444
Abstract: Suppose $ f$ is a $2{\text{-to-}}1$ continuous map from the hereditarily indecomposable continuum $X$ onto a continuum $Y$. In order for it to be the case that each proper subcontinuum $C$ in $Y$ has as its preimage two disjoint continua each of which $f$ maps homeomorphically onto $C$, it is obviously necessary that $ f$ satisfy the condition that each nondense connected subset of $Y$ has disconnected preimage. We show that this condition is also sufficient, and thus any $ 2{\text{-to-}}1$ continuous map from a hereditarily indecomposable continuum satisfying this condition must be confluent and have an image that is hereditarily indecomposable.
Polarized surfaces of $\Delta$-genus $3$
Maria Lucia
Fania;
Elvira Laura
Livorni
445-463
Abstract: Let $X$ be a smooth, complex, algebraic, projective surface and let $L$ be an ample line bundle on it. Let $\Delta = \Delta \,(X,L)= {c_1}{(L)^2} + 2 - {h^0}(L)$ denote the $\Delta$-genus of the pair $(X,L)$. The purpose of this paper is to classify such pairs under the assumption that $\Delta = 3$ and the complete linear system $ \vert L \vert$ contains a smooth curve. If $d \geq 7$ and $ g \geq \Delta$, Fujita has shown that $L$ is very ample and $g= \Delta$. If $d \geq 7$ and $ g < \Delta = 3$, then $ g= 2$ and those pairs have been studied by Fujita and Beltrametti, Lanteri, and Palleschi. To study the remaining cases we have examined the two possibilities of $L + tK$ being nef or not, for $t= 1,2$. In the cases in which $ L + 2K$ is nef it turned out to be very useful to iterate the adjunction mapping for ample line bundles as it was done by Biancofiore and Livorni in the very ample case. If $g > \Delta$ there are still open cases to solve in which completely different methods are needed.