Relative homology of bands: without critical points, and above a regular level #
Let f be a smooth real function on a boundaryless finite-dimensional manifold M, and let
a ≤ b be such that the band f⁻¹[a, b] is compact and contains no critical point of f. The
sublevel {f ≤ a} is a strong deformation retract of {f ≤ b}
(TauCeti.exists_sublevelDeformation), so the relative singular homology of the sublevel pair
({f ≤ b}, {f ≤ a}) vanishes in every degree, for coefficients in any preadditive category with
coproducts and homology (TauCeti.isZero_singularHomology_sublevelPair_of_mfderiv_ne_zero).
The band f⁻¹[a, b] itself, relative to its lower level f⁻¹(a), has the same relative homology
as the sublevel pair, whatever the critical points inside the band, as soon as a is a regular
value and the band is compact: the inclusion of pairs (f⁻¹[a, b], f⁻¹(a)) ⟶ ({f ≤ b}, {f ≤ a})
induces isomorphisms (TauCeti.isIso_singularHomologyMap_bandPairInclusion). When b is regular
as well, the band is a compact cobordism between the levels f⁻¹(a) and f⁻¹(b), so statements
about sublevel pairs, such as the Morse inequalities, apply to the cobordism relative to its
incoming boundary. The proof uses a collar f⁻¹[a, a'] without critical points above the
regular level: the flow lowering f at unit speed through it retracts the collar onto f⁻¹(a)
and {f ≤ a'} onto {f ≤ a}, so both subspaces may be enlarged by the collar, and then excision
of {f < a} from ({f ≤ b}, {f ≤ a'}) leaves (f⁻¹[a, b], f⁻¹[a, a']).
Consequently, regular collars can be cut off a band. If a ≤ s ≤ t ≤ b and the bands
f⁻¹[a, s] and f⁻¹[t, b] are compact and free of critical points, the long exact sequences of
the triples {f ≤ a} ⊆ {f ≤ t} ⊆ {f ≤ b} and {f ≤ a} ⊆ {f ≤ s} ⊆ {f ≤ t} identify the relative
homology of ({f ≤ b}, {f ≤ a}) with that of ({f ≤ t}, {f ≤ s}), for coefficients in an
abelian category with coproducts (TauCeti.singularHomologySublevelPairIsoOfCollars). Both
identifications are induced by inclusions of pairs
(TauCeti.singularHomologyMap_innerToTotal_comp_singularHomologySublevelPairIsoOfCollars_hom).
Main results #
TauCeti.isZero_singularHomology_sublevelPair_of_mfderiv_ne_zero: a compact band without critical points carries no relative homology.TauCeti.isIso_singularHomologyMap_bandPairInclusion: above a regular value, a compact band relative to its lower level has the relative homology of the sublevel pair.TauCeti.singularHomologySublevelPairIsoOfCollars: regular collars can be cut off a band without changing the relative homology of the sublevel pair.
References #
- J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Theorem 3.1.
- J. Milnor, Lectures on the h-cobordism theorem, Princeton University Press, 1965, §3.
A band without critical points carries no relative homology. If the band f⁻¹[a, b] is
compact and contains no critical point of the smooth function f, the relative singular homology
of the sublevel pair ({f ≤ b}, {f ≤ a}) vanishes in every degree.
The relative homology of a band over a regular level (Milnor, Lectures on the h-cobordism
theorem, §3). If a ≤ b, the band f⁻¹[a, b] is compact and a is a regular value of the
smooth function f, then the inclusion of pairs (f⁻¹[a, b], f⁻¹(a)) ⟶ ({f ≤ b}, {f ≤ a})
induces isomorphisms on relative singular homology in every degree. The band may contain critical
points, and b need not be a regular value.
Removing regular collars from a band. If the bands f⁻¹[a, s] and f⁻¹[t, b] are
compact and contain no critical points, then the relative homology of ({f ≤ b}, {f ≤ a}) is that
of ({f ≤ t}, {f ≤ s}): both are identified with the relative homology of ({f ≤ t}, {f ≤ a}) by
the long exact sequences of the triples {f ≤ a} ⊆ {f ≤ t} ⊆ {f ≤ b} and
{f ≤ a} ⊆ {f ≤ s} ⊆ {f ≤ t}.
Instances For
The identification TauCeti.singularHomologySublevelPairIsoOfCollars of
Hₙ({f ≤ b}, {f ≤ a}) with Hₙ({f ≤ t}, {f ≤ s}) is induced by inclusions of pairs: precomposed
with the map induced by ({f ≤ t}, {f ≤ a}) ⟶ ({f ≤ b}, {f ≤ a}), it is the map induced by
({f ≤ t}, {f ≤ a}) ⟶ ({f ≤ t}, {f ≤ s}).
The identification TauCeti.singularHomologySublevelPairIsoOfCollars of
Hₙ({f ≤ b}, {f ≤ a}) with Hₙ({f ≤ t}, {f ≤ s}) is induced by inclusions of pairs: precomposed
with the map induced by ({f ≤ t}, {f ≤ a}) ⟶ ({f ≤ b}, {f ≤ a}), it is the map induced by
({f ≤ t}, {f ≤ a}) ⟶ ({f ≤ t}, {f ≤ s}).