This is a following up question of Sphere spectrum, Character dual and Anderson dual.
What are the differences and the significances of the following:
(1). Homotopy classes of maps from a Thom spectrum to a shift of the Anderson dual to the sphere spectrum?
(2). Homotopy classes of maps from a Madsen-Tillmann bordism spectrum to a shift of the Anderson dual to the sphere spectrum?
p.s. I suppose my Journal Article links above/below for two spectra are 100% correct. Please correct me if I am imprecise or I miss the Refs.
R. Thom, Commentarii Mathematici Helvetici 28, 17 (1954).
S. Galatius, I. Madsen, U. Tillmann, and M. Weiss, Acta Math. 202, 195 (2009)