Manifolds with boundary that admit submersions with definite folds into R^n have restricted diffeomorphism types and Euler characteristics when boundary folds satisfy round or image-simple conditions.
Hempel,Some3-manifold groups with the same finite quotients, arXiv:1409.3509v2, preprint
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On submersions with definite folds of manifolds with boundary into Euclidean spaces
Manifolds with boundary that admit submersions with definite folds into R^n have restricted diffeomorphism types and Euler characteristics when boundary folds satisfy round or image-simple conditions.