It is well known that odd and even dimensions work differently.

Waves propagation in odd dimensions is unlike propagation in even dimensions.

A parity operator is a rotation in even dimensions, but cannot be represented as a rotation in odd dimensions.

Even dimensional spheres do not admit continuous nonvanishing vector fields, but odd dimensional spheres do.

Is there some unifying reason for these differences? Why is dimensional parity so important?

(If you know of any more interesting differences, please leave a comment and I will add them to the list.)

**Answer**

For *real* vector spaces, i.e. $\mathbb{R}^N$, one important difference between odd and even $N$ is that every real polynomial with *odd* order has a real zero, while there are real polynomials with *even* order that only have complex zeros.

Thus, a real invertible $N\times N$ matrix *always* has a non-zero eigenvector if $N$ is odd (because the characteristic polynomial has order $N$, and by the above has a zero). For even $N$, however, there are invertible real matrices with no non-zero eigenvector.

This means that in even-dimensional spaces, there are invertible linear mappings without an invariant one-dimensional subspace, while in odd-dimensional spaces, invertible linear mappings always have a one-dimensional invariant subspace. If you look only at rotations, you get that rotations in an odd-dimensional space always keep at least a line fixed, while in an even-dimensional space they do not.

That’s a pretty huge geometric difference between odd- and even-dimensional spaces, that should explain a least some of the items in the question.

**Attribution***Source : Link , Question Author : Potato , Answer Author : fgp*