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Morse theory for manifolds with boundary

Maciej Borodzik, András Némethi and Andrew Ranicki

Algebraic & Geometric Topology 16 (2016) 971–1023

We develop Morse theory for manifolds with boundary. Beside standard and expected facts like the handle cancellation theorem and the Morse lemma for manifolds with boundary, we prove that under suitable connectedness assumptions a critical point in the interior of a Morse function can be moved to the boundary, where it splits into a pair of boundary critical points. As an application, we prove that every cobordism of connected manifolds with boundary splits as a union of left product cobordisms and right product cobordisms.

Morse theory, manifold with boundary, cobordism, bifurcation of singular points
Mathematical Subject Classification 2010
Primary: 57R19
Secondary: 58E05, 58A05
Received: 23 March 2015
Accepted: 14 August 2015
Published: 26 April 2016
Maciej Borodzik
Institute of Mathematics
University of Warsaw
ul. Banacha 2
02-097 Warszawa
András Némethi
Alfréd Rényi Institute of Mathematics
Reáltanoda u. 13-15.
1053 Budapest
Andrew Ranicki
School of Mathematics
University of Edinburgh
Edinburgh EH9 3JZ