How does the parity of $n$ affect the properties of $\mathbb{R}^n$? 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?
 A: The hairy ball theorem states that there is no nonvanishing continuous tangent vector field on even-dimensional spheres.
A: 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.)

A: Let me try to focus the question as follows: Suppose you wake up in a dark and empty space. Which properties can help you decide whether the space is even or odd-dimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odd-dimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharply defined wavefronts need an odd number of spatial dimensions, in even-dimensional space the wavefront decays with a long tail. (The study of this difference goes back to Volterra and Hadamard.)
So to test for even/odd dimensionality you only need to shout in empty space. If you hear an echo you live in an even number of spatial dimensions.
