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?

It looks to me that Madsen-Tillmann bordism spectrum is a close relative of Thom spectrum. So what will be the comparison, differences, similarity between the two Homotopy classes above?

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)