Filtered and Intersection Homology Jon Woolf, work in progress with - PowerPoint PPT Presentation

Filtered and Intersection Homology Jon Woolf, work in progress with Ryan Wissett April, 2016

Part I Review of intersection homology

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.

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 .

Intersection homology and Poincar´ e duality Intersection homology A perversity picks out a subcomplex of intersection chains in S ∗ X : σ ∆ i ⇒ σ − 1 S ⊂ ( i − codim S + p ( S )) -skeleton − → X p -allowable ⇐ 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.

Intersection homology and Poincar´ e duality Intersection homology A perversity picks out a subcomplex of intersection chains in S ∗ X : σ ∆ i ⇒ σ − 1 S ⊂ ( i − codim S + p ( S )) -skeleton − → X p -allowable ⇐ 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) ⇒ I p H ∗ X topological invariant. X pseudomfld, p GM perversity =

Intersection homology and Poincar´ e duality Intersection homology A perversity picks out a subcomplex of intersection chains in S ∗ X : σ ∆ i ⇒ σ − 1 S ⊂ ( i − codim S + p ( S )) -skeleton − → X p -allowable ⇐ 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) ⇒ I p H ∗ X topological invariant. X pseudomfld, p GM perversity = Theorem (Goresky–MacPherson ’80) X compact, oriented n-dim pseudomfld, p , q complementary GM perversities = ⇒ ∃ intersection pairing I p H i X × I q H n − i X → Z which is non-degenerate over Q .

Part II Filtered homology

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 .

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 .

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.

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.

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. 2 1 1 0 2 2 1 The face maps ∆ i − 1 → ∆ i δ +1 ֒ δ are filtered.

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 } .

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 .

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 .

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 .

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 . There is a LES FS ∗ Y β FS ∗ X α · · · → FH ∗ X α → FH ∗ Y β → FH ∗ ( Y β , X α ) → FH ∗− 1 X α → · · ·

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 . There is a LES FS ∗ Y β FS ∗ X α · · · → 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 ) .

Simple examples of filtered homology Cones For [ x , t ] ∈ CX , the cone on X , and d > 1 have � � α ( x ) t > 0 i < d − 1 FH i X α ⇒ FH i CX β ∼ β [ x , t ] = t = 0 = = i ≥ d − 1 . d 0 When d ≤ 1 obtain homology of a point.

Simple examples of filtered homology Cones For [ x , t ] ∈ CX , the cone on X , and d > 1 have � � α ( x ) t > 0 i < d − 1 FH i X α ⇒ FH i CX β ∼ β [ x , t ] = t = 0 = = i ≥ d − 1 . d 0 When d ≤ 1 obtain homology of a point. Suspended torus Let X α = Σ T 2 where α ( x ) = 2 at suspension points and 0 elsewhere. Then  i = 0 Z     0 i = 1  FH i X α = Z 2 i = 2     i = 3 . Z 

Simple examples of filtered homology Cones For [ x , t ] ∈ CX , the cone on X , and d > 1 have � � α ( x ) t > 0 i < d − 1 FH i X α ⇒ FH i CX β ∼ β [ x , t ] = t = 0 = = i ≥ d − 1 . d 0 When d ≤ 1 obtain homology of a point. Suspended torus Let X α = Σ T 2 where α ( x ) = 3 at suspension points and 0 elsewhere. Then  i = 0 Z    Z 2  i = 1  FH i X α = 0 i = 2     i = 3 . Z 

Perversities and filtrations Given stratified X and perversity p define a depth function p ( x ) = codim S − p ( S ) ˆ for x ∈ S . The identity X ˆ p → X ˆ q is filtered ⇐ ⇒ p ≤ q .

Perversities and filtrations Given stratified X and perversity p define a depth function p ( x ) = codim S − p ( S ) ˆ for x ∈ S . The identity X ˆ p → X ˆ q is filtered ⇐ ⇒ p ≤ q . Setting X k = � codim S ≤ k S gives X 0 ⊂ X 1 ⊂ · · · ⊂ X k ⊂ · · · ⊂ X ◮ X ˆ � � 0 =

Perversities and filtrations Given stratified X and perversity p define a depth function p ( x ) = codim S − p ( S ) ˆ for x ∈ S . The identity X ˆ p → X ˆ q is filtered ⇐ ⇒ p ≤ q . Setting X k = � codim S ≤ k S gives X 0 ⊂ X 1 ⊂ · · · ⊂ X k ⊂ · · · ⊂ X ◮ X ˆ � � 0 = X 0 ⊂ X 1 ⊂ X ◮ X ˆ � � t =

Perversities and filtrations Given stratified X and perversity p define a depth function p ( x ) = codim S − p ( S ) ˆ for x ∈ S . The identity X ˆ p → X ˆ q is filtered ⇐ ⇒ p ≤ q . Setting X k = � codim S ≤ k S gives X 0 ⊂ X 1 ⊂ · · · ⊂ X k ⊂ · · · ⊂ X ◮ X ˆ � � 0 = X 0 ⊂ X 1 ⊂ X ◮ X ˆ � � t = X 0 ⊂ X 1 ⊂ X 2 ⊂ X 4 ⊂ · · · ⊂ X ◮ X ˆ � � m =

Perversities and filtrations Given stratified X and perversity p define a depth function p ( x ) = codim S − p ( S ) ˆ for x ∈ S . The identity X ˆ p → X ˆ q is filtered ⇐ ⇒ p ≤ q . Setting X k = � codim S ≤ k S gives X 0 ⊂ X 1 ⊂ · · · ⊂ X k ⊂ · · · ⊂ X ◮ X ˆ � � 0 = X 0 ⊂ X 1 ⊂ X ◮ X ˆ � � t = X 0 ⊂ X 1 ⊂ X 2 ⊂ X 4 ⊂ · · · ⊂ X ◮ X ˆ � � m = X 0 ⊂ X 1 ⊂ X 3 ⊂ X 5 ⊂ · · · ⊂ X ◮ X ˆ � � n =