Filtered and Intersection Homology
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1 Filtered and Intersection Homology Jon Woolf, work in progress with Ryan Wissett April, 2016
2 Part I Review of intersection homology
3 Singular intersection homology Perversities A perversity on a topologically stratified space X is a function p : {strata of X } Z. If 1. p(s) = p(codim S) for some p : N Z 2. p(k) = 0 for k 2 3. p(k + 1) = p(k) or p(k) + 1. then it is a Goresky MacPherson (GM) perversity.
4 Singular intersection homology Perversities A perversity on a topologically stratified space X is a function p : {strata of X } Z. If 1. p(s) = p(codim S) for some p : N Z 2. p(k) = 0 for k 2 3. p(k + 1) = p(k) or p(k) + 1. then it is a Goresky MacPherson (GM) perversity. Examples the zero perversity 0(k) = 0 the top perversity t(k) = max{k 2, 0} the lower middle perversity m(k) = max{ (k 2)/2, 0} the upper middle perversity n(k) = max{ (k 2)/2, 0} GM perversities p and q are complementary if p + q = t.
5 Intersection homology and Poincaré duality Intersection homology A perversity picks out a subcomplex of intersection chains in S X : i σ X p-allowable σ 1 S (i codim S + p(s)) -skeleton c S i X p-allowable all simplices in c are p-allowable Let I p S X = {c c, c are p-allowable} and I p H X its homology.
6 Intersection homology and Poincaré duality Intersection homology A perversity picks out a subcomplex of intersection chains in S X : i σ X p-allowable σ 1 S (i codim S + p(s)) -skeleton c S i X p-allowable all simplices in c are p-allowable Let I p S X = {c c, c are p-allowable} and I p H X its homology. Theorem (Goresky MacPherson 80) X pseudomfld, p GM perversity = I p H X topological invariant.
7 Intersection homology and Poincaré duality Intersection homology A perversity picks out a subcomplex of intersection chains in S X : i σ X p-allowable σ 1 S (i codim S + p(s)) -skeleton c S i X p-allowable all simplices in c are p-allowable Let I p S X = {c c, c are p-allowable} and I p H X its homology. Theorem (Goresky MacPherson 80) X pseudomfld, p GM perversity = I p H X topological invariant. Theorem (Goresky MacPherson 80) X compact, oriented n-dim pseudomfld, p, q complementary GM perversities = intersection pairing which is non-degenerate over Q. I p H i X I q H n i X Z
8 Part II Filtered homology
9 Filtered spaces and depth functions Filtered spaces A filtered space X α is a topological space with a filtration = X 1 X 0 X 1 X 2 X = X. A filtered map f : X α Y β is a map with f (X k ) Y k k N.
10 Filtered spaces and depth functions Filtered spaces A filtered space X α is a topological space with a filtration = X 1 X 0 X 1 X 2 X = X. A filtered map f : X α Y β is a map with f (X k ) Y k k N. Depth functions The filtration on X α is encoded in the depth function α: X N where α(x) = k x X k X k 1 so X k = α 1 {0,..., k} and f : X α Y β filtered α β f.
11 Examples of filtered spaces 1. A filtered space of depth 1 is a pair X 0 X 1 = X ; a filtered map of such is a map of pairs.
12 Examples of filtered spaces 1. A filtered space of depth 1 is a pair X 0 X 1 = X ; a filtered map of such is a map of pairs. 2. Filtering a CW complex by its skeleta fully faithfully embeds CW complexes and cellular maps into filtered spaces.
13 Examples of filtered spaces 1. A filtered space of depth 1 is a pair X 0 X 1 = X ; a filtered map of such is a map of pairs. 2. Filtering a CW complex by its skeleta fully faithfully embeds CW complexes and cellular maps into filtered spaces. 3. Let n δ be the standard simplex filtered by depth function δ(t 0,..., t n ) = #{i t i = 0}, e.g The face maps i 1 δ+1 i δ are filtered.
14 Filtered homology For filtered X α define S i X α = Z{ i δ X α}. Note : S i X α S i 1 X α 1 where (α 1)(x) = max{α(x) 1, 0}.
15 Filtered homology For filtered X α define S i X α = Z{ i δ X α}. Note : S i X α S i 1 X α 1 where (α 1)(x) = max{α(x) 1, 0}. Definition The filtered i-chains on X α are FS i X α = {c S i X α c S i 1 X α }. The filtered homology FH X α is the homology of FS X α.
16 Properties of filtered homology Functoriality Filtered f : X α Y β induces a chain map FS X α FS Y β and f : FH X α FH Y β.
17 Properties of filtered homology Functoriality Filtered f : X α Y β induces a chain map FS X α FS Y β and f : FH X α FH Y β. Filtered homotopy invariance If f and g are filtered homotopic then f = g : FH X α FH Y β.
