Small Triangulations of 4-Manifolds and the 4-Manifold Census

Abstract We present a framework to classify PL-types of large censuses of triangulated 4-manifolds, which we use to classify the PL-types of all triangulated 4-manifolds with up to six pentachora. This is successful except for triangulations homeomorphic to the 4-sphere, $$\mathbb {C}P^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>P</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , and the rational homology sphere $$QS^4(2)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Q</mml:mi> <mml:msup> <mml:mi>S</mml:mi> <mml:mn>4</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where we find at most four, three, and two PL-types respectively. We conjecture that they are all standard. In addition, we look at the cases resisting classification and discuss the combinatorial structure of these triangulations—which we deem interesting in their own rights.