arXiv subjects
Publications and source records attributed to John R. Klein.
We construct periodic families of Poincare complexes, partially solving a question of Hodgson that was posed in the proceedings of the 1982 Northwestern homotopy theory conference. We also construct infinite families of Poincare complexes whose top cell falls off after one suspension but which fail to embed in a sphere of codimension one. We give a homotopy theoretic description of the four-fold periodicity in knot cobordism.
Let $A$ be a unital $C^*$-algebra. Its unitary group, $UA$, contains a wealth of topological information about $A$. However, the homotopy type of $UA$ is out of reach even for $A = M_2(\CC)$. There are two simplifications which have been considered. The first, well-traveled road, is to pass to $π_*(U(A\otimes \KK ))$ which is isomorphic (with a degree shift) to $K_*(A)$. This approach has led to spectacular success in many arenas, as is well-known. A different approach is to consider $π_*(UA)\otimes\QQ $, the rational homotopy of $UA$. In joint work with G. Lupton and N. C. Phillips we have calculated this functor for the cases $A = C(X)\otimes M_n(\CC)$ and $A$ a unital continuous trace $C^*$-algebra. In this note we look at some concrete examples of this calculation and, in particular, at the $\ZZ$-graded map \[ π_*(UA)\otimes\QQ \longrightarrow K_{*+1}(A)\otimes\QQ . \]
Let ζbe an n-dimensional complex matrix bundle over a compact metric space X and let A_ζdenote the C*-algebra of sections of this bundle. We determine the rational homotopy type as an H-space of UA_ζ, the group of unitaries of A_ζ. The answer turns out to be independent of the bundle ζand depends only upon n and the rational cohomology of X. We prove analogous results for the gauge group and the projective gauge group of a principal bundle over a compact metric space X.
We show that the smooth torsion of bundles of manifolds constructed by Dwyer, Weiss, and Williams satisfies the axioms for higher torsion developed by Igusa. As a consequence we obtain that the smooth Dwyer-Weiss-Williams torsion is proportional to the higher torsion of Igusa and Klein.
This is the second in a series of papers. Here we develop here an intersection theory for manifolds equipped with an action of a finite group. As in our previous paper, our approach will be homotopy theoretic, enabling us to circumvent the specter of equivariant transversality. This theory has applications to embedding problems, equivariant fixed point theory and the problem of enumerating the periodic points of a self map of a compact smooth manifold.
Let K be a connected finite complex. This paper studies the problem of whether one can attach a cell to some iterated suspension S^j K so that the resulting space satisfies Poincare duality. When this is possible, we say that S^j K is a spine. We introduce the notion of quadratic self duality and show that if K is quadratically self dual, then S^j K is a spine whenever j is a suitable power of two. The powers of two come from the James periodicity theorem. We briefly explain how our main results, considered up to bordism, give a new interpretation of the four-fold periodicity of the surgery obstruction groups. We therefore obtain a relationship between James periodicity and the four-fold periodicity in L-theory.
We give new homotopy theoretic criteria for deciding when a fibration with homotopy finite fibers admits a reduction to a fiber bundle with compact topological manifold fibers. The criteria lead to a new and unexpected result about homeomorphism groups of manifolds. A tool used in the proof is a surjective splitting of the assembly map for Waldhausen's functor A(X). We also give concrete examples of fibrations having a reduction to a fiber bundle with compact topological manifold fibers but which fail to admit a compact fiber smoothing. The examples are detected by algebraic K-theory invariants. We consider a refinement of the Becker-Gottlieb transfer. We show that a version of the axioms described by Becker and Schultz uniquely determines the refined transfer for the class of fibrations admitting a reduction to a fiber bundle with compact topological manifold fibers. In an appendix, we sketch a theory of characteristic classes for fibrations. The classes are primary obstructions to finding a compact fiber smoothing.
We obtain multirelative connectivity statements about spaces of Poincare embeddings, as precursors to analogous statements about spaces of smooth embeddings. The latter are the key to convergence results in the functor calculus approach to spaces of embeddings.
In this note we study umkehr maps in generalized (co)homology theories arising from the Pontrjagin-Thom construction, from integrating along fibers, pushforward homomorphisms, and other similar constructions. We consider the basic properties of these constructions and develop axioms which any umkehr homomorphism must satisfy. We use a version of Brown representability to show that these axioms completely characterize these homomorphisms, and a resulting uniqueness theorem follows. Finally, motivated by constructions in string topology, we extend this axiomatic treatment of umkehr homomorphisms to a fiberwise setting.
In this paper we identify the ``nil-terms'' for Waldhausen's algebraic K-theory of spaces functor as the reduced K-theory of a category of equivariant spaces equipped with a homotopically nilpotent endomorphism.
For a space X, we define Frobenius and Verschiebung operations on the nil-terms NA^{fd} (X) in the algebraic K-theory of spaces, in three different ways. Two applications are included. Firstly, we obtain that the homotopy groups of NA^{fd} (X) are either trivial or not finitely generated as abelian groups. Secondly, the Verschiebung defines a Z[N_x]-module structure on the homotopy groups of NA^{fd} (X), with N_x the multiplicative monoid. We also we give a calculation of the homotopy groups of the nil-terms NA^{fd} (*) after p-completion for an odd prime p as Z_p[N_x]-modules up to dimension 4p-7. We obtain non-trivial groups only in dimension 2p-2, where it is finitely generated as a Z_p[N_x]-module, and in dimension 2p-1, where it is not finitely generated as a Z_p[N_x]-module.
To an inclusion topological groups H->G, we associate a naive G-spectrum. The special case when H=G gives the dualizing spectrum D_G introduced by the author in the first paper of this series. The main application will be to give a purely homotopy theoretic construction of Poincare embeddings in stable codimension.
Let M be a Poincare duality space of dimension at least four. In this paper we describe a complete obstruction to realizing the diagonal map M -> M x M by a Poincare embedding. The obstruction group depends only on the fundamental group and the parity of the dimension of M.
We give a new approach to intersection theory. Our "cycles" are closed manifolds mapping into compact manifolds and our "intersections" are elements of a homotopy group of a certain Thom space. The results are then applied in various contexts, including fixed point, linking and disjunction problems. Our main theorems resemble those of Hatcher and Quinn, but our proofs are fundamentally different.
We prove two kinds of fibering theorems for maps X --> P, where X and P are Poincare spaces. The special case of P = S^1 yields a Poincare duality analogue of the fibering theorem of Browder and Levine.
Almost forty years ago, C.T.C. Wall systematically analyzed the set of "thickenings" of a finite CW complex. Of the results he obtained, probably the most computationally important is the "suspension theorem," which is an exact sequence relating the n-dimensional thickenings of a finite complex to its (n+1)-dimensional ones. The object of this note is to fill in what we believe is a missing argument in the proof of that theorem.
For a closed PL manifold M, we consider the configuration space F(M,k) of ordered k-tuples of distinct points in M. We show that a suitable iterated suspension of F(M,k) is a homotopy invariant of M. The number of suspensions we require depends on three parameters: the number of points k, the dimension of M and the connectivity of M. Our proof uses a mixture of Poincare embedding theory and fiberwise algebraic topology.
We consider embeddings of a finite complex in a sphere. We give a homotopy theoretic classification of such embeddings in a wide range.