The Morse inequalities #

Let f be a Morse function on a boundaryless finite-dimensional manifold M, let V be a finite-dimensional module over a division ring k, and write M_a = {f ≤ a}. Let a ≤ b be values of f at no critical point, with compact band f⁻¹[a, b]. Write bᵢ = dim Hᵢ(M_b, M_a; V) for the dimensions of relative singular homology, and cᵢ for the number of critical points of index i with value in (a, b]. Then, for every n:

Hₙ(M_b, M_a; V) is finite-dimensional, and bₙ ≤ cₙ dim V (the weak Morse inequalities);
bₙ - bₙ₋₁ + ⋯ ± b₀ ≤ (cₙ - cₙ₋₁ + ⋯ ± c₀) dim V (the strong Morse inequalities);
if n ≥ dim M, then ∑ᵢ₌₀ⁿ (-1)ⁱ bᵢ = (∑ᵢ₌₀ⁿ (-1)ⁱ cᵢ) dim V (Euler's equality);
if n ≥ dim M, then ∑ᵢ₌₀ⁿ cᵢ dim V · tⁱ = ∑ᵢ₌₀ⁿ bᵢ tⁱ + (1 + t) Q(t) for a polynomial Q with natural number coefficients, the coefficient of tⁱ being the gap in the i-th strong inequality (the Morse polynomial identity);
b₀ + ⋯ + bₙ is at most dim V times the number of critical points with values in (a, b];
if no two of these critical points have indices differing by one, then bₙ = cₙ dim V (the lacunary principle).

The band W = f⁻¹[a, b] is then a compact cobordism from the level f⁻¹(a) to the level f⁻¹(b), and the inclusion of pairs (W, f⁻¹(a)) ⟶ (M_b, M_a) induces isomorphisms on relative homology (TauCeti.isIso_singularHomologyMap_bandPairInclusion). So the finiteness, the weak and strong inequalities, Euler's equality and the polynomial identity hold as well for the numbers bᵢ = dim Hᵢ(W, f⁻¹(a); V) of the cobordism relative to its incoming boundary.

Read from the other end, W is the band of -f between -b and -a, relative to its incoming boundary f⁻¹(b) (TauCeti.bandPair_neg). A critical point of index λ of f has index dim M - λ for -f. So the same statements hold for the numbers bᵢ = dim Hᵢ(W, f⁻¹(b); V) of the cobordism relative to its outgoing boundary, with cᵢ the number of critical points of index dim M - i with values in [a, b). Comparing the two Euler equalities, the Euler characteristics of (W, f⁻¹(b)) and (W, f⁻¹(a)) agree up to the sign (-1)^(dim M).

For compact M all of M is such a band, with M_a = ∅, and the same statements hold for the singular homology Hᵢ(M; V) and the numbers cᵢ of critical points of index i on M. In particular the singular homology of a compact manifold admitting a Morse function is finite-dimensional in every degree.

The proof is Milnor's. Across a single critical level the sublevel pair has the relative homology of a finite CW pair with one i-cell for each critical point of index i (TauCeti.IsMorse.exists_relCWComplex_isIso_singularHomologyMap_sublevelPair), where the three statements are the Morse inequalities for CW pairs. Across a band without critical points the relative homology vanishes, by the deformation retraction of M_b onto M_a (TauCeti.isZero_singularHomology_sublevelPair_of_mfderiv_ne_zero). A band is cut at the critical values it contains into finitely many such pieces (TauCeti.IsMorse.band_induction), and the long exact sequence of the triple M_a ⊆ M_m ⊆ M_b shows that dimensions and the alternating sums Sₙ are subadditive (TauCeti.TopTriple.finrank_singularHomology_totalPair_le and TauCeti.TopTriple.sum_range_neg_one_pow_mul_finrank_singularHomology_totalPair_le), while the critical counts are additive. Above the dimension of M the connecting map in degree n + 1 vanishes, since its source is zero by the weak inequalities, which turns the strong inequality into the equality. The polynomial identity and the lacunary principle are then the numerical consequences TauCeti.exists_sum_range_monomial_eq_add_mul and TauCeti.eq_of_forall_sum_range_neg_one_pow_mul_le of the strong inequalities and Euler's equality.