18 Properties of filtered homology Functoriality Filtered f : X α Y β induces a chain map FS X α FS Y β and f : FH X α FH Y β. Filtered homotopy invariance If f and g are filtered homotopic then f = g : FH X α FH Y β. Relative long exact sequence For filtered f : X α Y β where the underlying map is an inclusion we define FH i (Y β, X α ) = H i ( FS Y β / FS X α ). There is a LES FH X α FH Y β FH (Y β, X α ) FH 1 X α
19 Properties of filtered homology Functoriality Filtered f : X α Y β induces a chain map FS X α FS Y β and f : FH X α FH Y β. Filtered homotopy invariance If f and g are filtered homotopic then f = g : FH X α FH Y β. Relative long exact sequence For filtered f : X α Y β where the underlying map is an inclusion we define FH i (Y β, X α ) = H i ( FS Y β / FS X α ). There is a LES FH X α FH Y β FH (Y β, X α ) FH 1 X α Excision For Z α Y α X α with Z Y o there are isomorphisms FH (X α Z α, Y α Z α ) = FH (X α, Y α ).
20 Simple examples of filtered homology Cones For [x, t] CX, the cone on X, and d > 1 have { { α(x) t > 0 β[x, t] = d t = 0 = FH icx β FH i X α i < d 1 = 0 i d 1. When d 1 obtain homology of a point.
21 Simple examples of filtered homology Cones For [x, t] CX, the cone on X, and d > 1 have { { α(x) t > 0 β[x, t] = d t = 0 = FH icx β FH i X α i < d 1 = 0 i d 1. When d 1 obtain homology of a point. Suspended torus Let X α = ΣT 2 where α(x) = 2 at suspension points and 0 elsewhere. Then Z i = 0 0 i = 1 FH i X α = Z 2 i = 2 Z i = 3.
22 Simple examples of filtered homology Cones For [x, t] CX, the cone on X, and d > 1 have { { α(x) t > 0 β[x, t] = d t = 0 = FH icx β FH i X α i < d 1 = 0 i d 1. When d 1 obtain homology of a point. Suspended torus Let X α = ΣT 2 where α(x) = 3 at suspension points and 0 elsewhere. Then Z i = 0 Z 2 i = 1 FH i X α = 0 i = 2 Z i = 3.
23 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q.
24 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q. Setting X k = codim S k S gives Xˆ0 = ( X 0 X 1 X k X )
25 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q. Setting X k = codim S k S gives Xˆ0 = ( X 0 X 1 X k X ) Xˆt = ( X 0 X 1 X )
26 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q. Setting X k = codim S k S gives Xˆ0 = ( X 0 X 1 X k X ) Xˆt = ( X 0 X 1 X ) X ˆm = ( X 0 X 1 X 2 X 4 X )
27 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q. Setting X k = codim S k S gives Xˆ0 = ( X 0 X 1 X k X ) Xˆt = ( X 0 X 1 X ) X ˆm = ( X 0 X 1 X 2 X 4 X ) Xˆn = ( X 0 X 1 X 3 X 5 X )
28 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q. Setting X k = codim S k S gives Xˆ0 = ( X 0 X 1 X k X ) Xˆt = ( X 0 X 1 X ) X ˆm = ( X 0 X 1 X 2 X 4 X ) Xˆn = ( X 0 X 1 X 3 X 5 X ) p is a Goresky Macpherson perversity Xˆp is filtration by those X k with p(k) = p(k + 1)
29 Perversities and filtrations Given stratified X and perversity p define a depth function ˆp(x) = codim S p(s) for x S. The identity Xˆp Xˆq is filtered p q. Setting X k = codim S k S gives Xˆ0 = ( X 0 X 1 X k X ) Xˆt = ( X 0 X 1 X ) X ˆm = ( X 0 X 1 X 2 X 4 X ) Xˆn = ( X 0 X 1 X 3 X 5 X ) p is a Goresky Macpherson perversity Xˆp is filtration by those X k with p(k) = p(k + 1) Complementary perversities p and q give complementary filtrations: X k with k 2 appears in either Xˆp or Xˆq.
30 Intersection homology is filtered homology An elementary calculation gives i δ σ Xˆp filtered σ 1 S (i codim S + p(s)) -skeleton σ p-allowable
31 Intersection homology is filtered homology An elementary calculation gives i δ σ Xˆp filtered σ 1 S (i codim S + p(s)) -skeleton σ p-allowable Corollary FS Xˆp = I p S X and FH Xˆp = I p H X.
