# tetrahedron

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### any polyhedron having four plane faces

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The 3-simplices are pairs ([[phi].sub.123], [[phi].sub.023], [[phi].sub.013], [[phi].sub.012]| [[psi].sub.123], [[psi].sub.023], [[psi].sub.013], [[psi].sub.012]) consisting of a 3-simplex ([[phi].sub.123], [[phi].sub.023], [[phi].sub.013], [[phi].sub.012]) in [M.sub.1] and a 3-simplex ([[psi].sub.123], [[psi].sub.023], [[psi].sub.013],[[psi].sub.012]) in [M.sub.2] such that ([[phi].sub.ijk]|[[psi].sub.ijk]) is a 2-simplex in [M.sub.12] for each ijk, and the diagram
The case of [M.sub.12] is similar: it has the same 1-simplices as [M.sub.1], and once we impose commutativity of (3.5) on the 3-simplices, any 2-simplex ([phi]|[psi]) must obey (3.4) in order to have a 3-simplex with the boundary of [s.aub.1] ([phi]|[psi]).
consisting of a 3-simplex ([[phi].sub.123],[[phi].sub.023], [[phi].sub.013],[[phi].sub.012]|[[psi].sub.123],[[psi].sub.023],[[psi].sub.013],[[psi].sub.012]) of [M.sub.12] and a 3-simplex ([[phi]'.sub.123], [[phi]'.sub.023], [[phi]'.sub.013], [[phi]'.sub.012] | [[psi]'.sub.123], [[psi].sub.023], [[psi]'.sub.013], [[psi]'.sub.012]) of [M.sub.34] for which ([[phi].sub.ijk]|[[psi].sub.ijk]||[[phi]'.sub.ijk]|[[psi]'.sub.ijk]) is a 2-simplex in M for every 0 [less than or equal to] i < j < k [less than or equal to] 3 and the diagrams
For each [a.sub.01], [a.sub.02] and [a.sub.12] [member of] [[pi].sub.2](X) we fix a singular 3-simplex r([a.sub.01], [a.sub.02], [a.sub.12]):[[DELTA].sub.3] [right arrow] X such that r([a.sub.12]) and [r([a.sub.01], [a.sub.02], [a.sub.12].sub.|[0,1,2]] = [r([a.sub.01],[a.sub.02], [a.sub.12].sub.|[0,2,3] = r([a.sub.02]), [r([a.sub.01], [a.sub.02], [a.sub.12]).sub.|[0,2,3]] = [r([a.sub.12]) and r([a.sub.01], [a.sub.02], [a.sub.12]).sub.|[0,1,3]] = r([a.sub.01][a.sub.02][a.sup.-1,sub.12]).
Moreover each 3-simplex of [[DELTA].sub.5] appears exactly twice (once for each orientation) in the following element of [[pi].sub.3](X).
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