Hermitian structures on \(A_{\infty{}}\) ring spaces (Q1208678)

Let \(M\) be a compact manifold. Rationally, one can get geometric information about \(M\) from the Hermitian \(K\)-theory of the group ring \(\mathbb{Z}[\pi_ 1M]\) of the fundamental group \(\pi_ 1M\). This paper is the first of a series aimed at analogous results with only the prime 2 inverted. For this purpose the group ring \(\mathbb{Z}[\pi_ 1M]\) is replaced by the free infinite loop space \(Q(G_ +)\) on the loop group \(G\) of \(M\). This is an \(A_ \infty\) ring space with structures analogous to an involution and a central unit. The authors show that, after inverting 2, these structures can be replaced by genuine involutions with good properties. This produces what the authors call a Hermitian \(A_ \infty\) ring, which can be used as input for the machinery that they develop in later papers.