Tilted algebras
Dieter
Happel;
Claus Michael
Ringel
399-443
Abstract: Let $A$ be a finite dimensional hereditary algebra over a field, with $n$ simple $A$-modules. An $A$-module $T_A$ with $n$ pairwise nonisomorphic indecomposable direct summands and satisfying $ {\text{Ex}}{{\text{t}}^1}({T_A},\,{T_A}) = 0$ is called a tilting module, and its endomorphism ring $B$ is a tilted algebra. A tilting module defines a (usually nonhereditary) torsion theory, and the indecomposable $B$-modules are in one-to-one correspondence to the indecomposable $A$-modules which are either torsion or torsionfree. One of the main results of the paper asserts that an algebra of finite representation type with an indecomposable sincere representation is a tilted algebra provided its Auslander-Reiten quiver has no oriented cycles. In fact, tilting modules are introduced and studied for any finite dimensional algebra, generalizing recent results of Brenner and Butler.
Recognition of linear actions on spheres
Sören
Illman
445-478
Abstract: Let $G$ be a finite group acting smoothly on a homotopy sphere $\Sigma^m$. We wish to establish necessary and sufficient conditions for the given $G$-action on $\Sigma$ to be topologically equivalent to a linear action. That is, we want to be able to decide whether or not there exists a $G$-homeomorphism $\gamma:\Sigma\to S^m(\rho)$, where ${S^m}(\rho ) \subset {\mathbf{R}^{m + 1}}(\rho )$ denotes the unit sphere in an orthogonal representation space $\mathbf{R}^{m + 1}(\rho )$ for $G$. In order for a $G$-action on $\Sigma$ to be topologically equivalent to a linear action it is clearly necessary that: (i) For each subgroup $H$ of $G$ the fixed-point set $\Sigma^H$ is homeomorphic to a sphere, or empty. (ii) For any subgroups $H$ and $H \subsetneq {H_i},\,1 \leq i \leq k$, of $ G$ the pair $ (\Sigma^{H},\,\cup_{i=1}^{k}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $ (S^{n},\,\cup_{i=1}^{k}S_{i}^{n_{i}})$, where each $S_i^{{n_i}},\,1 \le i \le k$, is a standard $ n_i$-subsphere of $ S^n$. In this paper we consider the case where the fixed-point set $ \Sigma^G$ is nonempty and all other fixed-point sets have dimension at least 5. In giving efficient sufficient conditions we do not need the full strength of condition (ii). We only need: (ii)$^{\ast}$ For any subgroups $H$ and $H \subsetneq {H_i},\,1 \leq i \leq p$, of $ G$ such that $ {\operatorname{dim}}\,{\Sigma ^{{H_i}}} = {\operatorname{dim}}\,{\Sigma ^H} - 2$, the pair $ \Sigma^{H},\,\cup_{i=1}^{p}\Sigma^{H_{i}})$ is homeomorphic to a standard pair $({S^n},\, \cup _{i = 1}^pS_i^{n - 2})$, where each $ S_i^{n - 2},\,1 \le i \le p$, is a standard $(n-2)$-subsphere of $S^n$. Our main results are then that, in the case when $G$ is abelian, conditions (i) and (ii)$ ^{\ast}$ are necessary and sufficient for a given $G$-action on $\Sigma$ to be topologically equivalent to a linear action, and in the case of an action of an arbitrary finite group the same holds under the additional assumption that any simultaneous codimension 1 and 2 fixed-point situation is simple. Our results generalize, for actions of finite groups, a well-known theorem of Connell, Montgomery and Yang, and are the first to also cover the case where codimension 2 fixed-point situations occur.
Multiplicative properties of power maps. II
C. A.
McGibbon
479-508
Abstract: The notions of $ A_n$-maps and $ C_n$-forms can be regarded as crude approximations to the concepts of homomorphisms and commutativity, respectively. These approximations are used to study power maps on connected Lie groups and their localizations. For such groups the power map $ x \mapsto {x^n}$ is known to be an $A_2$-map if and only if $n$ is a solution to a certain quadratic congruence. In this paper, $A_3$-power maps are studied. For the Lie group Sp(l) it is shown that the $A_3$-powers coincide with solutions which are common to the quadratic congruence, mentioned earlier, and another cubic congruence. Other Lie groups, when localized so as to become homotopy commutative, are also shown to have infinitely many $ A_3$-powers. The proofs reflect the combinatorial nature of the obstructions involved.
Equivariant minimal models
Georgia V.
Triantafillou
509-532
Abstract: Let $G$ be a finite group. We give an algebraicization of rational $G$-homotopy theory analogous to Sullivan's theory of minimal models in ordinary homotopy theory.
On critical point theory for indefinite functionals in the presence of symmetries
Vieri
Benci
533-572
Abstract: We consider functionals which are not bounded from above or from below even modulo compact perturbations, and which exhibit certain symmetries with respect to the action of a compact Lie group. We develop a method which permits us to prove the existence of multiple critical points for such functionals. The proofs are carried out directly in an infinite dimensional Hilbert space, and they are based on minimax arguments. The applications given here are to Hamiltonian systems of ordinary differential equations where the existence of multiple time-periodic solutions is established for several classes of Hamiltonians. Symmetry properties of these Hamiltonians such as time translation invariancy or evenness are exploited.
