Mathematical Research Letters

Volume 28 (2021)

Number 6

Representing smooth $4$-manifolds as loops in the pants complex

Pages: 1703 – 1738



Gabriel Islambouli (Department of Mathematics, University of California, Davis, Calif., U.S.A.)

Michael Klug (Department of Mathematics, University of California, Berkeley, Calif., U.S.A.)


We show that every smooth, orientable, closed, connected $4$‑manifold can be represented by a loop in the pants complex. We use this representation, together with the fact that the pants complex is simply connected, to provide an elementary proof that such $4$‑manifolds are smoothly cobordant to a connected sum of complex projective planes, with either orientation. We also use this association to give information about the structure of the pants complex. Namely, given a loop in the pants complex, $L$, which bounds a disk, $D$, we show that the signature of the $4$‑manifold associated to $L$ gives a lower bound on the number of triangles in $D$.

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Received 6 January 2020

Received revised 26 June 2021

Accepted 20 July 2021

Published 29 August 2022