The pre-WDVV ring of physics and its topology

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf- Verlinde-Verlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex Δ _n is homotopy equivalent to a wedge of (n-2)! spheres of dimension n-4. We also verify the Cohen-Macaulay property. Additionally, recurrences are given for the face enumeration of the complex and the Hilbert series of the associated pre-WDVV ring.