nLab MTO

Idea

The orthogonal Thom spectrum $MO$ is obtained from the real universal line bundles, which arise as limits from the real tautological bundles?. Taking their orthogonal complement (which are transverse) with respect to the canonical scalar product then produces different Thom spaces and different suspension spectra $MTO(k)$.

Definition

Let $\gamma_\mathbb{R}^{k,n}\twoheadrightarrow Gr_k(\mathbb{R}^{n+k})$ be a real tautological vector bundle of rank $k$ and ${\gamma_\mathbb{R}^{k,n}}^\perp\twoheadrightarrow Gr_k(\mathbb{R}^{n+k})$ be its orthogonal complement of rank $n$ with respect to the standard scalar product, then:

$$MTO(k,n) \coloneqq\mathbf{Th}\left({\gamma_\mathbb{R}^{k,n}}^\perp\right) =\Sigma^{\infty-n}Th\left({\gamma_\mathbb{R}^{k,n}}^\perp\right)$$

(Gollinger 2016, Def. 1.2.1.)

There is a canonical vector bundle homomorphism ${\gamma_\mathbb{R}^{k,n}}^\perp\oplus\underline{\mathbb{R}}\rightarrow{\gamma_\mathbb{R}^{k,n+1}}^\perp$ (obtained as the pullback along the canonical inclusion $Gr_k(\mathbb{R}^{n+k})\hookrightarrow Gr_k(\mathbb{R}^{n+k+1}), V\mapsto V\times\{0\}$) and applying the Thom space yields a continuous map $\Sigma Th\left({\gamma_\mathbb{R}^{k,n}}^\perp\right)\rightarrow Th\left({\gamma_\mathbb{R}^{k,n+1}}^\perp\right)$ and further induces a spectrum homomorphism $MTO(k,n)\rightarrow MTO(k,n+1)$. Its limit is denoted:

$$MTO(k) \coloneqq\lim_{n\rightarrow\infty}MTO(k,n).$$

Galatius-Madsen-Tillmann-Weiss theorem

Theorem

(Galatius-Madsen-Tillmann-Weiss theorem) The loop space of the classifying space of the cobordism category $Cob_k$ is weakly homotopy equivalent to the infinite loop space of $MTO(k)$, hence:

$$\Omega|N Cob_k| \simeq\Omega^\infty MTO(k).$$

(Alternatively it is written as $|N Cob_k|\simeq\Omega^{\infty-1}MTO(k)$.)

(Galatius, Madsen, Tillmann & Weiss 2009, Main Theorem)

Related concepts

flavors of bordism homology theories/cobordism cohomology theories, their representing Thom spectra and cobordism rings:

bordism theory$\;$M(B,f) (B-bordism):

• MFr

• MO, MSO, MSpin, MSpin^c, MSpin^h MString, MFivebrane, M2-Orient, M2-Spin, MNinebrane (see also pin⁻ bordism, pin⁺ bordism, pinᶜ bordism, spin bordism, spinᶜ bordism, spinʰ bordism, string bordism, fivebrane bordism, 2-oriented bordism, 2-spin bordism, ninebrane bordism)

• MU, MSU, MΩΩSU(n)

• MP, MR

• MSp

• MTO, MTSO

• relative bordism theories: MOFr, MUFr, MSUFr

• equivariant bordism theory: equivariant MFr, equivariant MO, equivariant MU

• global equivariant bordism theory: global equivariant mO, global equivariant mU

• algebraic: algebraic cobordism

References

• Søren Galatius, Ib Madsen, Ulrike Tillmann, and Michael Weiss, The homotopy type of the cobordism category, Acta Math. 202 2 (2009) 195–239 [arXiv:math/0605249]
• William Gollinger, Madsen-Tillmann-Weiss spectra and a signature problem for manifolds (2016), pdf