Odd manifolds of small integral simplicial volume

Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal to $1$. Consequently, we obtain an elementary proof that, in general, the integral simplicial volume of (triangulated) manifolds is not computable.

2010 Mathematics Subject Classification

55N10, 57N65

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This work was supported by the CRC 1085 Higher Invariants (Universität Regensburg, funded by the DFG).

Received 15 March 2017

Received revised 31 July 2017

Accepted 26 September 2017

Published 24 May 2022