Sphere spectrum, Thom spectrum, and Madsen-Tillmann bordism spectrum

Question

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)

Answer 1

I'm a little confused by your question. You seem to be implying that the Madsen-Tillmann spectra are not Thom spectra, but this is not true: the definition of the spectrum $MTG(n)$ (for $G = O,SO,U$) is as the Thom spectrum of the virtual bundle $-\gamma_n$ over $BG(n)$, where $\gamma_n$ is the universal bundle. In general, computing $[X^\mu, I_\mathbf{Z}]$ for $X^\mu$ a Thom spectrum is equivalent to (the nontrivial task of) computing $\pi_\ast(X^\mu)$. Generally, one calculates this via the Adams spectral sequence (think back, e.g., to the original calculation of $\pi_\ast(MU)$). See https://arxiv.org/pdf/1801.07530.pdf and https://arxiv.org/pdf/1708.04264.pdf for nice expositions.