Necessary conditions for finite critical sets. Maps with infinite critical sets

Ioan Radu Peter, Cornel Pintea



We provide necessary conditions on a given map, between two compact differential manifolds, for its critical set to be finite. As consequences of these conditions we also provide several examples of pairs of compact differential manifolds such that every map between them has infinite critical set.


Critical points; homotopy groups; low dimensional manifolds

Full Text:



D. Andrica and L. Funar, On smooth maps with finitely many critical points, J. London Math. Soc. (2) 69 (2004), 783–800.

D. Andrica and L. Funar, On smooth maps with finitely many critical points. Addendum, J. London Math. Soc. (2) 73 (2006), 231–236.

P.T. Church and J.G. Timourian, Differentiable maps with 0-dimensional critical set I, Pacific J. Math. 41, No. 3 (1972), 615–630.

P.T. Church and J.G. Timourian, Maps with 0-dimensional critical set, Pacific J. Math. 57, No. 1 (1975), 59–66.

D.B.A. Epstein, Factorization of 3-manifolds, Comment. Math. Helv. 36 (1961), 91–102.

D.B.A. Epstein, Ends, Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice-Hall, Englewood Cliffs, N.J., 1962, 110–117.

L. Funar, Global classification of isolated singularities in dimensions (4, 3) and (8, 5), Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) Vol. X (2011), 819–861.

A. Hatcher, Algebraic Topology, Cambridge University Press, 2002.

J. Hempel, 3-Manifolds, Ann. of Math. Studies, no. 86. Princeton University Press, Princeton, N. J., 1976.

M.W. Hirsch, Differential Topology, Springer, 1976.

M.J. Micallef and J.D. Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227.

C. Pintea, Differentiable mappings with an infinite number of critical points, Proc. Amer. Math. Soc., Vol. 128, No. 11. (2000), 3435–3444.

C. Pintea, A measure of the deviation from there being fibrations between a pair of compact manifolds, Diff. Geom. Appl. 24 (2006), 579–587.

J.J. Rotman, An Introduction to Algebraic Topology, Springer Verlag, 1988.

J.-P. Serre, Homologie singulire des espaces fibrs. Applications, Ann. Math. 54 (3) (1951), 425–505.

S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399.

C.T.C. Wall, Classification problems in differential topology. V. On certain 6-manifolds, Invent. Math. 1 (1966), 355–374.


  • There are currently no refbacks.

Partnerzy platformy czasopism