nLab String bordism

Idea

String bordism theory, $MString$, is bordism homology theory for manifolds equipped with string structure.

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, MString, …

• MU, MSU, …

  MΩΩSU(n)

  MP, MR

• MSpin^c

• MSp

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

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]