We apply mapping class group techniques and trisections to study intersection forms of smooth
–manifolds.
Johnson defined a well-known homomorphism from the Torelli group
of a compact surface. Morita later showed that every homology
–sphere
can be obtained from the standard Heegaard decomposition of
by regluing
according to a map in the kernel of this homomorphism. We prove an analogous result for trisections
of
–manifolds.
Specifically, if
and
admit handle
decompositions without
–
or
–handles
and have isomorphic intersection forms, then a trisection
of can be obtained
from a trisection of
by cutting and regluing by an element of the Johnson kernel. We also describe how invariants of
homology
–spheres
can be applied, via this result, to obstruct intersection forms of smooth
–manifolds. As
an application, we use the Casson invariant to recover Rohlin’s theorem on the signature of spin
–manifolds.
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