arXiv subjects
Publications and source records attributed to John R. Klein.
For a finite CW pair $(K,L)$, we show how to construct a Poincaré triad $(P;\partial_0 P,\partial_1 P)$ and a weak homotopy equivalence $(K,L) \overset{{}_\sim}\to (P,\partial_0 P)$. Furthermore, the Poincaré pair $(P,\partial P)$ has a trivial Spivak normal fibration. The proof is homotopy-theoretic. We also prove a uniqueness result.
We exhibit a homotopy theoretic proof of the Fundamental Theorem of Poincaré surgery in the simply connected case. We also deduce the Poincaré transversality exact sequence.
We provide a simplified approach to the the stable Hopf invariant. We provide short elementary proofs of the Cartan Formula, the Composition Formula, and the Transfer formula. In addition, when $π$ is a discrete group, we show how to extend these results to the stable category of $π$-spaces. We also consider the extent to which the stable Hopf invariant is unique.
This paper characterizes the stabilized second James-Hopf invariant by means of three axioms. Specifically, we show that it is the unique natural transformation satisfying the Cartan formula, vanishing on suspensions. The proof combines the natural stable splitting of the James construction with Goodwillie calculus.
We establish Poincaré embedding results in the relative setting, generalizing previously known results in the absolute case. Our primary motivation comes from applications to non-simply connected Poincaré surgery, which will be developed in a forthcoming paper. Along the way, we introduce a new tool: the relative Hopf invariant in the equivariant setting.
Given an acyclic map $X\to Y$ of closed manifolds dimension $d$, we study the relationship between the embeddings of $Y$ in $S^{n}$ with those of $X$ in $S^{n}$ when $n-d \ge 3$. The approach taken here is to first solve the Poincaré duality space variant of the problem. We then apply the surgery machine to obtain the corresponding results for manifolds. Thereafter, we focus on the case when $X$ is a smooth homology sphere and deduce results about the homotopy type of the space of block embeddings of $X$ in a sphere.
For an $n$-qubit system, a rational function on the space of mixed states which is invariant with respect to the action of the group of local symmetries may be viewed as a detailed measure of entanglement. We show that the field of all such invariant rational functions is purely transcendental over the complex numbers and has transcendence degree $4^n - 2n-1$. An explicit transcendence basis is also exhibited.
We introduce the notion of an X-state on $n$-qubits. After taking the Zariski closure of the set of X-states in the space of all mixed states, we obtain a complex algebraic variety $\scr X$ that is equipped with the action of the Lie group of local symmetries $G$. We show that the field of $G$-invariant rational functions on $\scr X$ is purely transcendental over the complex numbers of degree $2^{2n-1}-n-1$.
We compute the field of rational local unitary invariants for locally maximally mixed states and symmetrically mixed states of two qubits. In both cases, we prove that the field of rational invariants is purely transcendental. We also construct explicit geometric quotients and prove that they are always rational. All the results are obtained by working over the field of real numbers, employing methods from classical and geometric invariant theory over arbitrary fields of characteristic zero.
We prove that the Becker-Gottlieb transfer is functorial up to homotopy, for all fibrations with finitely dominated fibers. This resolves a lingering foundational question about the transfer, which was originally defined in the late 1970s in order to simplify the proof of the Adams conjecture. Our approach differs from previous attempts in that we closely emulate the geometric argument in the case of a smooth fiber bundle. This leads to a "multiplicative'" description of the transfer, different from the standard presentation as the trace of a diagonal map.
We calculate the field of rational local unitary invariants for mixed states of two qubits, by employing methods from algebraic geometry. We prove that this field is rational (i.e. purely transcendental), and that it is generated by nine algebraically independent polynomial invariants. We do so by constructing a relative section, in the sense of invariant theory, whose Weyl group is a finite abelian group. From this construction, we are able to give explicit expressions for the generating invariants in terms of the Bloch matrix representation of mixed states of two qubits. We also prove similar rationality statements for the local unitary invariants of symmetric mixed states of two qubits. Our results apply to both complex-valued and real-valued invariants.
For a finitely dominated Poincaré duality space $M$, we show how the author's total obstruction to the existence of a Poincaré embedding of the diagonal map $M \to M \times M$ relates to the Reidemeister trace of the identity map of $M$. We also show that if the dimension of $M$ is even and at least four, and if $π_1(M)$ is a finite direct product of cyclic groups of order two, then the diagonal map admits a Poincaré embedding.
In this short paper, we give two proofs that the Euler characteristic is multiplicative, for fiber sequences of finitely dominated spaces. This is equivalent to proving that the Becker-Gottlieb transfer is functorial on $π_0$.
The object of this paper is to show that non-homotopy finite Poincaré duality spaces are plentiful. Let $π$ be finitely presented group. Assuming that the reduced Grothendieck group $\tilde K_0(\Bbb Z[π])$ has a non-trivial 2-divisible element, we construct a finitely dominated Poincaré space $X$ with fundamental group $π$ such that $X$ is not homotopy finite. The dimension of $X$ can be made arbitrarily large. Our proof relies on a result which says that every finitely dominated space possesses a stable Poincaré duality thickening.
We establish a number of foundational results on Poincaré spaces which result in several applications. One application settles an old conjecture of C.T.C. Wall in the affirmative. Another result shows that for any natural number n, there exists a finite CW pair $(X,Y)$ satisfying relative Poincaré duality in dimension n with the property that $Y$ fails to satisfy Poincaré duality. We also prove a relative version of a result of Gottlieb about Poincaré duality and fibrations.
The aim of this note is to construct a probability measure on the space of trajectories in a continuous time Markov chain having a finite state diagram, or more generally which admits a global bound on its degree and rates. Our approach is elementary. Our main intention is to fill a gap in the literature.
We introduce the notion of a protocol, which consists of a space whose points are labeled by real numbers indexed by the set of cells of a fixed CW complex in prescribed degrees, where the labels are required to vary continuously. If the space is a one-dimensional manifold, then a protocol determines a continuous time Markov chain. When a homological gap condition is present, we associate to each protocol a 'characteristic' cohomology class which we call the hypercurrent. The hypercurrent comes in two flavors: one algebraic topological and the other analytical. For generic protocols we show that the analytical hypercurrent tends to the topological hypercurrent in the low temperature limit. We also exhibit examples of protocols having nontrivial hypercurrent.
We generalize to higher algebraic $K$-theory an identity (originally due to Milnor) that relates the Reidemeister torsion of an infinite cyclic cover to its Lefschetz zeta function. Our identity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zeta function of such a family. We also exhibit several examples of families of endomorphisms having non-trivial endomorphism torsion.