nLab MTSO

Idea

The special orthogonal Thom spectrum $MSO$ is obtained from the orientable real universal line bundles, which arise as limits from the orientable 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 $MTSO(k)$.

Definition

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

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

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

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

Galatius-Madsen-Tillmann-Weiss theorem

Theorem

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

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

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

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

Homotopy groups

	$\pi_{-4}$	$\pi_{-3}$	$\pi_{-2}$	$\pi_{-1}$	$\pi_0$	$\pi_1$	$\pi_2$	$\pi_3$	$\pi_4$
MTSO(2)	$1$	$1$	$\mathbb{Z}$	$1$	$\mathbb{Z}$	$1$	$\mathbb{Z}$	$\mathbb{Z}/24\mathbb{Z}$	$\mathbb{Z}$
MTSO(3)	$1$	$\mathbb{Z}$	$1$	$1$	$1$	$\mathbb{Z}$	?	?	?

(Gollinger 2016, Thrm. 3.0.8.)

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