Eisenstein series of weight $3/2$. I
Ting Yi
Pei
573-606
Abstract: We prove that in the space of elliptic modular forms with weight 3/2, the orthogonal complement of the subspace of cusp forms with respect to the Petersson inner product is generated by Eisenstein series in some special cases.
Independent families in complete Boolean algebras
B.
Balcar;
F.
Franěk
607-618
Abstract: We present a proof (without any set-theoretical assumptions) that every infinite complete Boolean algebra includes a free subalgebra of the same cardinality. It follows that the set of all ultrafilters on an infinite complete Boolean algebra $B$ has power $2^{\vert B\vert}$.
Partition algebras for almost-disjoint families
James E.
Baumgartner;
Martin
Weese
619-630
Abstract: A set $a \subseteq \omega$ is a partitioner of a maximal almost-disjoint faculty $F$ of subsets of $\omega$ if every element of $F$ is almost contained in or almost-disjoint from $ a$. The partition algebra of $F$ is the quotient of the Boolean algebra of partitioners modulo the ideal generated by $F$ and the finite sets. We show that every countable algebra is a partition algebra, and that CH implies every algebra of cardinality $\leq {2^{{\aleph _0}}}$ is a partition algebra. We also obtain consistency and independence results about the representability of Boolean algebras as partition algebras.
On extending free group actions on spheres and a conjecture of Iwasawa
Frank
Connolly;
Robert
Geist
631-640
Abstract: A transfer map for Reidemeister torsion is defined and used to determine whether free actions of $ \mathbf{Z}/k$ on $ S^{2n+1}$, $n > 1$, extend to free actions of $\mathbf{Z}/hk$. It is shown that for $ k$ odd, every free $\mathbf{Z}/k$ action on $S^{2n+1}$, $n > 1$, extends to a free $\mathbf{Z}/2k$ action. For prime $p$, extension of an arbitrary free $\mathbf{Z}/p$ action to a free $\mathbf{Z}/p^{2}$ action is reduced to a long-standing conjecture of Iwasawa.
$2$-sided embeddings of projective planes into $3$-manifolds
Mitsuyuki
Ochiai
641-650
Abstract: Let $M$ be a nonorientable closed $ 3$-manifold which admits a $ 2$-sided embedding of a projective plane. Then we first prove the following theorem: If $M$ has a Heegaard splitting of genus two, then $ M$ is homeomorphic to ${P^{2}}\times {S^{1}}$. Next, let $ M$ be a nonorientable $ 3$-manifold whose fundamental group is abelian. We verify that if $ M$ has a Heegaard splitting of genus two, then $M$ is either the nonorientable $2$-sphere bundle over the circle or ${P^{2}}\times {S^{1}}$.
On the group of real analytic diffeomorphisms of a compact real analytic manifold
J.
Leslie
651-669
Abstract: In this paper we exhibit a Lie group structure on the group of real analytic diffeomorphisms of a compact real analytic manifold. Further, we show that a variant of the Kupka-Smale theorem holds for the underlying topology of the above mentioned Lie group.
Whitney levels in hyperspaces of certain Peano continua
Jack T.
Goodykoontz;
Sam B.
Nadler
671-694
Abstract: Let $X$ be a Peano continuum. Let $ {2^x}$ (resp., $ C(X)$) be the space of all nonempty compacta (resp., subcontinua) of $ X$ with the Hausdorff matric. Let $\omega$ be a Whitney map defined on $\mathcal{H}={2^{X}}$ or $C(X)$ such that $\omega$ is admissible (this requires the existence of a certain type of deformation of $ \mathcal{H}$). If $\mathcal{H}=C(X)$, assume $X$ contains no free arc. Then, for any $ {t_0} \in (0,\omega (X))$, it is proved that $ {\omega ^{ - 1}}({t_0}),\,{\omega ^{ - 1}}([0,\,{t_0}])$, and ${\omega ^{ - 1}}([{t_0},\,\omega (X)])$ are Hilbert cubes. This is an analogue of the Curtis-Schori theorem for $ \mathcal{H}$. A general result for the existance of admissible Whitney maps is proved which implies that these maps exist when $ X$ is starshaped in a Banach space or when $X$ is a dendrite. Using these results it is shown, for example that being an AR, an ANR, an LC space, or an $ {\text{L}}{{\text{C}}^n}$ space is not strongly Whitney-reversible.
Unstable bordism groups and isolated singularities
David
Ellis
695-708
Abstract: An isolated singularity of an embedded submanifold can be topologically smoothed if and only if a certain obstruction element in ${\pi _ \ast }(MG)$ vanishes, where $ G$ is the group of the normal bundle. In fact this obstruction lies in a certain subgroup which is referred to here as the unstable $ G$-bordism group. In this paper some of the unstable $SO$-bordism groups are computed; the obstruction to smoothing the complex cone on an oriented submanifold $X \subset \mathbf{C}{P^n}$ at $\infty$ is computed in terms of the characteristic numbers of $X$. Examples of nonsmoothable complex cone singularities are given using these computations.
