DECOMPOSING STRATIFIED SPACES

Dfn. Stratified space: space X, loc. finite partition into loc. closed subsets X_{i} , each \overline{X}_{i} is union of some X_{j} .

Terminology. X_{i} ≥ X_{j} if X_{i} ⊂ \overline{X}_{j}. γ: [0, c] → X nonincreasing if, for s ≤ t, stratum of γ(s) is ≥ stratum of γ(t).

Dfn. Category C_{X}: space of objects ∐_{i} X_{i} , space of morphisms ∐_{j,i} Lk(j, i) where Lk(j, i) is space of nonincr. paths from X_{j} to X_{i} . Standard decomposition: Expect X ≃ BC_{X} for “good” stratified X. Observe

BC_{X} ≅ hocolim Lk(i_{0}, i_{1}, …, i_{n}) where Lk(i_{0}, i_{1}, …, i_{n}) is space of nonincr paths γ: [0, c] → X with subdivision

0 = c_{0} ≤ c_{1} ≤ ⋯ ≤ c_{n−1} ≤ c_{n} = c

and γ(c_{k}) ∈ X_{i_{k}} , all k. 1

Ex. EI–category C (all endomorphisms are isomorphisms). Order isomorphism classes: [c] ≤ [d] if ∃ morphism c → d. (Want finiteness condition on resulting poset.) Full subcategories C_{[d]} : objects e with [d] ≤ [e]. Stratify Y = BC with strata Y_{[d]} so that

closure of Y_{[d]} = B(C_{[d]}).

Dfn. Given p: X → Y where Y stratified, X any space. Say p is σ–fibration if p has lifting property for homotopies in Y with nonincreasing tracks.

“Other” decomposition: Given EI–category C, space X and σ–fibration p: X → BC. Then

X ≃ hocolim F

for some F: C^{op} → Spaces with F (c) ≃ p^{−1}(c).

Rmk. Stratification of BC pulls back to a stratification of X.

Dfn. EI–category K. Objects: finite sets over {0, 1, 2, 3}. Morphism from S to T:

– injective map e: S → T over {0, 1, 2, 3}

– map λ from T ∖ e(S) to {−1, +1}.

Goal. Let B(mo) classify “families” (submersions) of oriented smooth 3–manifolds with proper Morse function. Make σ–fibration B(mo) → BK. Use “other decomposition” to decompose B(mo).

Problem. B(mo) not clearly def’d. Escape. Focus on “families”. New prblm. What families does BK classify ?

Fact. C any small category. Then BC classifies sheaves of C–sets (or C^{op}–sets) with representable stalks.

Details. C–set: functor C^{op} → Sets. Classification up to concordance.

Ex. Given proper smooth (π, f): E → X × ℝ where π submersion with oriented 3–dim’l fibers, all maps f_{x} = f| π^{−1}(x) Morse. Make sheaf Γ = Γ^{(π,f)} of K^{op}–sets on X. For open U ⊂ X and S in K, elts of Γ(S, U) are all

(λ_{x}) ∈ ∏_{x∈U} mor_{K}(f^{−1}(0) ∩ crit(f_{x}) , S)

whose two adjoints, from a subset of π^{−1}(U) to S and from a subset of U × S to {±1}, are cts. Note representable stalks.

Plan. Verify σ–fibration property in “family” setting. Given (π, f) and Γ as above and extension of Γ to sheaf Γ^{♯} on X × [0, 1]. Assume cardinality of representing objects for stalks of Γ^{♯} at (x, t) nonincreasing function of t. Find matching extension (π^{♯}, f^{♯}) of family (π, f) to base X × [0, 1].

Method. Easy. Take π^{♯} = π × [0, 1]. Get f^{♯} by perturbing f × [0, 1]. Insist

crit(f_{(x,t)}^{♯}) = crit(f_{x}) , ∀t ∈ [0, 1].

Conclusion. B(mo) ≃ hocolim_{S in K^{op}} B(mo_{S}).

Remark. B(mo_{S}) classifies “families” of oriented 3-mflds with proper Morse f and isomorphism

f^{−1}(0) ∩ crit(f) → S .

Lem. B(mo_{S}) also classifies fam. of ortd 3-mflds with proper Morse f and isomorphism crit(f) → S where

crit(f) ⊂ f^{−1}(0) .

Proof. For closed/cpct parameter mfld X, conjugate all Morse functions with embedding e: ℝ → ℝ isotopic to id, where im(e) small nhd of 0. For noncompact X, use family (e_{x}) of such, x ∈ X. Concordance class unchanged.

Rmk. “Family” as in lemma is bundle (though only assumed submersion).

Proof. Given such family (π, f): E → X × ℝ,

⋃_{x∈X} crit(f_{x}) ⊂ f^{−1}(0).

Split into f^{−1} of [1, ∞), (−∞, −1] and [−1, +1]. Get proper submer. to X × [1, ∞), X × (−∞, −1] and X, respectively. Proper submersions: bundles.

Rmk. Ortd 3-mfld with proper Morse function, unique critical value 0, is trace of simult. surgeries on ortd closed surface. Hence B(mo_{S}) classifying space for bundle of ortd closed surfaces with surgery data indexed by S.

M.Weiss (joint work with I.Madsen)