String can be viewed as higher version of Spin, while the latter plays a fundamental role in Atiyah-Singer index theory. People try to develop parallel theory for String, the whole story of which is still mystery. Geometry and topology of String manifolds then attract increasing attentions and interests, while the ones of dimensional 24 are quite special among the String manifolds. In this talk, following Hirzebruch, Mahowald-Hopkins we will discuss the topological aspect of 24-dimensional String manifolds. In particular, we will give an integral basis for the String cobordism groups at dimension 24, and show divisibility of various characteristic numbers.

This is joint work with Fei Han (NUS) in progress. The initial motivation for this work is a question of Teichner, which will be also addressed and discussed.