On loop spaces of configuration spaces (Q2781380)

The authors analyse topological and homological decompositions for loop spaces of configuration spaces, as well as the twisting between the factors. They give explicit descriptions of loop spaces homologies as Hopf algebras.NEWLINENEWLINENEWLINEThe central object is the Lie algebra \(L_k(q)\). Denote by \(V_n(q)\) the free graded abelian group of rank \(n\) concentrated in degree \(q\) and generated \(\{B_{n+1,1},\dots, B_{n+1,n}\}\), and by \(L_k(q)\) the free graded Lie algebra generated by \(\bigoplus^{k-1}_{n=1} V_n(q)\) modulo the infinitesimal braid relations \(([B_{ij}, B_{st}]=0\) if \(\{i,j\}\cap \{s,t\}= \emptyset\), \([B_{ij},B_{it}+(-1)^q B_{tj}]=0\) and \([B_{tj}, B_{ij}+B_{it}] =0)\). Then, \(H_*(\Omega F(\mathbb{R}^m,k)\); \(\mathbb{Z})\cong\bigcup L_k(m-2)\) as a Hopf algebra. Let now \(M\) be a simply-connected punctured manifold \((M=M'\smallsetminus \{\text{point}\})\) of dimension \(\geq 3\) and let \(E_*\) be a homology theory that satisfies the strong form of the Künneth theorem for \((\Omega M)^k\). The embedding \(e:\mathbb{R}^n\to M\) induces then a monomorphism \(E_*\Omega F (\mathbb{R}^m, k)\to E_*\Omega F(M,k)\). Denote by \(s_i:M\to F(M,k)\) the natural sections, then \(E_*\Omega F(M,k)\) is generated as an algebra by the image of \(E(\Omega e)_*\) and the images of the \(E(\Omega s_i)_*\). The authors describe the relations between the respective images. A simply connected manifold \(M\) is called braidable if there exists a subspace \(A\subset M\) with two isotopic embeddings \(e_1,e_2:A\to M\) with disjoint images and such that the induced maps \(\Omega (e_i)\) induce surjections in homology. Suppose now that \(M\) is 1-connected, punctured, braidable, with primitively generated loop space homology and some additional hypothesis related to the characteristic, then \(H_*\Omega F(M,k)\) is a primitively generated Hopf algebra generated by the \(B_{ij}\) and the elements \(m_i=H_* (\Omega s_i)(m)\), \(m\in H_*(\Omega M)\) subject to some explicit relations. The authors deduce explicit decompositions for product spaces and Lie groups.