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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Handle attaching on generic maps


Author: Youn W. Lee
Journal: Trans. Amer. Math. Soc. 279 (1983), 77-94
MSC: Primary 57R40; Secondary 57R42, 57R65
DOI: https://doi.org/10.1090/S0002-9947-1983-0704603-0
MathSciNet review: 704603
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Abstract: Using the handle attaching technique along the singular value set of generic maps in the stable range together with the handle subtraction of Haefliger, smooth immersions and embeddings are studied. We generalize Whitney's immersion theorem, and Haefliger and Hirsh's result on embedding and classification of embeddings of $ k$-connected ($ (k + 1)$-connected for the classification) smooth $ n$-manifolds into $ {{\mathbf{R}}^{2n - k}}$. For example, we obtain the following as a generalization of Whitney's immersion theorem. If $ f: {V^n} \to {M^m}, {3n} < {2m}$, is a generic map such that each component of its double point set is either a closed manifold or diffeomorphic to the $ (2n - m)$-disk, then $ f$ is homotopic to an immersion.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0704603-0
Article copyright: © Copyright 1983 American Mathematical Society