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Stable Maps of 3‐Manifolds Into the Plane and their Quotient Spaces
Author(s) -
Motta Walter,
Porto Paulo,
Saeki Osamu
Publication year - 1995
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/s3-71.1.158
Subject(s) - mathematics , quotient , quotient space (topology) , pure mathematics , polyhedron , manifold (fluid mechanics) , space (punctuation) , plane (geometry) , type (biology) , combinatorics , mathematical analysis , geometry , mechanical engineering , ecology , linguistics , philosophy , engineering , biology
We study stable maps f : M → R 2 of closed orientable 3‐manifolds M into the plane using their quotient spaces, which are defined to be the spaces of the connected components of f ‐fibres and which are known to be 2‐dimensional polyhedra. We show that every 3‐manifold admits a stable map whose quotient space is homeomorphic to that of a stable map of S 3 We also deduce a Morse‐type inequality for stable maps, which implies that there is no universal stable map of S 3 in the above sense. Certain moves in the quotient spaces corresponding to generic homotopies of stable maps are also studied.