Polynomials on affine manifolds
David
Fried
709-719
Abstract: For a closed affine manifold $M$ of dimension $m$ the developing map defines an open subset $ D(\tilde M) \subset {{\mathbf{R}}^m}$. We show that $ D(\tilde M)$ cannot lie between parallel hyperplanes. When $m \le 3$ we show that any nonconstant polynomial $p:{{\mathbf{R}}^m} \to {\mathbf{R}}$ is unbounded on $D(\tilde M)$. If $D(\tilde M)$ lies in a half-space we show $ M$ has zero Euler characteristic. Under various special conditions on $ M$ we show that $ M$ has no nonconstant functions given by polynomials in affine coordinates.
On the variety of invariant subspaces of a finite-dimensional linear operator
Mark A.
Shayman
721-747
Abstract: If $V$ is a finite-dimensional vector space over $ \mathbf{R}$ or $\mathbf{C}$ and $A \in {\operatorname {Hom}}(V)$, the set $ {S_A}(k)$ of $ k$-dimensional $ A$-invariant subspaces is a compact subvariety of the Grassmann manifold $ {G^k}(V)$, but it need not be a Schubert variety. We study the topology of $ {S_A}(k)$. We reduce to the case where $A$ is nilpotent. In this case we prove that $ {S_A}(k)$ is connected but need not be a manifold. However, the subset of $ {S_A}(k)$ consisting of those subspaces with a fixed cyclic structure is a regular submanifold of ${G^k}(V)$.
Local analyticity in weighted $L\sp{1}$-spaces and applications to stability problems for Volterra equations
G. S.
Jordan;
Olof J.
Staffans;
Robert L.
Wheeler
749-782
Abstract: We study the qualitative properties of the solutions of linear convolution equations such as $x \ast \mu = f$. We are especially concerned with finding conditions which ensure that these equations have resolvents which belong to, or are determined up to a term belonging to, certain weighted ${L^1}$-spaces. Our results are obtained as consequences of more general Banach algebra results on functions that are locally analytic with respect to the elements of a weighted ${L^1}$-space. In particular, we derive a proposition of Wiener-Lévy type for weighted $ {L^1}$-spaces which underlies all subsequent results. Our method applies equally well to equations more general than those mentioned above. We unify and sharpen the results of several recent papers on the asymptotic behavior of Volterra convolution equations of the types mentioned above, and indicate how many of them can be extended to the Fredholm case. In addition, we give necessary and sufficient conditions on the perturbation term $f$ for the existence of bounded or integrable solutions $x$ in some critical cases when the corresponding limit equations have nontrivial solutions.
Disproof of a coefficient conjecture for meromorphic univalent functions
Anna
Tsao
783-796
Abstract: Let $\Sigma$ denote the class of functions $g(z) = z + {b_0} + {b_1}{z^{ - 1}} + \cdots$ analytic and univalent in $\vert z\vert> 1$ except for a simple pole at $ \infty$. A well-known conjecture asserts that $ \vert{b_n}\vert\, \leq 2/(n + 1)\qquad (n = 1,2, \ldots )$ with equality for $g(z) = {(1 + {z^{n + 1}})^{2/(n + 1)}}/z = z + 2{z^{ - n}}/(n + 1) + \cdots$. Although the conjecture is true for $n=1,2$ and certain subclasses of the class $ \Sigma$, the general conjecture is known to be false for all odd $n\ge 3$ and $n=4$. In $\S 2$, we generalize a variational method of Goluzin and develop second-variational techniques. This enables us in $\S 3$ to construct explicit counterexamples to the conjecture for all $n > 4$. In fact, the conjectured extremal function does not even provide a local maximum for ${\text{Re}}\{ {b_n}\}$, $n > 4$.
On some open problems of P. Tur\'an concerning Birkhoff interpolation
A. K.
Varma
797-808
Abstract: In 1974 P. Turán (see [6]) raised many interesting open problems in Approximation Theory some of which are on Birkhoff Interpolation. The object of this paper is to answer some of these problems (XXXVI-XXXIX). We obtain some new quadrature formulas where function values and second derivatives are only prescribed on the zeros of $\displaystyle {\pi _n}(x) = {c_n}\int_{ - 1}^x {{P_{n - 1}}(t)\,dt}$ , ${P_n}(x)$ being Legendre polynomial of degree $ n$.
An elementary integral solution operator for the Cauchy-Riemann equations on pseudoconvex domains in ${\bf C}\sp{n}$
R. Michael
Range
809-816
Abstract: An integral representation formula for $(0,q)$ forms is constructed on a strictly pseudoconvex domain $D$ in $ \mathbf{C}^n$ by using only the local geometry of the boundary of $D$. By combining this representation with elementary results about compact operators in Banach spaces, one obtains the solution of the Levi problem and, more importantly, an integral solution operator for $ \bar{\partial}$ on $ D$. The construction does not need any a priori knowledge of the solvability of $ \bar{\partial}$ and thus allows us to establish fundamental global results by a direct and elementary method.