TITLE: How does the parity of $n$ affect the properties of $\mathbb{R}^n$?
QUESTION [8 upvotes]: Does the parity of the dimension of $\mathbb{R}^n$ affect its structure/properties? As in, does it make a difference if $n$ is even or odd?

REPLY [4 votes]: 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.)