The free loop space and the algebraic K-theory of spaces
Let \(\Lambda(X)\) denote the free loop space of a space X, let \(\Sigma\) denote suspension, and \(Q=\Omega^{\infty}\Sigma^{\infty}\). Define \(B(X)=Q\Sigma (ES^ 1\times_{S^ 1} \Lambda (X))\), and \(\tilde B(X)=fibre(B(X)\to B(point))\). Let A(X) denote the algebraic K-theory of the space X, and \(\tilde A(X)=\text{fibre}(A(X)\to A(\text{point})).\) Let finally Y denote a connected space of the (based) homotopy type of a (based) CW complex. With this notation, the paper is addressed to the following three theorems. Theorem 1. There is a homotopy equivalence of infinite loop spaces, \[ \tilde B(\Sigma Y)=Q\Sigma (ES^ 1_+ \wedge_{S^ 1} \Lambda \Sigma Y)\simeq \tilde A(\Sigma Y). \] Theorem 2. There is a homotopy equivalence of infinite loop spaces, \[ \prod_{n\geq 1}Q(EZ_{n^+} \wedge_{Z_ n} Y^{(n)})\simeq Q(ES^ 1_+ \wedge_{S^ 1} \Lambda (\Sigma Y)) \] where \(Y^{(n)}\) denotes the n-fold smash product of Y, which is acted upon by \(Z_ n\) by cyclic permutation of coordinates. Theorem 3. There is a homotopy equivalence of infinite loop spaces, \[ \prod_{n\geq 1}Q(EZ_{n^+} \wedge_{Z_ n} Y^{(n)})\simeq \Omega \tilde A(\Sigma Y). \] Of these, theorem 1 results by combining the other two theorems. Theorem 2 was obtained by the first two authors [Comment. Math. Helv. 62, 423-449 (1987; Zbl 0632.57028)], and theorem 3 has been proved by the third author [unpublished; an account will prescordingly. Suppose L is a Lie algebra over k. Then the following are equivalent: (a) L is central simple of index I. (b) L is isomorphic to a Lie algebra \(K(A,-)\) associated to a division algebra \((A,-)\) which is a form of an \((8,m)\)-product algebra.'' The algebra \(K(A,-)\) is a \({\mathbb{Z}}\)-graded Lie algebra constructed from A, its set of skew elements and a subalgebra of the Lie algebra of the endomorphisms of A. Necessary and sufficient conditions are given for two algebras \(K(A,-)\) and \(K(A',-)\) to be isomorphic by means of the so called Albert forms of the algebras \((A,-)\) and \((A',-)\). Finally, the main result is used to obtain new proofs of some known descriptions of the central simple Lie algebras of index I over real closed fields, local fields, number fields and to describe these algebras over the fields of iterated Laurent series over the real field.
scientific article; zbMATH DE number 4060296
scientific article