On the spin bordism of \(B(E_ 8\times{} E_ 8)\) (Q1182732)

The investigation is motivated by needs of heterotic string theory [see \textit{E. Witten} ``Topological tools in ten dimensional physics'', and ``Unification in ten dimensions'' in Unified string theories Workshop, Santa Barbara/Calif. 1985, 400-437, 438-456 (1986; Zbl 0645.00001)]. Using the equivalence between cohomology of \(BE_ 8\) and \(K(Z,4)\) through dimension 15, the author calculates spin bordism groups of \(K(Z,4)\wedge K(Z,4)\) in dimensions less than 12. Basic (for the calculations) Corollary 5 is proved by application of the Atiyah- Hirzebruch spectral sequence and Steenrod operations. Finally, Corollary 7 states that \(\Omega_{11}^{\text{Spin}}\bigl(B(E_ 8\times E_ 8)\bigr)=0\), which is a simple consequence of the previous calculations.