One way to define the cup product on a finite simplicial complex $K$ is as follows.
i) Choose a partial ordering on the vertex set of $K$ which induces a total ordering on the vertex set of any simplex. The $p$-th chain group $C_p(K)$ is then the free abelian group on the $p$-simplices of $K$ with vertices listed in increasing order and has the usual differential.
ii) The simplicial cup product $C^p(K) \times C^q(K) \xrightarrow{\cup} C^{p+q}(K)$ is given by the formula
$$(\phi \cup \psi)[v_{i_0},…,v_{i_{p+q}}] = \phi(v_{i_0},…,v_{i_p}) \cdot \psi(v_{i_p},…,v_{i_{p+q}})$$
where $[v_{i_0},…,v_{i_{p+q}}]$ is a basis element of $C_{p+q}(K)$ and again the indices $i_j$ satisfy $i_0 < i_1 < … < i_{p+q}$. We then extend by bilinearity.
Suppose instead that I wish to work with a more flexible set up with regards to vertex orderings. A standard way to do this is the notion of an orientation – two choices of orderings of the vertices of a simplex are equivalent if they differ by an even permutation, and an orientation of a simplex is a choice of one of the two equivalence classes of vertex orderings. If the vertex set is finite and we label is as $\{0,1,..,n\}$ then we could agree that the standard orientation of each simplex is the one where the vertices are listed in increasing order.
The above formula for the cup product would no longer be well-defined on the cochain level since it would not necessarily remain invariant under even permutations of the vertices $v_{i_0}, v_{i_1}, … ,v_{i_{p+q}}$. Are there any books which deal with this issue? Most if the books I have consulted have worked only with fixed vertex orderings and the more general orientations.
Sorry, I mean: Most if the books I have consulted have worked only with fixed vertex orderings and not the more general orientations approach.
Best Answer
I think the usual fix is to define $$ (\phi\cup\psi)[v_0,\ldots,v_{p+q}] \;=\; \frac{1}{(p+q+1)!}\sum_{\sigma\in S_{p+q+1}} (-1)^\sigma \phi[v_{\sigma(0)},\ldots,v_{\sigma(p)}]\cdot\psi[v_{\sigma(p)},\ldots,v_{\sigma(p+q)}] $$ on the level of cochains, where $(-1)^\sigma$ denotes the sign of the permutation $\sigma$.