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Tsinghua-BIMSA Symplectic Geometry Seminar
Tsinghua-BIMSA Symplectic Geometry Seminar
Equivariant Transversality in Morse Homology via Stabilization
Equivariant Transversality in Morse Homology via Stabilization
Organizers
Speaker
Erkao Bao
Time
Monday, June 1, 2026 4:50 PM - 5:50 PM
Venue
Shuangqing
Abstract
An obstacle in defining equivariant Morse homology is the failure of
transversality: for a compact Lie group acting on a closed manifold,
an invariant Morse–Bott function f may not admit any invariant metric
g such that the pair (f,g) is Morse–Bott–Smale. As a result, the
moduli spaces of certain gradient flow lines are not transversely cut
out, preventing a well-defined count.
We resolve this problem via a stabilization technique: by making a
C^1-small perturbation f' of f near critical points, we show that a
generic invariant metric now makes (f',g) Morse–Bott–Smale. This
restores transversality and allows us to follow Austin–Braam’s
approach to define equivariant cohomology over real coefficients. In
the finite group case, the construction lifts to integer coefficients,
and also yields both invariant and coinvariant Morse homology for
orbifolds.
Motivated by this framework, we discuss how to compute equivariant
Morse homology without perturbing f, by taking a limit of the
perturbed theory and analyzing the resulting gluable infinitesimal
cascades.
Finally, if time permits, we apply this to get a localization result
for Lagrangian Floer homology under finite group action, which is a
work in progress.
This talk is based on several joint works with Tyler Lawson, Lina Liu,
Robi Huq, Shengzhen Ning, and Cheuk-Yu Mak.
transversality: for a compact Lie group acting on a closed manifold,
an invariant Morse–Bott function f may not admit any invariant metric
g such that the pair (f,g) is Morse–Bott–Smale. As a result, the
moduli spaces of certain gradient flow lines are not transversely cut
out, preventing a well-defined count.
We resolve this problem via a stabilization technique: by making a
C^1-small perturbation f' of f near critical points, we show that a
generic invariant metric now makes (f',g) Morse–Bott–Smale. This
restores transversality and allows us to follow Austin–Braam’s
approach to define equivariant cohomology over real coefficients. In
the finite group case, the construction lifts to integer coefficients,
and also yields both invariant and coinvariant Morse homology for
orbifolds.
Motivated by this framework, we discuss how to compute equivariant
Morse homology without perturbing f, by taking a limit of the
perturbed theory and analyzing the resulting gluable infinitesimal
cascades.
Finally, if time permits, we apply this to get a localization result
for Lagrangian Floer homology under finite group action, which is a
work in progress.
This talk is based on several joint works with Tyler Lawson, Lina Liu,
Robi Huq, Shengzhen Ning, and Cheuk-Yu Mak.