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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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closed as too broad by R. van Dobben de Bruyn, abx, Tom De Medts, Mateusz Kwaśnicki, Todd Trimble Oct 1 '18 at 13:19

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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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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