\magnification=\magstep2

{\bf Exercise} The map that assigns to an ordered simplex
$(u_0,...,u_m)$ the oriented simplex $u_0 \wedge ... \wedge u_m$
is a chain equivalence.

{\bf Solution}
For simplicity, let us write $u_0 ... u_m$ instead of $(u_0,...,u_m)$.
Put any total order on the set of vertices.
Define a map in the opposite direction by sending
$s_0 \wedge ... \wedge s_m$ to $s_0 ... s_m$ for
every strictly increasing sequence $s_0, ..., s_m$.

We need a chain homotopy $e$ such that
$$(de+ed)(u_0 ... u_m) = u_0 ... u_m - (-1)^\sigma s_0 ... s_m$$
when $s_0, ..., s_m$ is strictly increasing and
$u_0 \wedge ... \wedge u_m = (-1)^\sigma s_0 \wedge ... \wedge s_m$.

Let $$e(u_0,...,u_m) = \sum_{i=0}^m (-1)^{i+f(i)}
s_0 ... s_i (u_0 ... u_m \setminus s_0 ... s_{i-1})$$
where $(u_0 ... u_m \setminus s_0 ... s_{i-1})$ is the sequence
obtained from $u_0 ... u_m$ by cancelling the elements $s_0 ... s_{i-1}$,
and $f(i) = g(0) + ... + g(i-1)$, where $g(i)$ is the number of $j$
such that $j < i$ and $u_j > s_i$.

This choice works, as one easily verifies.

Example: 
$(de+ed)(3142) = d(13142+12342-12334+12344)+e(142-342+312-314) =
(3142-1142+1342-1312+1314)+(2342-1342+1242-1232+1234)-
(2334-1334+1234-1234+1233)+(2344-1344+1244-1234+1234)+
(1142-1242-1244)-(2342-2334+2344)+(1312+1232+1233)-(1314+1334-1344)
= 3142 + 1234$.

And $(de+ed)(121) = d(1121+1112-1122)+e(21-11+12) =
(121-121+111-112)+(112-112+112-111)-(122-122+112-112)+
(121+122)-(111-111)+(112-122) = 121+112$.

\bye

