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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?

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    $\begingroup$ What kinds of properties specifically are you interested in? $\endgroup$ – j.c. Oct 1 '18 at 5:43
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    $\begingroup$ 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
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    $\begingroup$ Even dimensional manifolds are the basic framework for symplectic geometry, odd for contact geometry. $\endgroup$ – Parschallen Oct 1 '18 at 10:39
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    $\begingroup$ This question seems a little broad. There are all kinds of differences one can point to. For example, the one-point compactifications $S^n$ behave differently: the odd-dimensional ones have non-vanishing 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
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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.

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    $\begingroup$ In odd dimensions $-1$ is orientation reversing. $\endgroup$ – Liviu Nicolaescu Oct 1 '18 at 8:30
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    $\begingroup$ 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
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    $\begingroup$ This sounds like a great premise for a new Saw-like 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
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The hairy ball theorem states that there is no nonvanishing continuous tangent vector field on even-dimensional spheres.

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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.)
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