Integral Homology under Singer–Frobenius Actions
This writing sample studies the integral first homology of the point–line
incidence graph of the projective plane $\mathrm{PG}(2,q)$ under its
Singer–Frobenius action. The calculation compares homology coinvariants
with the ordinary homology of the quotient graph. An explicit integral
cycle basis reduces the family calculation to a finite-index inclusion of
lattices; Smith normal form then records the torsion and the failure of an
equivariant integral splitting.
The case $q=4$ is the first example in which invariant cycles occur while
the comparison map remains non-primitive. The final section gives a
composite-rank obstruction to a prime-to-rank vanishing statement by using
the fixed subbuilding of a Sylow subgroup.