1 6 N ov 2 00 5 The pre - WDVV ring of physics and its topology ∗

We show how a simplicial complex arising from the WDVV (Witten-Dijkgraaf-VerlindeVerlinde) equations of string theory is the Whitehouse complex. Using discrete Morse theory, we give an elementary proof that the Whitehouse complex ∆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.