Relative homology across a critical level of a Morse function #
Let f be a Morse function and c a value with a compact band f⁻¹[c - δ, c + δ]. Milnor's
cell attachment theorem in CW form (TauCeti.IsMorse.exists_relCWComplex_sublevel_deformation)
gives a small ε > 0 and a finite relative CW complex C over the lower sublevel
M_{c-ε} = {f ≤ c - ε}, with one m-cell for each critical point of index m on the level c,
onto which the upper sublevel M_{c+ε} strong deformation retracts. Since the deformation fixes
C ⊇ M_{c-ε}, the relative singular homology of the sublevel pair
TauCeti.sublevelPair f : (M_{c+ε}, M_{c-ε}) is the relative homology of the CW pair
(C, M_{c-ε}), by the map induced by the inclusion of pairs
(TauCeti.IsMorse.exists_relCWComplex_isIso_singularHomologyMap_sublevelPair).
Every cell of C is attached directly along M_{c-ε}, so all cellular differentials of C
vanish, and the relative homology is free on the critical points
(TauCeti.IsMorse.exists_singularHomology_sublevelPair_iso_sigma): Hₙ(M_{c+ε}, M_{c-ε}; R) is
the coproduct of one copy of R for each critical point of index n on the level c, for
coefficients in any abelian category with coproducts. Over a division ring, it has dimension the
number of these critical points times the dimension of the coefficients
(TauCeti.IsMorse.exists_finrank_singularHomology_sublevelPair_eq). This is the ordinary Morse
case of the identification of the relative homology across a critical level with the homology of
the critical set shifted by the index. In particular, if every critical point on the level has
index at most d, the relative homology of the sublevel pair is concentrated in degrees at most
d (TauCeti.IsMorse.exists_isZero_singularHomology_sublevelPair_of_manifoldMorseIndex_le).
The width ε can be removed: if a < c < b, the band f⁻¹[a, b] is compact and c is its only
critical value, then the collars f⁻¹[a, c - ε] and f⁻¹[c + ε, b] carry no relative homology,
and the long exact sequences of triples identify Hₙ(M_b, M_a) with Hₙ(M_{c+ε}, M_{c-ε})
(TauCeti.IsMorse.nonempty_singularHomology_sublevelPair_iso_sigma and
TauCeti.IsMorse.finrank_singularHomology_sublevelPair_eq). These are the groups
Hₙ(M_{aⱼ₊₁}, M_{aⱼ}) of a filtration of M by sublevels at values aⱼ separating the critical
values.
Main results #
TauCeti.IsMorse.exists_relCWComplex_isIso_singularHomologyMap_sublevelPair: across a critical level, the inclusion of a finite CW pair with one cell for each critical point, of dimension its index, induces isomorphisms on relative homology with the sublevel pair.TauCeti.IsMorse.exists_singularHomology_sublevelPair_iso_sigmaandTauCeti.IsMorse.exists_finrank_singularHomology_sublevelPair_eq: the relative homology across a critical level is free on the critical points of the level, graded by their index.TauCeti.IsMorse.exists_isZero_singularHomology_sublevelPair_of_manifoldMorseIndex_le: vanishing above a bound on the indices of the critical points on the level.TauCeti.IsMorse.nonempty_singularHomology_sublevelPair_iso_sigmaandTauCeti.IsMorse.finrank_singularHomology_sublevelPair_eq: the same for a compact band with a single critical value.
References #
- J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Theorem 3.2 and Section 5.
- A. Hatcher, Algebraic Topology, Section 2.2, Theorem 2.35.
The relative homology across a critical level is cellular. Let f be a Morse function and
c a value with a compact band f⁻¹[c - δ, c + δ]. For some 0 < ε ≤ δ there is a finite
relative CW complex C over {f ≤ c - ε}, whose m-cells correspond to the critical points of
index m on the level c, such that for every coefficient object the inclusion of pairs
(C, {f ≤ c - ε}) ⟶ ({f ≤ c + ε}, {f ≤ c - ε}) induces isomorphisms on relative singular
homology.
The relative homology across a critical level is free on the critical points. Let f be
a Morse function and c a value with a compact band f⁻¹[c - δ, c + δ]. For some 0 < ε ≤ δ,
the relative singular homology Hₙ({f ≤ c + ε}, {f ≤ c - ε}; R), with coefficients in an object
R of an abelian category with coproducts, is the coproduct of one copy of R for each critical
point of index n on the level c. In particular it vanishes when no critical point of the level
has index n.
The Morse numbers of a critical level. Let f be a Morse function and c a value with a
compact band f⁻¹[c - δ, c + δ]. For some 0 < ε ≤ δ, the relative singular homology
Hₙ({f ≤ c + ε}, {f ≤ c - ε}; V), with coefficients in a finite-dimensional module V over a
division ring, is finite-dimensional, of dimension dim V times the number of critical points of
index n on the level c.
Vanishing above the indices across a critical level. Let f be a Morse function whose
critical points on the level c all have index at most d, and let the band f⁻¹[c - δ, c + δ]
be compact. For some 0 < ε ≤ δ, the relative singular homology Hₙ({f ≤ c + ε}, {f ≤ c - ε}; R)
vanishes for every n > d and every coefficient object R of an abelian category with
coproducts.
Bands with a single critical value #
The relative homology of a band with one critical value is free on its critical points.
Let f be a Morse function and a < c < b, such that the band f⁻¹[a, b] is compact and c is
the only critical value of f in [a, b]. Then Hₙ({f ≤ b}, {f ≤ a}; R), with coefficients in
an object R of an abelian category with coproducts, is the coproduct of one copy of R for each
critical point of index n on the level c.
The Morse numbers of a band with one critical value. Let f be a Morse function and
a < c < b, such that the band f⁻¹[a, b] is compact and c is the only critical value of f in
[a, b]. Then, for a finite-dimensional module V over a division ring, Hₙ({f ≤ b}, {f ≤ a}; V)
has dimension dim V times the number of critical points of index n on the level c.