Celebratio Mathematica — Ranicki

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M. Borodzik, A. Némethi, and A. Ranicki: Codimension 2 embeddings, algebraic surgery and Seifert forms. Preprint, November 2012. ArXiv 1211.5964 techreport

We study the cobordism of manifolds with boundary, and its applications to codimension 2 embeddings \( M^m \subset N^{m+2} \), using the method of the algebraic theory of surgery. The first main result is a splitting theorem for cobordisms of algebraic Poincaré pairs, which is then applied to describe the behaviour on the chain level of Seifert surfaces of embeddings \( M^{2n-1} \subset S^{2n+1} \) under isotopy and cobordism. The second main result (update: which is false) is that the \( S \)-equivalence class of a Seifert form is an isotopy invariant of the embedding, generalizing the Murasugi–Levine result for knots and links. The third main result is a generalized Murasugi–Kawauchi inequality giving an upper bound on the difference of the Levine–Tristram signatures of cobordant embeddings.

András Némethi	Related
Maciej Borodzik	Related

M. Borodzik, A. Némethi, and A. Ranicki: “On the semicontinuity of the \( \mod 2 \) spectrum of hypersurface singularities,” J. Algebr. Geom. 24 : 2 (2015), pp. 379–398. MR 3311588 Zbl 1318.14005 ArXiv 1210.0798 article

András Némethi	Related
Maciej Borodzik	Related

M. Borodzik, A. Némethi, and A. Ranicki: “Morse theory for manifolds with boundary,” Algebr. Geom. Topol. 16 : 2 (2016), pp. 971–1023. MR 3493413 Zbl 1342.57018 ArXiv 1207.3066 article

András Némethi	Related
Maciej Borodzik	Related