Torsion Invariants Research Articles

REIDEMEISTER TORSION AND SIMPLICIAL QUANTUM GRAVITY

We show that three-dimensional simplicial quantum gravity, as described by dynamically triangulated manifolds, is connected with a Gaussian model determined by the simple homotopy types of the underlying manifolds. By exploiting this result it is shown that the partition function of three-dimensional simplicial quantum gravity is well defined in a convex region in the plane of the gravitational and cosmological coupling constants. Such a region is determined by the Reidemeister–Franz torsion invariants associated with orthogonal representations of the fundamental groups of the set of manifolds considered. The system shows a critical behavior and undergoes a first order phase transition at a well-defined value of the couplings, again determined by the torsion invariants. On the critical line the partition function can be explicitly related to a Gaussian measure on the general linear group GL (∞, R), showing evidence of a well-defined thermodynamical limit of the theory, with a stable (vacuum) configuration corresponding to three-dimensional (homology) manifolds. The first order nature of the transition yielding such a configuration seems to support the belief in the absence of a continuum limit of the theory. More generally, the approach presented here provides further analytical support for the picture of three-dimensional simplicial quantum gravity which has been abstracted from numerical simulations.

Determinants, torsion, and strings

We apply the results of [BF1, BF2] on determinants of Dirac operators to String Theory. For the bosonic string we recover the “holomorphic factorization” of Belavin and Knizhik. Witten's global anomaly formula is used to give sufficient conditions for anomaly cancellation in the heterotic string (for arbitrary background spacetimes). To prove the latter result we develop certain torsion invariants related to characteristic classes of vector bundles and to index theory.

A Condition In Constructing Chain Homotopies

In this note we show by an example that the answer is negative, namely that f. —g, does not always hold. We also consider the case R = K[G^ where K is a field, and show that the answer is negative if and only if the characteristic of the field K divides the order of G when G is a finite group. We also obtain some condition for an infinite group G. If 1. 1 were true, then the arguments of Morimoto in [3], which computes a homology class of the torsion invariant (cf, Theorem 8. 4 of Dovermann-Rothenberg [2]), would be considerably simplified. Thus the negativeness of 1. 1 suggests that we cannot do without delicate arguments as in [3] in computing torsion invariants. The author would like to thank M. Morimoto, T. Matumoto and K. Motose for valuable conversations. He would also like to thank

Waldhausen's theory of k-fold end structures: a survey

Let R + be the space of nonnegative real numbers. F. Waldhausen defines a k-fold end structure on a space X as an ordered k-tuple of continuous maps x f : X → R +, 1 ⩽ j ⩽ k, yielding a proper map x: X → ( R +) k . The pairs ( X, x) are made into the category E k of spaces with k-fold end structure. Attachments and expansions in E k are defined by induction on k, where elementary attachments and expansions in E 0 have their usual meaning. The category E k / Z consists of objects ( X, i) where i: Z → X is an inclusion in E k with an attachment of i( Z) to X, and the category E k ‖ Z consists of pairs ( X, i) of E k / Z that admit retractions X → Z. An infinite complex over Z is a sequence X = { X 1 ⊂ X 2 ⊂ … ⊂ X n …} of inclusions in E k ‖ Z. The abelian grou p S 0( Z) is then defined as the set of equivalence classes of infinite complexes dominated by finite ones, where the equivalence relation is generated by homotopy equivalence and finite attachment; and the abelian group S 1( Z) is defined as the set of equivalence classes of X 1, where X ∈ E k / Z deformation retracts to Z. The group operations are gluing over Z. This paper presents the Waldhausen theory with some additions and in particular the proof of Waldhausen's proposition that there exists a natural exact sequence 0 → S 1( Z × R)→ π S 0( Z) by utilizing methods of L.C. Siebenmann. Waldhausen developed this theory while seeking to prove the topological invariance of Whitehead torsion; however, the end structures also have application in studying the splitting of a noncompact manifold as a product with R[1].

Torsion invariants and actions of finite groups.

Torsion invariants and actions of finite groups.

Invariance of Torsion and the Borsuk Conjecture

The following results of Whitehead and Wall are well-known applications of the algebraic K-theoretic functors K0 and K1 to basic homotopy questions in topology.THEOREM 1 [20]. If f : X → Y is a homotopy equivalence between compact CW complexes, then there is a torsion τ(ƒ) in the algebraically-defined Whitehead group Wh π1(Y) which vanishes if and only if f is a simple homotopy equivalence.THEOREM 2 [18]. If X is an arbitrary space which is finitely dominated (i.e., homotopically dominated by a compact polyhedron), then there is an obstruction σ(X) in the algebraically-defined reduced projective class group which vanishes if and only if X is homotopy equivalent to some compact polyhedron.If we direct sum over components, then the above statements make good sense even if the spaces involved are not connected.

Reidemeister torsion and group invariants of three-dimensional manifolds

Two formulas are presented in this note. The first is purely algebraic and expresses the first elementary ideal of a finitely generated group in terms of the module of elementary derivatives. The second formula expresses the module of elementary derivatives of the fundamental group of a connected compact three-dimensionalpl -manifold with zero Euler characteristic in terms of the Reidemeister torsion of this manifold.

Compact Hilbert cube manifolds and the invariance of Whitehead torsion

Compact Hilbert cube manifolds and the invariance of Whitehead torsion

Generalized product theorems for torsion invariants with applications to flat bundles

Generalized product theorems for torsion invariants with applications to flat bundles

Geometric groups and Whitehead torsion

The purpose of this paper is to define geometric and to relate them to various problems in topology. This relation is exhibited through Conjectures I and II. It will be shown that Conjecture I implies Conjecture II and that Conjecture II implies the topological invariance of Whitehead torsion. Conjecture II is true for 2-complexes, and this implies that if K and N are finite connected complexes, L' K is a subcomplex with dim L $2, and f: K -> N is a homeomorphism with f IK-L p.w.l., then f is a simple homotopy equivalence. Another corollary is that if K is a 2-complex contained in a p.w.l. manifold Mn, Un is a compact p.w.l. submanifold, K' U' M, and U e-deforms to K, then KG U is a simple homotopy equivalence and, thus, if n > 65 U is a regular neighborhood of K. Finally, for any finitely presented group with Wh (-a) #0, ] an h-cobordism W with -a (W) = which is not topologically trivial. Geometric groups are related to other problems in topology and some of these are mentioned without proof in the appendix. For example, Conjecture I implies that compact ANRs of finite dimension have the homotopy type of finite complexes. Conjecture II has a noncompact analogue and since the difficulties are local, there is essentially nothing new here (Conjecture II is true for infinite 2-complexes). This noncompact form of Conjecture 11 implies the following: If f: Rn -> Rn is a homeomorphism (n >5) such that fx Id: Rn x Rk Rn x Rk is stable, then f is stable. The final note of the appendix implies the following: Suppose (ai,j) is an infinite matrix with integer entries, and that it and its inverse are band matrices, i.e., bounded about the diagonal. Then (ai,j) can be diagonalized by row operations. This is a nongeometric analogue of Conjecture II for the infinite complex R1.