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?

1$\begingroup$ What kinds of properties specifically are you interested in? $\endgroup$ – j.c. Oct 1 '18 at 5:43

4$\begingroup$ There are many differences between even and odd dimensions, but without more details it is not clear what direction you want answers. Examples: Complex/symplectic structures do not exist on odd dimensional spaces. Contact structures do not exist on even dimensional spaces. $\endgroup$ – Thomas Rot Oct 1 '18 at 5:54

1$\begingroup$ Even dimensional manifolds are the basic framework for symplectic geometry, odd for contact geometry. $\endgroup$ – Parschallen Oct 1 '18 at 10:39

1$\begingroup$ This question seems a little broad. There are all kinds of differences one can point to. For example, the onepoint compactifications $S^n$ behave differently: the odddimensional ones have nonvanishing vector fields. (Also: why is number theory a tag, precisely?) $\endgroup$ – Todd Trimble♦ Oct 1 '18 at 13:17

$\begingroup$ I was originally thinking of the result that a polynomial of odd degree has at least one real root. In a certain sense, the structure of the object depends on the space it is defined in.This got me curious about the relationship between the number of a dimension and the types of structures it can support.I was asking the question in regard to the tags on the question; i.e, how does dimension of space come up in Linear Algebra, Topology, etc. Number theory tagged specifically because I was curious if the number of the dimension being composite or prime would have any impact. $\endgroup$ – user2192320 Oct 1 '18 at 15:00
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 odddimensional?
In the context of wave propagation, there is the fundamental difference that Huygens principle is only valid in odddimensional space, see Wave Propagation in Even and Odd Dimensional Spaces. Sharply defined wavefronts need an odd number of spatial dimensions, in evendimensional 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.

5$\begingroup$ In odd dimensions $1$ is orientation reversing. $\endgroup$ – Liviu Nicolaescu Oct 1 '18 at 8:30

2$\begingroup$ The situation is a little bit more complicated. For instance, Huyghens principle does not apply in $1$space dimension, although $1$ is odd. $\endgroup$ – Denis Serre Oct 1 '18 at 10:29

$\begingroup$ @DenisSerre  indeed, the case $n = 1$ is special because a pulse has only one path to take, so the interference that produces the sharp wavefront does not happen; thanks for noting/correcting. $\endgroup$ – Carlo Beenakker Oct 1 '18 at 10:45

2$\begingroup$ This sounds like a great premise for a new Sawlike horror movie. A group of strangers wake up in a dark, empty space. They need to determine the dimension of the ambient space, but their time is running out! $\endgroup$ – Somatic Custard Oct 1 '18 at 13:07
The hairy ball theorem states that there is no nonvanishing continuous tangent vector field on evendimensional spheres.
If the dimension of a vector space is odd, then all (orientationpreserving) rotations in odd dimensions fix some axis. Many of the differences between evendimensional and odddimensional geometry relate to this fact. For example,
 The lack of symplectic structure in odd dimensions follows from the Liealgebra version of the above statement: all odddimensional antisymmetric maps are degenerate.
 The $1$ map doesn't fix any axis, so it cannot be orientationpreserving 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.)