We outline the main features of the definitions and applications of crossed complexes and cubical #-groupoids with connections.

Recently there has been a surge of interest in the Seiberg-Witten invariants of 3-manifolds, see [3], [4], [7]. The Seiberg-Witten invariant of a closed oriented 3-manifold M is a function SW from the set of Spin c-structures on M to Z. This function is defined under the assumption b1(M) ≥ 1 where b1(M) is the

Dedicated to Hvedri Inassaridze for his 70th birthday The category of crossed complexes gives an algebraic model of CW-complexes and cellular maps. Free crossed resolutions of groups contain information on a presentation of the group as well as higher homological information. We relate this to the problem of calculating non-abelian extensions. We show how the strong properties of this category allow for the computation of free crossed resolutions for amalgamated sums and HNN-extensions of groups, and so obtain computations of higher homotopical syzygies in these cases. 1

We give an interpretation of Yetter’s Invariant of manifolds M in terms of the homotopy type of the function space TOP(M,B(G)), where G is a crossed module and B(G) is its classifying space. From this formulation, there follows that Yetter’s invariant depends only on the homotopy type of M, and the weak homotopy type of the crossed module G. We use this interpretation to define a twisting of Yetter’s Invariant by cohomology classes of crossed modules, defined

(communicated by Hvedri Inassaridze) Global actions were introduced by A. Bak to give a combinatorial

We present a new approach to simple homotopy theory of polyhedra using finite topological spaces. We define the concept of collapse of a finite space and prove that this new notion corresponds exactly to the concept of a simplicial collapse. More precisely, we show that a collapse X ց Y of finite spaces induces a simplicial collapse K(X) ց K(Y) of their associated simplicial complexes. Moreover, a simplicial collapse K ց L induces a collapse X(K) ց X(L) of the associated finite spaces. This establishes a one-to-one correspondence between simple homotopy types of finite simplicial complexes and simple equivalence classes of finite spaces. We also prove a similar result for maps: We give a complete characterization of the class of maps between finite spaces which induce simple homotopy equivalences between the associated polyhedra. Furthermore, this class describes all maps coming from simple homotopy equivalences at the level of complexes. The advantage of this theory is that the elementary move of finite spaces is much simpler than the elementary move of simplicial complexes: It consists of removing (or adding) just a single point of the space.

Crossed complexes are shown to have an algebra sufficiently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex. This leads to the fundamental crossed complex of a simplicial set. The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes. We start with some historical background, and give a survey of the required basic facts on crossed complexes.

Abstract. We study the Reidemeister torsion and the analytic torsion of the m dimensional disc in the Euclidean m dimensional space, using the base for the homology defined by Ray and Singer in [20]. We prove that the Reidemeister torsion coincides with a power of the volume of the disc. We study the additional terms arising in the analytic torsion due to the boundary, using generalizations of the Cheeger-Müller theorem. We use a formula proved by Brüning and Ma [2], that predicts a new anomaly boundary term beside the known term proportional to the Euler characteristic of the boundary [16]. Some of our results extend to the case of the cone over a sphere, in particular we evaluate directly the analytic torsion for a cone over the circle and over the two sphere. We compare the results obtained in the low dimensional cases. We also consider a different formula for the boundary term given by Dai and Fang [9], and we show that the result obtained using this formula is inconsistent with the direct calculation of the analytic torsion. 1.

this paper all matroids will have ground set [n], and we shall frequently omit the symbol n from our notation. Define a partial ordering on M + n by M # # M if M # is a quotient of M . Let# + n be the simplicial complex of chains in M + n ; every simplex s ## + n can be written as s = #M 1 , . . . , M r #, with all M i # M + n and M 1 # . . . # M r . 1.1 Convention. A superscript number on a matroid will denote its rank. Subscripts will be used for general indexing purposes. According to this convention, in a context where all matroids are on [n], M 0 and M n will denote the unique matroids of rank 0 and rank n respectively. For another example of the use of this convention, when we write s =# M i 0 , M i 1 , . . . , M i r # in# + n , the notation will imply that M i 0 # M i 1 # . . . # M i r and that rank(M i 0 ) = i 0 , rank(M i 1 ) = i 1 etc. 1.2 For any subset N # M + n let #(N ) be the subcomplex of# + n generated by N : #(N ) consists of all simplices of# + n all of whose vertices belong to N . In particular let M n = {M # M + n : 1 #rank(M) # n - 1}, and let# n = #(M n )

The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical ω-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.