Collar neighbourhood

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In topology, a branch of mathematics, a collar neighbourhood of a manifold with boundary M {\displaystyle M} is a neighbourhood of its boundary  ∂ M {\displaystyle \partial M} that has the same structure as  ∂ M × [ 0 , 1 ) {\displaystyle \partial M\times [0,1)} .

Formally, if  M {\displaystyle M} is a differentiable manifold with boundary,  U ⊂ M {\displaystyle U\subset M} is a collar neighbourhood of  M {\displaystyle M} whenever there is a diffeomorphism f : ∂ M × [ 0 , 1 ) → U {\displaystyle f:\partial M\times [0,1)\to U} such that for every  x ∈ ∂ M {\displaystyle x\in \partial M} ,  f ( x , 0 ) = x {\displaystyle f(x,0)=x} .[1]: p. 222  Since  [ 0 , 1 ) {\displaystyle [0,1)} is diffeomorphic to  [ 0 , ∞ ) {\displaystyle [0,\infty )} , it is equivalent to take a diffeomorphism  f : ∂ M × [ 0 , ∞ ) → U {\displaystyle f:\partial M\times [0,\infty )\to U} .[2]: §6

Every differentiable manifold has a collar neighbourhood.[1]: Th. 9.25 [2]: Th. 4.6.1

## References

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1. ^ a b Lee, John (2012), Introduction to Smooth Manifolds, Graduate Texts in Mathematics, vol. 218, Springer, ISBN 9781441999825
2. ^ a b Hirsch, Morris W. (1976). Differential topology. New York Heidelberg Berlin: Springer-Verlag. ISBN 978-1-4684-9449-5.

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