On manifolds homotopy equivalent to the total spaces of $S^7$-bundles over $S^8$

Ajay Raj and Tibor Macko

Address:
Department of Algebra and Geometry, FMFI, Comenius University, Bratislava, SK-84248, Slovakia
Department of Algebra and Geometry, FMFI, Comenius University, Bratislava, SK-84248, Slovakia, and Institute of Mathematics, Slovak Academy of Sciences, Štefánikova 49, Bratislava, SK-81473, Slovakia

E-mail: tibor.macko@fmph.uniba.sk

Abstract: We calculate the structure sets in the sense of surgery theory of total spaces of bundles over eight-dimensional sphere with fibre a seven-dimensional sphere, in which manifolds homotopy equivalent to the total spaces are organized, and we investigate the question, which of the elements in these structure sets can be realized as such bundles.

AMSclassification: primary 19J25; secondary 55R25, 55R40, 57N55.

Keywords: vector bundle, sphere bundle over sphere, microbundle, homotopy equivalence, homeomorphism, surgery, characteristic class.

DOI: 10.5817/AM2024-3-125