nLab 2-oriented bordism

Idea

A 2-oriented bordism is a B-bordism for the tangential structure ((B,f)-structure) being the 2-orientation. Its bordism homology theory and cobordism cohomology theory are described by the Thom spectrum M2-Orient. Its definition is fully analogous to that of similar bordisms like oriented bordism. Similarily, there are 2-oriented bordism groups and the 2-oriented bordism ring:

$$\Omega_n^{2\text{-}Orient} \coloneqq\pi_n M2\text{-}Orient =\lim_{k\rightarrow\infty}\pi_{n+k} M2\text{-}Orient_k;$$

$$\Omega^{2\text{-}Orient} \coloneqq\bigoplus_{n\in\mathbb{N}}\Omega_n^{2\text{-}Orient}.$$

Properties

Proposition

Every $n$-dimensional 2-orient manifold is 2-orient bordant to a $\min\left\{9,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected 2-orient manifold, equivalently meaning that every 2-orient bordism homology class in $\Omega_n^{2\text{-}Orient}$ can be represented by such a 2-orient manifold. (For $n\geq 19$, the result stabilizes at a 9-connected 2-orient manifold.)

(Botvinnik & Labbi 14, Lem. 3.2 (1))

Proposition

For $n$-dimensional $\min\left\{9,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected 2-orient manifolds $M$ and $N$, a 2-orient bordism $W\colon M\rightsquigarrow N$ exists with $M\hookrightarrow W$ also $\min\left\{9,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected.

(Botvinnik & Labbi 14, Lem. 3.2 (2))

Proposition

If a $n$-dimensional $k$-connected compact 2-orient manifold $M$ with $k\leq 8$ and $n\geq 2k+3$ is 2-orient bordant to another compact 2-orient manifold $N$, then $M$ can be obtained from $N$ by surgery of codimension at least $k+2$.

(Botvinnik & Labbi 14, Prop. 3.4)

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

• Hisham Sati, Ninebrane structures (arXiv:1405.7686)

• Boris Botvinnik, Mohammed Labbi, Highly connected manifolds of positive $p$-curvature, Transactions of the AMS, Trans. Amer. Math. Soc. 366 (2014), 3405-3424 [arXiv:1201.1849, doi:10.1090/S0002-9947-2014-05939-4]