Kiyoshi Igusa

A characterization of finite Auslander—Reiten quivers

(a) We give necessary and sufficient conditions for a finite k-modulated translation quiver to be an Auslander-Reiten quiver in terms of certain homology groups associated to the quiver (Section 3). (b) We show that whether a finite k-modulated translation quiver is an Auslander-Reiten quiver depends only on the underlying valued translation quiver and the characteristic of k and give necessary and sufficient conditions only in terms of the valued quiver and char k. We also show that any valued quiver with valuations (1, n) or (n, l), n Q 3, admits a modulation over any prime field (Sections 5,6, 7). (c) To a valued translation quiver r we associate an unvalued translation quiver r (i.e., all valuations are 1) and show that a finite k-modulated translation quiver is an Auslander-Reiten quiver if and only if the associated unvalued translation quiver with the trivial K-modulation (K = prime field of k) is an Auslander-Reiten quiver (Section 8).

Notes on the no loops conjecture

We use algebraic K-theory to resolve the ‘no loops conjecture’ for finite dimensional algebras of finite global dimension over an algebraically closed field. We also prove two special cases of a new conjecture concerning automorphisms of such algebras.

The stability theorem for smooth pseudoisotopies

Key w o r d s , Pseudoisotopy, stability, Morse theory. iSupported by NSF Grant No. MCS-85-02317.

Radical layers of representable functors

(i) Let 0 --f A --t4 B +O C + 0 be an almost split sequence. Then the induced sequence O+ G( , A)-+“* G( ,B)+“* G( , C)-+yS~ 0 is exact where S = GS = ( , C)/r( , C). The maps a*, /?* have degree 1 and y has degree 0. (ii) Let I--t I/sot I be a left minimal almost split map where I is an indecomposable injective. Let k < 03 be maximal such that im( , sot I) c rk( , I). If k < co then the induced sequence 0 + G( , sot I) -+ G( , I) + G( , I/sot Z) is exact where the first map has degree k and the second has degree 1. If k = co then the induced map G( , I) --t G( , Z/sot I) is a monomorphism of degree 1. Dual results hold for covariant functors (X, ).

The space of framed functions

Introduction. The purpose of this paper is to prove that the space of framed functions on a compact smooth manifold N is (dim N 1)-connected. The precise definitions and statements are given in §1. The purpose of this introduction is to explain why we are interested in framed functions, in particular we explain how they are related to pseudoisotopy theory. Suppose that Wn+1 is a cobordism from Mo to M1 (all manifolds being compact and smooth). Thus aw= Mo U M1 U (8Mo x I). Also suppose we have a Morse function t: W I. Then we get a fi ite relative cell c mplex X= Mo U e1 U e2

Combinatorial Miller–Morita–Mumford classes and Witten cycles

We obtain a combinatorial formula for the Miller-Morita-Mum- ford classes for the mapping class group of punctured surfaces and prove Wittens conjecture that they are proportional to the dual to the Witten cycles. The proportionality constant is shown to be exactly as conjectured by Arbarello and Cornalba (1). We also verify their conjectured formula for the leading coefficient of the polynomial expressing the Kontsevich cycles in terms of the Miller-Morita-Mumford classes. AMS Classification 57N05; 55R40, 57M15

CLUSTER COMPLEXES VIA SEMI-INVARIANTS

We define and study virtual representation spaces for vectors having both positive and negative dimensions at the vertices of a quiver without oriented cycles. We consider the natural semi-invariants on these spaces which we call virtual semi-invariants and prove that they satisfy the three basic theorems: the first fundamental theorem, the saturation theorem and the canonical decomposition theorem. In the special case of Dynkin quivers with n vertices, this gives the fundamental interrelationship between supports of the semi-invariants and the tilting triangulation of the ( n −1)-sphere.

Axioms for higher torsion invariants of smooth bundles

This paper attempts to explain the relationship between various characteristic classes for smooth manifold bundles which are known as ‘higher torsion’ classes. We isolate two fundamental properties that these cohomology classes may or may not have: additivity and transfer. We show that higher Franz-Reidemeister torsion and higher Miller-Morita-Mumford classes satisfy these axioms. Conversely, any characteristic class of smooth bundles satisfying the two axioms must be a linear combination of these two examples.We also show how any higher torsion invariant, that is, any characteristic class satisfying the two axioms, can be computed for a smooth bundle with a fiberwise Morse function with distinct critical values. Finally, we explain the statements of the conjectured formulas relating higher analytic torsion classes, higher Franz-Reidemeister torsion and Dwyer-Weiss-Williams smooth torsion.

On the homotopy type of the space of generalized Morse functions

LET N BE A compact smooth (Cm) n-manifold and let g: N--, R be a fixed Morse function without critical points on 8N. We shall say that a functionf: N + R is admissible if it is smooth and agrees with g near 3N. Let 9(N, 8N) denote the space of all admissible functions on N. Then F(N, 8N) is convex and thus contractible. (We use the strong C”-topology[2] for all function spaces.) Let A(N, 8N) denote the space of all admissible Morse functions on N. Then A(N, 8N) has many components since functions in a single component have the same number of critical points. To obtain a connected space of functions we must allow cancellation of critical points. Thus let &‘(N, 8N) denote the space of all admissible functions of N whose critical points are either Morse or birth-death singularities (defined in 2.1). In this paper we compute the (n I)-homotopy type of S’(N, dN). Let f: N + IF! be an admissible function. Then the derivative of f gives a map N/aN + T(r) where T(t) is the Thorn space of the tangent bundle r of N. This map is given as follows. We choose a fixed Riemannian metric on N by embedding N in R” and we let E = min (1, inf { IlVg(x)llIx E aN}). Then the composition of l/~Vfz N -+ E with the collapsing map E + T(T) where E is the total space of T sends aN to *, so it induces a map N/aN + T(r). Suppose that f E A(N, 8N). Then we can choose E > 0 a continuous function off so that U = {x E NI/Vf(x)II < L } c int N = N 8N and U is the union of disjoint contractible neighborhoods Vi of the critical points yi off. Using this E we again get a map N/aN + T(z). But this map now lifts to a map N/aN + T@ *r ) where p *z is the pull-back of r along the projection p: BO x N-P N. The lifting is given by sending x E Vi to (Pi, l/cvf(x)) where Pi is the plane in IF” spanned by the negative eigenvectors of P~(JJJ considered as an element of BO. (The complement of U goes to *.) In the special case when N is embedded in R” this gives an element of Q”P((BO x N),) where the lower “ + ” means add a disjoint base point. In other cases we get an element of CPS”((B0 x N),) by passing to the normal disk bundle of N by a construction called suspension (3.1.a). In any case we get a map J/: .&(N, cYN)+CPS~((BO x N),). In

Mixed cobinary trees

5 we show how the negative eigenplanes of cancelling Morse singularities can be made to match up at a birth-death singularity thus allowing us to extend the function I,+ to H(N, 8N). The composition &‘(N, dN)42”S”((BO x N)+)-+fPSm(N+) is null homotopic since it extends to all of P(N, 8N). Consequently we get a map 6: H(N, aN)+ W’Soc(BO A (N,)) N fiber [QmSm((BO x N)+)--,R”S”(N+)]. The main theorem of this paper is that this map is n-connected. The (n l)-connectivity of