Field theory configuration spaces for connective ko-theory
This paper is based on the author's thesis [``Connective one-dimensional Euclidean field theories'', Ph.D. Thesis, University of Notre Dame (2005)]. The author presents a new \(\Omega\)-spectrum for connective \(ko\)-theory by constructing connective covers (denoted by \(\text{Inf}_n\) below) of the spaces \(\mathcal{EFT}_n\) of certain types of euclidean fileld theories defined by \textit{S. Stolz} and \textit{P. Teichner} [London Mathematical Society Lecture Note Series 308, 247--343 (2004; Zbl 1107.55004)], which are weakly homotopy equivalent to the \((-n)\)-th space in an \(\Omega\)-spectrum associated to \(KO\). The key to the author's approach is reformulating these spaces \(\mathcal{EFT}_n\) in terms of their generating operators, or certain configurations associated to these operators. Let \(\mathcal{C}_n\) be the standard Clifford algebra associated to the positive definite inner product on \({\mathbb R}^n\) with generators \(e_1, \cdots, e_n\). For \(n > 0\), define \(\mathcal{C}_{-n}\) to be the Clifford algebra with respect to the negative definite inner product with generators \(f_1, \cdots, f_n\) which anti-commute pairwise and square to \(+1\). Also fix a separable real Hilbert space \(H_n\) with compatible right \(\mathcal{C}_{n-1}\)-module structure, which contains all irreducible \(\mathcal{C}_{n-1}\)-modules infinitely many times. Then when viewing \(\mathcal{C}_n\) as a \({\mathbb Z}/2\)-graded algebra such that the even part \(\mathcal{C}^{\text{ev}}_n\) is isomorphic to \(\mathcal{C}_{n-1}\), one obtains a \({\mathbb Z}/2\)-graded \(\mathcal{C}_n\)-module \(\mathcal{H}_n= H_n\otimes_{\mathcal{C}^{\text{ev}}_n}\mathcal{C}_n\). Denote by \(\mathrm{Inf}_n\) the set of all odd, self-adjoint operators on the closures of their domains in \(\mathcal{H}_n\) which have compact resolvent and are \(\mathcal{C}_n\)-linear. Firstly it is shown that for all \(n\), \(\mathcal{EFT}_n\) become homotopy equivalent to \(\mathrm{Inf}_n\) (nicely topologized) (Lemma 3.7). Here one considers the right \(\mathcal{C}_{n+k}\)-module \(\mathcal{H}_{n+k}\) as \(\mathcal{C}_{-k}-\mathcal{C}_n\)-bimodule (denoted by \(\mathcal{H}^k_n\)) by replacing the last \(k\) generators \(e_{k+1}, \cdots, e_{n+k}\) of \(\mathcal{C}_{n+k}\) acting from the right by generators \(f_{k+1}, \cdots, f_{n+k}\) of \(\mathcal{C}_{-k}\) acting from the left. Denote then by \(\mathrm{Inf}^k_n\) the space consisting of all \(\mathcal{C}_n\)-linear operators \(G\) of type \(\mathrm{Inf}_n\) defined on \(\mathcal{H}^k_n\) such that their squares \(G^2\) also are \(\mathcal{C}_{-k}-\mathcal{C}_n\)-linear. The author's main theme is to prove that for \(k\geq 0\) and all \(n\), there are weak homotopy equivalences \(\mathrm{Inf}^k_n \sim \Omega \mathrm{Inf}^{k+1}_{n-1}\) (Theorem 4.1). From this it follows that \(\mathrm{Inf}^k_n\) become \((k-1)\)-connected covers of \(\mathrm{Inf}_n\), that is, \(\mathrm{Inf}^k_n \sim \mathrm{Inf}_n\langle k \rangle\) (Corollary 4.3). Thus one can obtain the desired spaces by setting \(\mathrm{inf}_n = \mathrm{Inf}^n_n\). The proof is done by constructing quasi-fibrations with contractible total spaces. Such a quasi-fibration \(F \to E \to B\) has a property such that \(F \sim \Omega B\). The weak homotopy equivalences above are obtained by applying this formula to the quasi-fibrations given there.
scientific article; zbMATH DE number 5731285
scientific article