The coefficients of Q are ranks of boundary maps. Cut the band at regular values a = s 0 ≤ s 1 ≤ ⋯ ≤ s (N + 1) = b so that each piece f⁻¹[s j, s (j + 1)] contains at most one critical value; on a compact manifold such values exist (TauCeti.IsMorse.exists_regular_sublevel_partition). Each piece then has bᵢ = cᵢ dim V, and summing the defects of subadditivity of Sₙ over the triples (M_{s (j + 2)}, M_{s (j + 1)}, M_{s 0}) (TauCeti.sum_range_finrank_range_singularHomologyδ_eq) shows that the coefficient of tⁿ in Q is the sum over j < N of the ranks of the connecting maps Hₙ₊₁(M_{s (j + 2)}, M_{s (j + 1)}) ⟶ Hₙ(M_{s (j + 1)}, M_{s 0}) (TauCeti.IsMorse.coeff_eq_sum_range_finrank_range_singularHomologyδ_sublevelPair).

Main results #

TauCeti.IsMorse.finrank_singularHomology_sublevelPair_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_sublevelPair_le and TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_sublevelPair_eq: the weak and strong Morse inequalities and Euler's equality for a compact band between regular values.
TauCeti.IsMorse.exists_sum_range_monomial_sublevelPair_eq_add_mul, TauCeti.IsMorse.sum_range_finrank_singularHomology_sublevelPair_le and TauCeti.IsMorse.finrank_singularHomology_sublevelPair_eq_of_forall_ne_add_one: the Morse polynomial identity, the bound on the total Betti number and the lacunary principle for such a band.
TauCeti.IsMorse.finite_singularHomology_bandPair, TauCeti.IsMorse.finrank_singularHomology_bandPair_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_bandPair_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_bandPair_eq and TauCeti.IsMorse.exists_sum_range_monomial_bandPair_eq_add_mul: the same statements for the cobordism f⁻¹[a, b] relative to its incoming boundary f⁻¹(a).
TauCeti.IsMorse.finite_singularHomology_upperBandPair, TauCeti.IsMorse.finrank_singularHomology_upperBandPair_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_upperBandPair_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_upperBandPair_eq and TauCeti.IsMorse.exists_sum_range_monomial_upperBandPair_eq_add_mul: the same statements for the cobordism relative to its outgoing boundary f⁻¹(b), with critical points counted by their complementary index.
IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_upperBandPair_eq_neg_one_pow_mul: the Euler characteristics of the cobordism relative to its two boundaries agree up to the sign (-1)^(dim M).
TauCeti.IsMorse.sum_range_finrank_range_singularHomologyδ_sublevelPair_eq and TauCeti.IsMorse.coeff_eq_sum_range_finrank_range_singularHomologyδ_sublevelPair: for a band cut at regular values into pieces with at most one critical value each, the gap in the n-th strong inequality, which is the coefficient of tⁿ in Q, is a sum of ranks of connecting maps of triples of sublevels.
TauCeti.IsMorse.finrank_singularHomology_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_le, TauCeti.IsMorse.sum_range_neg_one_pow_mul_finrank_singularHomology_eq, TauCeti.IsMorse.exists_sum_range_monomial_eq_add_mul, TauCeti.IsMorse.sum_range_finrank_singularHomology_le and TauCeti.IsMorse.finrank_singularHomology_eq_of_forall_ne_add_one: the same for a compact manifold, with the numbers TauCeti.criticalCount of critical points of each index and their total TauCeti.totalCriticalCount.
TauCeti.IsMorse.sum_range_finrank_range_singularHomologyδ_eq and TauCeti.IsMorse.coeff_eq_sum_range_finrank_range_singularHomologyδ: on a compact manifold, the coefficients of Q as sums of ranks of connecting maps along a cut of the values of f into pieces with at most one critical value each, which exists by TauCeti.IsMorse.exists_regular_sublevel_partition.

References #

J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Section 5.
R. Bott, Lectures on Morse theory, old and new, Bulletin of the American Mathematical Society 7 (1982), 331--358, Section 1.

The Morse inequalities #

Main results #

References #

The quotient polynomial as a sum of ranks #

The cobordism relative to its outgoing boundary #