K. Varadarajan

Study of modules over formal triangular matrix rings

Abstract In this paper we carry out a systematic study of modules over a formal triangular matrix ring T= A 0 M B . Using the alternative description of right T -modules as triples (X,Y) f with X ∈ Mod−A , Y ∈ Mod−B and f : Y ⊗ B M → X in Mod−A, we shall characterize respectively uniform, hollow, finitely embedded, projective, generator or progenerator modules over T . For projective modules an explicit method for constructing a dual basis is described. Also necessary and sufficient conditions are found for a T -module to admit a projective cover. When the conditions are fulfilled we give an explicit method for constructing a projective cover.

On Hopfian Rings

The main results proved in this paper are:(i) If R is a boolean hopfian ring then the polynomial ring R[T] is hopfian.(ii) Let R and S be hopfian rings. Suppose the only central idempotents in S are 0 and 1 and that S is not a homomorphic image of R. Then R × S is a hopfian ring.

Group actions on flag manifolds and cobordism

In this paper we show that any a G %n (respectively a G Qm) can be represented by a closed smooth (respectively closed, oriented smooth) manifold M admitting a smooth (Z/2) (respectively S^-action with a finite stationary set. We also completely determine the Grassman manifolds CGnk, HGnk and Gnk which are oriented boundaries as well as those which represent non-torsion elements in f2*.

ACYCLICITY OF CERTAIN HOMEOMORPHISM GROUPS

Introduction. The concept of a mitotic group was introduced in [3] by Baumslag, Dyer and Heller who showed that mitotic groups were acyclic. In [8] one of the authors introduced the concept of a pseudo-mitotic group, a concept weaker than that of a mitotic group, and showed that pseudo-mitotic groups were acyclic and that the group Gn of homeomorphisms of R n with compact support is pseudo-mitotic. In our present paper we develop techniques to prove pseudomitoticity of certain other homeomorphism groups. In [5] Kan and Thurston observed that the group of set theoretic bijections of Q with bounded support is acyclic. A natural question is to decide whether the group of homeomorphisms of Q (resp. the irrationals / ) with bounded support is acyclic or not. In the present paper we develop techniques to answer this question in the affirmative. Also the techniques developed here enable us to show that the group of homeomorphisms of the Cantor set which are identity in a neighbourhood of 0 and 1 is pseudo-mitotic and hence acyclic. It is worth noticing the contrast between our results and the results of R. D. Anderson in [1] and [2]. Anderson, using his techniques shows that the group of all homeomorphisms of Q, / or the Cantor set is a simple group. He also shows that the group of orientation preserving homeomorphisms of S or S is simple.

On certain homeomorphism groups

Abstract Let R , Q , I and C denote respectively the reals, the rationals, the irrationals and the Cantor set endowed with their usual topologies. For any topological space X let G(X) denote the group of homeomorphisms of X. Let H( R ) denote the group of orientation preserving homeomorphisms of R , Sω the group of permutations of the set N of natural numbers and G0( R n) the group of homeomorphisms of R n with compact support for any integer n ≤ 1. We show that G ∗ F can be imbedded in G when G is any one of the groups G( Q ), G( I ), G( C ), H( R ), Sω or G0( R n) where F is a free group of rank c = # R .

Criteria for the asphericity of 2-complexes

We obtain criteria for a finite 2-complex to be aspherical in terms of Euler characteristics.

A note on the homotopy type of posets

Abstract For any poset P let J ( P ) denote the complete lattice of order ideals in P . J ( P ) is a contravariant functor in P . Any order-reserving map f : P → Q can be regarded as an isotone (=order-preserving) map of either P ∗ into Q or P into Q ∗ . The induced map of J ( Q ) to J ( P ∗ )(resp. J ( Q ∗ ) into J ( P )) will be denoted by J l ( f )(resp. J r ( f )). Our first result asserts that if f : P → Q , g : Q → P are maps of a Galois connection, then (a) J r ( f ): J ( Q ∗ )→ J ( P ) ∗ , J l ( g ): J ( P ∗ )→ J ( Q ∗ ) and (b) J l ( f ): J ( Q ) ∗ → J ( P ∗ ), J r ( g ): J ( P ∗ )→ J ( Q ) ∗ are Galois connections. For any lattice L , we denote the poset L - {0,1} by L . We analyse conditions which will imply that J r (f) (J (Q∗)) ∋ J (P)∗ and J l (g) (J (P)∗) ∋ J (Q)∗ . Under these conditions, from Walkers results [3] it will follow that J r (f)/ (J (Q∗)) : (J (Q∗)) → (J (P)∗ is a homotopy equivalence with J l (g) (J (P)∗ : (J (P)∗ → (J (Q∗)) as its homotopy inverse. Given an isotone map f : P → Q it is easy to find the necessary and sufficient conditions for J ( f ) to satisfy J(f) (J (Q)) ∋ J (P) . When these conditions are fulfilled, we also find a sufficient condition that ensures that J(f)/ J (Q) : J (Q) → J (P) is a homotopy equivalence. We give examples to show that the homotopy type of P neither determines nor is determined by the homotopy type of J (P) .

Chains, multichains and Mo¨bius numbers

Abstract For any finite poset P and any integer k ⩾0, let α k ( P ) denote the number of k -chains (i.e. chains of cardinality k or length k −1) in P . The polynomial α ( P , X )=∑; k ⩾0 α k ( P ) X k will be referr ed to as the chain generating polynomial of P . Our first results determine the chain generating polynomial α ( P ⊗ Q , X ) and the multichain generating series m ( P ⊗ Q , X ) of the ordinal product P ⊗ Q , for any two finite posets P and Q . Using this we determine the Mobius function for certain ordinal products. For any integer n ⩾1 let [ n ], B n , D n , L n ( q ) and Π n denote the lattices defined by Stanley (1986). When P is one of the posets [ n ], B n or D n the values of α k ( P ) for any k ⩾0 are well known and can easily be determined. When P = L n ( q ) o Π n we will establish recurrence relations in this paper which will effectively determine α k ( P ). For example, we explicitly determine α k ( L n ( q )) for all k ⩾0 when n ⩽3. We also obtain recurrence relations for the number m k ( P ) of k -multichains of P when P = L n ( q ) or Π n . Using these we explicitly determine m k ( L n ( q )) for n ⩽4.