32 Intersection homology is filtered homology An elementary calculation gives i δ σ Xˆp filtered σ 1 S (i codim S + p(s)) -skeleton σ p-allowable Corollary FS Xˆp = I p S X and FH Xˆp = I p H X. Remarks Functoriality of FH = known functoriality of IH
33 Intersection homology is filtered homology An elementary calculation gives i δ σ Xˆp filtered σ 1 S (i codim S + p(s)) -skeleton σ p-allowable Corollary FS Xˆp = I p S X and FH Xˆp = I p H X. Remarks Functoriality of FH = known functoriality of IH Intersection homology is a filtered homotopy invariant
34 Intersection homology is filtered homology An elementary calculation gives i δ σ Xˆp filtered σ 1 S (i codim S + p(s)) -skeleton σ p-allowable Corollary FS Xˆp = I p S X and FH Xˆp = I p H X. Remarks Functoriality of FH = known functoriality of IH Intersection homology is a filtered homotopy invariant Filtered homology LES gives relative LES for IH, and obstruction sequence for change of perversities.
35 Part III Spectral sequence of a filtered space
36 The spectral sequence For filtered X α the singular complex S X has natural filtration 0 S X α S X α 1 S X
37 The spectral sequence For filtered X α the singular complex S X has natural filtration 0 S X α S X α 1 S X yielding a spectral sequence with E 0 -page S 0 X α S 1 X α S 2 X α 0 S 0 X α 1 S 0 X α S 1 X α 1 S 1 X α 0 0 S 0 X α 2 S 0 X α 1 converging to Gr H X where Gr i H j X = {[c] H jx c S j X α i } {[c] H j X c S j X α i+1 }.
38 The spectral sequence The singular complex S X of filtered X α has natural filtration 0 S X α S X α 1 S X yielding a spectral sequence with E 1 -page FS 0 X α FS 1 X α FS 2 X α 0 FS 0 X α 1 S 0 X α+ S 1 X α FS 1 X α 1 S 1 X α+ S 2 X α 0 0 FS 0 X α 2 S 0 X α 1 + S 1 X α 1 converging to Gr H X where Gr i H j X = {[c] H jx c S j X α i } {[c] H j X c S j X α i 1 }.
39 The spectral sequence The singular complex S X of filtered X α has natural filtration 0 S X α S X α 1 S X yielding a spectral sequence with E 2 -page FH 0 X α FH 1 X α FH 2 X α 0 FH 0 (X α 1, X α ) FH 1 (X α 1, X α ) 0 0 FH 0 (X α 2, X α 1 ) converging to Gr H X where Gr i H j X = {[c] H jx c S j X α i } {[c] H j X c S j X α i 1 }.
40 The spectral sequence The singular complex S X of filtered X α has natural filtration 0 S X α S X α 1 S X yielding a spectral sequence with E -page Gr 0 H 0 X Gr 0 H 1 X Gr 0 H 2 X 0 Gr 1 H 0 X Gr 1 H 1 X converging to Gr H X where 0 0 Gr 2 H 0 X Gr i H j X = {[c] H jx c S j X α i } {[c] H j X c S j X α i 1 }.
41 Examples of the spectral sequence X α CW-complex with skeletal filtration E 2 -page is cellular chain complex: FH 0 X α FH 1 (X α 1, X α ) 0 0 0
42 Examples of the spectral sequence X α CW-complex with skeletal filtration E 3 = E -page is cellular homology: H cell 0 X H cell 1 X 0 0 0
43 Examples of the spectral sequence X α = ΣT 2 with α(suspension points) = 3 E 2 -page: Z Z 2 0 Z Z
44 Examples of the spectral sequence X α = ΣT 2 with α(suspension points) = 3 E 3 = E -page: Z 0 0 Z Z
45 Part IV Cap products and Poincaré Duality?
46 Alternative filtration for simplices Let n δ denote the n-simplex with filtration δ (t 0,..., t n ) = min{i t n i 0} and FS X α the associated complex of filtered chains Proposition There is a homotopy equivalence FS X α FS X α provided by composition with id : n δ n δ and barycentric subdivision. So filtered homology can be computed using either complex.
47 Cap products Filtered homology as a module The inclusions of the back faces of n δ are filtered. The usual cap product formula restricts to S i X S j X α S j i X α inducing H i X FH j X α FH j i X α, so that FH X α is an H X -module.
48 Cap products Filtered homology as a module The inclusions of the back faces of n δ are filtered. The usual cap product formula restricts to S i X S j X α S j i X α inducing H i X FH j X α FH j i X α, so that FH X α is an H X -module. Generalised Poincaré duality? A more refined approach should yield a cap product where p + q = t. FH i Xˆp FH j Xˆ0 FH j ixˆq 1
49 Cap products Filtered homology as a module The inclusions of the back faces of n δ are filtered. The usual cap product formula restricts to S i X S j X α S j i X α inducing H i X FH j X α FH j i X α, so that FH X α is an H X -module. Generalised Poincaré duality? A more refined approach should yield a cap product FH i Xˆp FH j Xˆ0 FH j ixˆq 1 where p + q = t. If we can improve this to FH i Xˆp FH j Xˆ0 FH j ixˆq then generalised Poincaré duality would follow.
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