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

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)

• If $X$ is any spectrum, $[X, \Sigma^n I\mathbb Z]$ fits into a short exact sequence as described in this MathOverflow answer -- in particular, it can be quickly determined from the homotopy groups of $X$. Therefore the question reduces to the difference between the homotopy groups of a Thom spectrum and the homotopy groups of a Madsen-Tillmann spectrum, and this MathOverflow question and answer is probably a good place to start for that. – Arun Debray Oct 12 '18 at 3:23
• A Madsen-Tillmann spectrum is'' a Thom spectrum! You may look at Section 2 of the following paper for a general construction: Søren Galatius and Oscar Randal-Williams. Stable moduli spaces of high-dimensional manifolds. Acta Math., 212(2):257–377, 2014. – user51223 Oct 13 '18 at 4:00

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.