A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper “Assembly maps in bordism-type theories”, especially Sections 3 and 6.

Laures and McClure explain how to promote these simplicial sets to strictly commutative symmetric ring spectra. See their papers “Multiplicative properties of Quinn spectra” and “Commutativity properties of Quinn spectra”.

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper “Assembly maps in bordism-type theories”, especially Sections 3 and 6.

Laures and McClure explain how to promote these simplicial sets to strictly commutative symmetric ring spectra. See their papers “Multiplicative properties of Quinn spectra” and “Commutativity properties of Quinn spectra”.

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper “Assembly maps in bordism-type theories”, especially Sections 3 and 6.

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis.

Very roughly, Ω^∞MG is a simplicial set whose n-simplices are n-dimensional G-manifolds with corners, with corners of codimension i assigned to simplicial faces of Δ^n of codimension i.

For precise definitions and statements see Quinn's paper “Assembly maps in bordism-type theories”, especially Sections 3 and 6.

Laures and McClure explain how to promote these simplicial sets to strictly commutative symmetric ring spectra. See their papers “Multiplicative properties of Quinn spectra” and “Commutativity properties of Quinn spectra”.