Homotopical Topology by Anatoly Fomenko & Dmitry Fuchs

Author:Anatoly Fomenko & Dmitry Fuchs
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

B: Differential Topology Interpretations of the Euler Class

For a closed oriented manifold X, the value of the Euler class of the tangent bundle e(X) = e(τ X ) on the fundamental class [X] is equal to the Euler characteristic χ(X) of X (this is Proposition 2 of Sect. 18.5). This implies that a closed manifold possesses a nonvanishing vector field if and only if its Euler characteristic is zero (corollary in Sect. 18.5).

Some other properties of the Euler class are given here as exercises.

Exercise 19.

Prove that a closed manifold X (orientable or not) possesses a continuous family of tangent lines (equivalently: The tangent bundle τ(X) possesses a one-dimensional subbundle) if and only if χ(X) = 0.

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