Cell and handle attachment across a critical level #
Let f be a smooth function on a boundaryless finite-dimensional manifold, and let c be a
critical value at which f has finitely many critical points, all nondegenerate, such that the
band f⁻¹[c - δ, c + δ] is compact for some δ > 0. Milnor's Theorem 3.2 says that passing the
level c attaches one cell at each of these critical points: for small ε > 0, the sublevel
M_{c+ε} = {f ≤ c + ε} deformation retracts onto the lower sublevel M_{c-ε} = {f ≤ c - ε} with
one λ-cell attached at each critical point of index λ.
The cells are explicit. In a normalized Morse chart ψ at a critical point of index λ,
f = c - Σ_{j < λ} yⱼ² + Σ_{j ≥ λ} yⱼ²,
and the cell is the disc {‖y‖² ≤ ε, yⱼ = 0 for j ≥ λ} of the negative coordinates, read in M.
Its boundary sphere lies on the level f⁻¹(c - ε).
Following Milnor, write u for the negative and v for the positive coordinates of a chart. The
function F = f - μ(‖u‖² + 2‖v‖²) agrees with f outside the ellipsoids ‖u‖² + 2‖v‖² ≤ 2ε,
where μ is a smooth nonincreasing cutoff with μ = 2ε on [0, ε] and μ = 0 on [2ε, ∞)
(TauCeti.exists_contDiff_antitone_cutoff). It has the same sublevel {F ≤ c + ε} = M_{c+ε}.
When the band f⁻¹[c - ε, c + ε] is compact and its critical points lie in the charts, F has no
critical points in the band F⁻¹[c - ε, c + ε]: outside the ellipsoids it agrees with f, whose
critical points in the band are the chart centres; along the positive directions F strictly
increases away from v = 0, and on v = 0 it lies below c - ε. So M_{c+ε} deformation
retracts onto {F ≤ c - ε} (TauCeti.exists_sublevelDeformation), which in turn retracts onto
M_{c-ε} with the cells attached (TauCeti.exists_inter_sublevel_union_cells_deformation), since
lowering the positive coordinates lowers F. The two retractions compose.
Milnor's Remark 3.3 refines this to a handle attachment, up to a diffeomorphism rather than a
homotopy. Since F has no critical points in the compact band, the descending flow of
TauCeti.exists_diffeomorph_sublevel is a diffeomorphism of M carrying M_{c+ε} = {F ≤ c + ε}
onto {F ≤ c - ε}, and the level f⁻¹(c + ε) onto F⁻¹(c - ε). For a cutoff positive on
(-∞, 2ε), the sublevel {F ≤ c - ε} is M_{c-ε} together with one handle in each chart, the
points of {F ≤ c - ε} on or above the level f = c - ε in Milnor's ellipsoid
‖u‖² + 2‖v‖² ≤ 2ε. In the coordinates (u, v) it is
Milnor's handle TauCeti.quadraticHandle ε μ, which is homeomorphic to D^λ × D^(m-λ) with its
attaching region {f = c - ε} carried onto ∂D^λ × D^(m-λ) (TauCeti.quadraticHandleHomeomorph).
Main results #
TauCeti.exists_sublevel_union_cells_deformation: for normal-form charts with disjoint sources around the critical points of a compact bandf⁻¹[c - ε, c + ε], the sublevelM_{c+ε}strong deformation retracts ontoM_{c-ε}with the cells of the charts attached.TauCeti.IsMorse.exists_sublevel_union_cells_deformation: Milnor's cell attachment theorem for a Morse function with a compact band around a critical value, with the charts and the widthε₀supplied byTauCeti.IsMorse.exists_admissible_morseCharts, valid for all smaller widths too.TauCeti.exists_diffeomorph_sublevel_eq_union_handles: for such charts, a diffeomorphism ofMcarriesM_{c+ε}ontoM_{c-ε}with one handleD^λ × D^(m-λ)attached in each chart, along∂D^λ × D^(m-λ).TauCeti.IsMorse.exists_diffeomorph_sublevel_eq_union_handles: Milnor's handle attachment theorem for a Morse function with a compact band around a critical value.
References #
- J. Milnor, Morse Theory, Annals of Mathematics Studies 51, Princeton University Press, 1963, Theorem 3.2 and Remark 3.3.
