TY - JOUR

T1 - Stable maps of 3-manifolds into the plane and their quotient spaces

AU - Motta, Walter

AU - Porto, Paulo

AU - Saeki, Osamu

N1 - Funding Information:
The first author is partially supported by CNPg. The third author is partially supported by CCInt and FAPESP. 1991 Mathematics Subject Classification: 57R45.

PY - 1995/7

Y1 - 1995/7

N2 - We study stable maps f: M → R2of 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 S3We also deduce a Morse-type inequality for stable maps, which implies that there is no universal stable map of S3in the above sense. Certain moves in the quotient spaces corresponding to generic homotopies of stable maps are also studied.

AB - We study stable maps f: M → R2of 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 S3We also deduce a Morse-type inequality for stable maps, which implies that there is no universal stable map of S3in the above sense. Certain moves in the quotient spaces corresponding to generic homotopies of stable maps are also studied.

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U2 - 10.1112/plms/s3-71.1.158

DO - 10.1112/plms/s3-71.1.158

M3 - Article

AN - SCOPUS:84959802067

SN - 0024-6115

VL - s3-71

SP - 158

EP - 174

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 1

ER -