The Cayley Plane and the Witten Genus

Elliptic cohomology is at the heart of many recent developments in topology. What led to its discovery was Ochanine's observation in the 1980s that there are many more multiplicative genera for spin fiber bundles than for oriented fiber bundles, one for each elliptic curve with a marked point of order 2. Given that multiplicative genera for spin manifolds have led to such unexpectedly rich developments, it seems reasonable to investigate multiplicative genera for other types of fiber bundles, for example O<8> fiber bundles. My investigation has led to new genera, one of which reveals a surprising synergy between the elliptic genus and the Witten genus. My investigation has also led to what turned out to be a rediscovery of a result first discovered by Rainer Jung in unpublished work in the early 1990s: the Witten genus equals bordism modulo Cayley plane bundles. This is analogous to Kreck-Stolz's result that the elliptic genus equals bordism modulo HP^2 bundles and Ochanine's result that the signature equals bordism modulo CP^2 bundles.