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Note on the mappings of complex projective spaces. (English) Zbl 0958.57029
Let \(\mathbb CP^s\) denote complex projective \(s\)-space. Let \(n\) be even and \(2k\leq n\). This note contributes to the study of singularities of smooth maps \(\mathbb CP^n \rightarrow \mathbb CP^{n+k-1}\).
The homotopy classification of (continuous) maps \(\mathbb CP^n \rightarrow \mathbb CP^{n+k-1}\) is given by the integral cohomology group \(H^2(\mathbb CP^n; \mathbb Z)\). Given any smooth map \(f\: \mathbb CP^n \rightarrow \mathbb CP^{n+k-1}\) such that \(f^\ast (\operatorname {generator}) \neq \pm \operatorname {generator}\), it is known [S. Feder, Topology 4, 143-158 (1965; Zbl 0151.32301)] that \(f\) must have singular points (hence is not an immersion). It is proved here that not all of its singularities are the simplest possible, and that the same can be said about singularities of maps bordant to a nonzero (rational) multiple of \(f\).
The proof uses an earlier result on immersions due to the author [Math. Proc. Camb. Philos. Soc. 103, No. 1, 89-95 (1988; Zbl 0644.57015)].
MSC:
57R45 Singularities of differentiable mappings in differential topology
57N70 Cobordism and concordance in topological manifolds
57R20 Characteristic classes and numbers in differential topology
57N35 Embeddings and immersions in topological manifolds
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References:
[1] BOARDMAN J. M.: Singularities of differentiable maps. Inst. Hautes Études Sci. Publ. Math. 33 (1967), 21-57. · Zbl 0165.56803
[2] CONNER P. E.-FLOYD E. E.: Differentiable Periodic Maps. Ergeb. Math. Grenzgeb., Neue Folge, Band 33, Springer-Verlag, Berlin, 1964. · Zbl 0125.40103
[3] FEDER S.: Immersions and embeddings in complex projective spaces. Topology 4 (1965), 143-158. · Zbl 0151.32301
[4] SZŰCS A.: Immersions in bordism classes. Math. Proc. Cambridge Philos. Soc. 103 (1988), 89-95. · Zbl 0644.57015
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