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12 - Tubular Neighbourhoods and Transversality
Gereon Quick, Norwegian University of Science and Technology, Trondheim
Book:

Differential Topology

Published online:

08 August 2026

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03 September 2026, pp 200-226
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Summary

This chapter introduces the normal bundle of a submanifold and shows that it is a vector bundle. We then prove the existence of tubular neighbourhoods, demonstrating that every submanifold has an open neighbourhood diffeomorphic to an open neighbourhood of the zero section in its normal bundle, a fundamental result in differential topology. Several applications follow: we prove Whitney’s Approximation Theorem, showing that continuous maps between smooth manifolds can be approximated by smooth maps; we establish Brouwer’s Invariance of Domain for smooth injective maps; and we prove Ehresmann’s Fibration Theorem, which states that submersions form locally trivial fibrations. Finally, we study Thom’s transversality theorems on the existence of smooth maps transverse to given submanifolds.

A Vanishing Theorem for the Twisted Normal Bundle of Curves in $\mathbb{P}^{n}$, $n\geqslant 8$
Part of
- Curves
E. Ballico
Journal:

Canadian Mathematical Bulletin / Volume 63 / Issue 1 / March 2020

Published online by Cambridge University Press:

30 August 2019, pp. 1-5

Print publication:

March 2020
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We prove the existence of a smooth and non-degenerate curve $X\subset \mathbb{P}^{n}$, $n\geqslant 8$, with $\deg (X)=d$, $p_{a}(X)=g$, $h^{1}(N_{X}(-1))=0$, and general moduli for all $(d,g,n)$ such that $d\geqslant (n-3)\lceil g/2\rceil +n+3$. It was proved by C. Walter that, for $n\geqslant 4$, the inequality $2d\geqslant (n-3)g+4$ is a necessary condition for the existence of a curve with $h^{1}(N_{X}(-1))=0$.

(Z 2)k -Actions Fixing a Product of Spheres and a Point
Pedro L. Q. Pergher
Journal:

Canadian Mathematical Bulletin / Volume 38 / Issue 3 / 01 September 1995

Published online by Cambridge University Press:

20 November 2018, pp. 366-372

Print publication:

01 September 1995
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In the paper we identify up to bordism all manifolds with (Z 2)k -action whose fixed point set is Sn × Sm U point.

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