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In trying to begin to learn basic homological algebra, i am confronted with orientation of simplices. The definition seems unmotivated and unintuitive: for $n$-simplices with $n \in \{-1,0,1,2\}$, it is intuitive to visualize: intuitively, given a 2-simplex, we can walk to the left or to the right. For $n=3$ and above, the definition seems arbitrary. Is there a "geometric intuition" for this?

Definition of orientation: two (ordered) simplices with same vertices have same orientation if one is the componentwise image of an even permutation of the other. Else they are said to have opposite orientation.

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  • $\begingroup$ Could you please add to the question your definition of orientation? (This will presumably entail including the definition of a simplex as well.) $\endgroup$ Dec 8, 2021 at 1:12
  • $\begingroup$ @AndrewD.Hwang added $\endgroup$ Dec 8, 2021 at 15:51
  • $\begingroup$ Here's an attempt at geometric intuition: Assuming the standard $n$-simplex is the convex hull of the standard basis vectors in $\mathbf{R}^{n+1}$, swapping one pair of vertices corresponds to a reflection, which "reverses orientation," while swapping two pairs of vertices corresponds to a rotation. Since a composition of rotations is a rotation, and the composition of a rotation and a reflection is a reflection, a vertex permutation preserves orientation iff the permutation is even. <> All this presumes a certain underlying belief that permutations have an invariant parity.... $\endgroup$ Dec 8, 2021 at 22:52
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    $\begingroup$ @AndrewD.Hwang This makes me realize i don't have intuition for orientation in $\mathbb R^n$ for $n>1$. Maybe i should ask another question about that. It still seems like a strange thing to do, to associate an orientation to simplices, but for $n=1$ it makes sense to me because of analogy with closed paths. $\endgroup$ Dec 9, 2021 at 16:50
  • $\begingroup$ Possibly of interest: geometric meaning of orientation and higher-dimensional analogues of handedness $\endgroup$ Dec 9, 2021 at 21:00

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