If the dimension of a vector space is odd, then all (orientation-preserving) rotations in odd dimensions fix some axis. Many of the differences between even-dimensional and odd-dimensional geometry relate to this fact. For example,

• The lack of symplectic structure in odd dimensions follows from the Lie-algebra version of the above statement: all odd-dimensional antisymmetric maps are degenerate.
• The $-1$ map doesn't fix any axis, so it cannot be orientation-preserving in odd dimensions.
• Synge's theorem states that if $M$ is compact, Riemannian, and has positive sectional curvature, then there is a conclusion which depends on the pairity of its dimension. The proof makes essential use of the above fact. (See Lemma 3.8 in "Riemannian Geometry" by do Carmo.)