On characteristic numbers of 24 dimensional string manifolds

In this paper, we study the Pontryagin numbers of 24 dimensional String manifolds. In particular, we find representatives of an integral basis of the String cobrodism group at dimension 24, based on the work of Mahowald and Hopkins (The structure of 24 dimensional manifolds having normal bundles which lift to BO [8], from “Recent progress in homotopy theory” (D. M. Davis, J. Morava, G. Nishida, W. S. Wilson, N. Yagita, editors), Contemp. Math. 293 , Amer. Math. Soc., Providence, RI, 89-110, 2002), Borel and Hirzebruch (Am J Math 80: 459–538, 1958) and Wall (Ann Math 75:163–198, 1962). This has immediate applications on the divisibility of various characteristic numbers of the manifolds. In particular, we establish the 2-primary divisibilities of the signature and of the modified signature coupling with the integral Wu class of Hopkins and Singer (J Differ Geom 70:329–452, 2005), and also the 3-primary divisibility of the twisted signature. Our results provide potential clues to understand a question of Teichner.