The Projective Class Group and the Whitehead Group

Let M n + 3 {M^{n + 3}} be a compact orientable manifold which is the total space of a fiber bundle over a compact orientable manifold K 3 {K^3} with an effective circle action of hyperbolic type. Assume that the fiber N n {N^n} in this bundle is a closed orientable manifold with Noetherian integral group ring, with vanishing projective class and Whitehead groups, and such that the structure set S TOP ( N n × D k , ∂ ) {S_{\text {TOP}}}\,({N^n} \times {D^k},\partial ) of topological surgery vanishes for sufficiently large k k . Then the projective class and Whitehead groups of M M vanish and S TOP ( M n + 3 × D k , ∂ ) = 0 {S_{\text {TOP}}}\,({M^{n + 3}}\, \times \, {D^k},\partial ) = 0 if n + k ⩾ 3 n + k \geqslant 3 or if K 3 {K^3} is closed and n = 2 n = 2 . The UNil \text {UNil} groups of Cappell are the main obstacle here, and these results give new examples of generalized free products of groups such that UNil j {\text {UNil}}_j vanishes in spite of the failure of Cappell’s sufficient condition.

We give a characterization of k-dimensional (k > 1) Menger manifolds admitting boundaries in the sense of Chapman and Siebenmann.In [5] Chapman and Siebenmann considered the problem of putting a boundary on a Hilbert cube manifold (similar problems in the cases of smooth and piecewise linear manifolds were considered in [2,14]; a parametrical version of the above problem was considered in [13]).It was proved there that if a Q-manifold M satisfies certain minimal necessary homotopy-theoretic conditions (finite type and tameness at oo ), then there are two obstructions o0O(M) and Too(M) to M having a boundary.The first one is an element of the group lim{J?67ri(M -A) : A c M compact}, where %nx is the projective class group functor.If Ooo(M) = 0, then the second obstruction can be defined as an element of the first derived limit of the inverse system lim {W hnx (M -A) : A M compact} , where Whnx is the Whitehead group functor.Further, it was proved in [5] that the different boundaries that can be put on M constitute a whole shape class and that a classification of all possible ways of putting boundaries on M can be done in terms of the group lim {Whnx (M -A) : Ac.M compact}.In the present paper we carry out a similar program for the problem of putting boundaries on pn+x -manifolds, where pn+x denotes the (n + 1 )-dimensional universal Menger compactum (a p"+x -manifold M admits a boundary if there exists a compact p"+x-manifold N such that M = N-Z , where Z is a Z-set in N ; in this case we shall say that N is a compactification of M corresponding to the boundary Z , and conversely, Z is a boundary of M corresponding to the compactification N. We recall also that a closed subset Z of a space X is said to be a Z-set if for each open cover ^ e cov(X) there is a map f:X-*X-Z ^-close to the identity map of X ).Having in mind a deep analogy between the theories of pn+x -manifolds and Q-manifolds [1,[6][7][8][9] it is not surprising that the corresponding results are valid in the case of pn+xmanifolds as well.However, it should be observed that the situation in the last case is much simpler.For example, the analogies of the above-described

Let M be a compact connected Riemannian manifold whose sectional curvature values are all nonpositive. Let T denote the fundamental group of M. We prove that any homotopy equivalence : N -> M from a compact closed manifold TV is homotopic to a homeomorphism, provided that m > 5 where m = dim M . We show that the surgery L-group L k+m (T, w { ) is isomorphic to the set of homotopy classes of k k

Chapter 26 - Wall's Finiteness Obstruction

Projective modules and class groups

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.

Let K be the field of algebraic functions on an elliptic curve that can be described by an equation of the form y 2 = f ( x ) {y^2} = f(x) where f ( x ) f(x) is a quartic polynomial over a field k. Moreover, assume that the Riemann surface for K contains no points rational over k. When k is the field of real numbers it is well known that K may also be expressed as a quadratic extension of a function field L = k ( u , v ) L = k(u,v) of algebraic functions on a conic whose Riemann surface also contains no points rational over k. We extend this result to p-adic ground fields k. Moreover, we describe the various subfields of index two and genus zero (conic subfields) in terms of the k-rational points on the Jacobian of K. This is done for arbitrary ground fields. In particular, the embedding of the projective class group of K (over k) is seen to describe exactly those conic subfields that possess k-rational points.