nLab string bordism

Idea

A string bordism is a B-bordism for the tangential structure ((B,f)-structure) being the string structure. Its bordism homology theory and cobordism cohomology theory are described by the Thom spectrum MString.

Definition

Let $M$ and $N$ be $n$-dimensional string manifolds with respective string structures $\tau_M\colon M\rightarrow BString(n)$ and $\tau_N\colon N\rightarrow BString(n)$. A $n+1$-dimensional string manifold $W$ with string structure $\tau_W\colon W\rightarrow BString(n+1)$ together with inclusions $i\colon M\hookrightarrow\partial W$ and $j\colon N\hookrightarrow\partial W$ so that:

$$\partial W =i(M)+j(N);$$

$$\mathcal{B}k\circ\tau_M =\tau_W\circ i;$$

$$\mathcal{B}k\circ\tau_N =\tau_W\circ j$$

with the canonical inclusion $k\colon String(n)\rightarrow String(n+1)$ is a string bordism between $M$ and $N$. It is fully denoted by $(W,M,N,i,j)$, but usually $W$ is sufficient from context.

String bordism groups

Under the equivalence relation of string bordism, all $n$-dimensional closed string manifolds form the string bordism group $\Omega_n^String$, which has the disjoint union as composition, the empty manifold as neutral element and the inversion of orientation as inversion. According to Thom's theorem, string bordism groups are exactly the stable homotopy groups of the Thom spectrum MString:

$$\Omega_n^String \cong\pi_n MString =\lim_{k\rightarrow\infty}\pi_{n+k}MString_k.$$

Since $BString=BO\langle 8\rangle$ is $7$-connected, the first seven string bordism groups ($0\leq n\leq 6$) coincide with the framed bordism groups?:

• $\Omega_0^String\cong\Omega_0^fr\cong\mathbb{Z}$
• $\Omega_1^String\cong\Omega_1^fr\cong\mathbb{Z}_2$
• $\Omega_2^String\cong\Omega_2^fr\cong\mathbb{Z}_2$
• $\Omega_3^String\cong\Omega_3^fr\cong\mathbb{Z}_24$
• $\Omega_4^String\cong\Omega_4^fr\cong 1$
• $\Omega_5^String\cong\Omega_5^fr\cong 1$
• $\Omega_6^String\cong\Omega_6^fr\cong\mathbb{Z}_2$

Further string bordism groups include:

• $\Omega_7^String\cong 1$
• $\Omega_8^String\cong\mathbb{Z}\oplus\mathbb{Z}_2$
• $\Omega_9^String\cong\Theta_9/bP_10\cong\mathbb{Z}_2^2$ with the Kervaire-Milnor group $\Theta_9\cong\mathbb{Z}_2^2$ of exotic 9-spheres and the subgroup $bP_10\cong 1$ of those bounding stably parallelizable 10-manifolds
• $\Omega_10^String\cong\mathbb{Z}_6$, generated by an exotic 10-sphere
• $\Omega_11^String\cong 1$, which is used in M-theory
• $\Omega_12^String\cong\mathbb{Z}$
• $\Omega_13^String\cong\mathbb{Z}_3$, generated by an exotic 13-sphere
• $\Omega_14^String\cong\mathbb{Z}_2$, generated by the exotic 14-sphere
• $\Omega_15^String\cong\mathbb{Z}_2$, generated by an exotic 15-sphere
• $\Omega_16^String\cong\mathbb{Z}^2$
• $\Omega_27^String\not\cong 1$

String bordism ring

All string bordism groups in a direct sum form the string bordism ring:

$$\Omega^String \coloneqq\bigoplus_{n\in\mathbb{N}}\Omega_n^String,$$

which has the cartesian product as additional composition and the singleton as an additional neutral element.

Properties

Proposition

Every $n$-dimensional string manifold is string bordant to a $\min\left\{7,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected string manifold, equivalently meaning that every string bordism homology class in $\Omega_n^String$ can be represented by such a string manifold. (For $n\geq 15$, the result stabilizes at a 7-connected string manifold.)

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

Proposition

For $n$-dimensional $\min\left\{7,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected string manifolds $M$ and $N$, a string bordism $W\colon M\rightsquigarrow N$ exists with $M\hookrightarrow W$ also $\min\left\{7,\left\lceil\frac{n}{2}-1\right\rceil\right\}$-connected.

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

Proposition

If a $n$-dimensional $k$-connected compact string manifold $M$ with $k\leq 6$ and $n\geq 2k+3$ is string bordant to another compact string manifold $N$, then $M$ can be obtained from $N$ by surgery of codimension at least $k+2$.

(Botvinnik & Labbi 14, Prop. 3.4 & Crl. 3.5)

Related concepts

• Witten genus

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

• Manifold Atlas, String bordism

About general $BO\langle\ell\rangle$ bordisms:

• 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]

Discussion of relation to the Witten genus:

• Carl McTague, The Cayley plane and String bordism, Geom. Topol. 18 (2014) 2045-2078 (arXiv:1111.4520)

Discussion of secondary characteristic classes in string cobordism cohomology theory and in tmf:

• Ulrich Bunke, Niko Naumann, Secondary Invariants for String Bordism and tmf, Bulletin des Sciences Mathématiques, Volume 138, Issue 8, December 2014, Pages 912-970 (doi:10.1016/j.bulsci.2014.05.002)

String bordism of the classifying space of $E_8$:

• Michael Hill, The $String$ bordism of $B E_8$ and $B E_8 \times B E_8$ through dimension 14, Ill. J. Math. 53 1 (2009) 183-196 [doi:10.1215/ijm/1264170845]

Discussion of geometric string bordism in degree 3 as a means to speak (via the Pontryagin-Thom theorem) about the third stable homotopy group of spheres:

• Domenico Fiorenza, Eugenio Landi, Integrals detecting degree 3 string cobordism classes [arXiv:2209.12933]

• Domenico Fiorenza, String bordism invariants in dimension 3 from $U(1)$-valued TQFTs, talk at QFT and Cobordism, CQTS (Mar 2023) [web]