Cell attachment in disjoint Morse charts. Let f be smooth, and let ψ i be charts of
M with pairwise disjoint sources, each containing the closed ball of radius √(2ε) about 0
in its target, in which f = c - Σ_{j < k i} yⱼ² + Σ_{j ≥ k i} yⱼ². Suppose the band
f⁻¹[c - ε, c + ε] is compact and its critical points lie in the chart sources. The cell of the
chart ψ i is the disc {‖y‖² ≤ ε, yⱼ = 0 for j ≥ k i} of the first k i coordinates, read in
M.
Then the lower sublevel {f ≤ c - ε} with these cells attached is a strong deformation retract
of {f ≤ c + ε}: there is a retraction of {f ≤ c + ε} onto it, and a homotopy in
{f ≤ c + ε} from this retraction to the identity which fixes it pointwise. The cells lie in
{f ≤ c}, so intersecting with {f ≤ c + ε} does not change the retract.
Handle attachment in disjoint Morse charts (Milnor, Remark 3.3). Let f be smooth, and let
ψ i be charts of M with pairwise disjoint sources, each containing the closed ball of radius
√(2ε) about 0 in its target, in which f = c - Σ_{j < k i} yⱼ² + Σ_{j ≥ k i} yⱼ². Suppose the
band f⁻¹[c - ε, c + ε] is compact and its critical points lie in the chart sources.
Then there are handles H i in the chart sources, on or above the level f⁻¹(c - ε), and a
diffeomorphism Φ of M carrying the upper sublevel {f ≤ c + ε} onto the lower sublevel
{f ≤ c - ε} with the handles attached. Each handle is homeomorphic to the product
D^{k i} × D^{n - k i} of closed unit balls in the first k i and in the remaining coordinates, by
a homeomorphism e i carrying:
- its intersection with
{f ≤ c - ε}onto∂D^{k i} × D^{n - k i}; - its core, the points whose coordinates beyond the first
k ivanish, ontoD^{k i} × 0.
Φ carries the level f⁻¹(c + ε) onto the boundary of the attached space: the points on the top
D^{k i} × ∂D^{n - k i} of every handle containing them, and on the level f⁻¹(c - ε) outside the
handles.
Milnor's cell attachment theorem. Let f be a Morse function on a Hausdorff manifold whose
band f⁻¹[c - δ, c + δ] is compact, for some 0 < δ. Then there are a width 0 < ε₀ ≤ δ and
normalized Morse charts ψ p with pairwise disjoint sources, indexed by the critical points p
on the level f⁻¹(c), with ψ p p = 0, with the closed ball of radius √(2ε₀) in the target of
ψ p, and in which f = c - Σ_{i < λ} yᵢ² + Σ_{i ≥ λ} yᵢ², where λ is the Morse index of f
at p. For every width 0 < ε ≤ ε₀, the cell at p is the disc
{‖y‖² ≤ ε, yᵢ = 0 for i ≥ λ} of its negative coordinates, read in M through ψ p.
The lower sublevel {f ≤ c - ε} with these cells attached is a strong deformation retract of
{f ≤ c + ε}, for every such ε: there is a retraction of {f ≤ c + ε} onto it, and a
homotopy in {f ≤ c + ε} from this retraction to the identity which fixes it pointwise.
Milnor's handle attachment theorem (Milnor, Remark 3.3). Let f be a Morse function on a
Hausdorff manifold of dimension m whose band f⁻¹[c - δ, c + δ] is compact, for some 0 < δ.
Then there are a width 0 < ε ≤ δ and normalized Morse charts ψ p with pairwise disjoint sources,
indexed by the critical points p on the level f⁻¹(c), with ψ p p = 0, with the closed ball of
radius √(2ε) in the target of ψ p, and in which f = c - Σ_{i < λ} yᵢ² + Σ_{i ≥ λ} yᵢ², where
λ is the Morse index of f at p.
Attaching one handle D^λ × D^(m-λ) at each critical point turns the lower sublevel
{f ≤ c - ε} into the upper one {f ≤ c + ε}, up to a diffeomorphism Φ of M: the conclusion
of TauCeti.exists_diffeomorph_sublevel_eq_union_handles holds for these charts.