M Naeem Ahmad - math.ksu.edunaeem/slideseminar2.pdf · Relative Cobordism M Naeem Ahmad Kansas...

Relative Cobordism

M Naeem Ahmad

Kansas State University

naeem@math.ksu.edu

KSU Graduate Student Topology Seminar

M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 1 / 13

Introduction

We will discuss the relative cobordism semigroup of two cobordismcategories. In order to describe it we will need to go systematicallythrough the definitions of Grothendieck group, category P, group S/ ∼,and homomorphisms β and α.

Introduction

We will discuss the relative cobordism semigroup of two cobordismcategories. In order to describe it we will need to go systematicallythrough the definitions of Grothendieck group, category P, group S/ ∼,and homomorphisms β and α.

Grothendieck Group

Let X be a category having finite sums whose isomorphism classes ofobjects form a set. Define two pairs (X1,X2), (Y1,Y2) ∈ X× X to beequivalent if there exists an A ∈ ob(X) such thatX1 + Y2 + A ∼= X2 + Y1 + A.

Definition

The set of equivalence classes of pairs in X× X, denoted by K (X), iscalled the Grothendieck group of X.

Remark

K (X) is an abelian group under the operation induced by the sum in X.

Grothendieck Group

Definition

Remark

Grothendieck Group

Definition

Remark

Grothendieck Group

Definition

Remark

Grothendieck Group

Definition

Remark

Grothendieck Group

Definition

Remark

The Category P

Let (C, ∂, i) and (C′, ∂′, i ′) be two cobordism categories, F : C → C′ an

additive functor, and t : ∂′F∼= // F∂ a natural equivalence of additive

functors such that the diagram

∂′F (A)

i ′(F (A)) $$HHHH

t(A)// F (∂A)

F (i(A))vvvv

is commutative.

Notation

The subcategory of closed objects of C will be denoted by Ccl.

The Category P

∂′F (A)

i ′(F (A)) $$HHHH

t(A)// F (∂A)

F (i(A))vvvv

is commutative.

Notation

The Category P

∂′F (A)

i ′(F (A)) $$HHHH

t(A)// F (∂A)

F (i(A))vvvv

is commutative.

Notation

The Category P

∂′F (A)

i ′(F (A)) $$HHHH

t(A)// F (∂A)

F (i(A))vvvv

is commutative.

Notation

The Category P

∂′F (A)

i ′(F (A)) $$HHHH

t(A)// F (∂A)

F (i(A))vvvv

is commutative.

Notation

The Category P

∂′F (A)

i ′(F (A)) $$HHHH

t(A)// F (∂A)

F (i(A))vvvv

is commutative.

Notation

The Category P

Definition

The category P is defined as the category with

ob(P) = (X ,Y , f ) | X ∈ C′,Y ∈ Ccl, f : ∂′X∼= // FY

and Map((X1,Y1, f1), (X2,Y2, f2)) the set of(φ, ψ) ∈ Map(X1,X2)×Map(Y1,Y2) such the diagram

∂′X1f1 //

∂′φ

∂′X2f2 // FY2

commutes.

The Category P

Definition

∂′X1f1 //

∂′φ

∂′X2f2 // FY2

commutes.

The Category P

Definition

∂′X1f1 //

∂′φ

∂′X2f2 // FY2

commutes.

The Category P

Definition

∂′X1f1 //

∂′φ

∂′X2f2 // FY2

commutes.

The Category P

Definition

∂′X1f1 //

∂′φ

∂′X2f2 // FY2

commutes.

The Category P

Definition

∂′X1f1 //

∂′φ

∂′X2f2 // FY2

commutes.

The Category P

Remark

The category P has finite sums and a small category P0(X ∈ C′0,Y ∈ C0)such that each element of ob(P) is isomorphic to an element in ob(P0).

The Category P

Remark

The category P has finite sums and a small category P0(X ∈ C′0,Y ∈ C0)such that each element of ob(P) is isomorphic to an element in ob(P0).

Relative Cobordism Semigroup

Let (x1, x2) and (y1, y2) be two elements of ob(P)× ob(P). We say that(x1, x2) ∼ (y1, y2) if there exist elements u, v ∈ ob(P) such thatx1 + u ∼= y1 + v and x2 + u ∼= y2 + v .

Remark

Let S = ((X1,Y1, f1), (X2,Y2, f2)) ∈ ob(P)× ob(P) | Y1∼= Y2. The set

of equivalence classes S/ ∼ together with the operation induced by thesum forms an abelian group.

Define the homomorphism

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

where ∅ is an initial object of C and j1, j2 are the unique homomorphismsof the initial object.

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

Let S = ((X1,Y1, f1), (X2,Y2, f2)) ∈ ob(P)× ob(P) | Y1∼= Y2.

The setof equivalence classes S/ ∼ together with the operation induced by thesum forms an abelian group.

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

where ∅ is an initial object of C

and j1, j2 are the unique homomorphismsof the initial object.

Remark

β : K (C′cl)→ S/ ∼

by(X1,X2) 7→ ((X1, ∅, j1), (X2, ∅, j2))

Suppose that we are given a homomorphism

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

such that α β = q, where q is the quotient homomorphism

K (C′cl)→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

then for two objects (X1,Y1, f1) and (X2,Y2, f2) of P, we write(X1,Y1, f1) ≡ (X2,Y2, f2) if there exist U1,U2 ∈ ob(C) such thatY1 + ∂U1

∼= Y2 + ∂U2, and

α((X1 + FU1,Y1 + ∂U1, f1 + tU1), (X2 + FU2,Y2 + ∂U2, f2 + tU2)) = 0.

Remark

The relation ≡ is an equivalence relation.

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

such that

α β = q, where q is the quotient homomorphism

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

such that α β = q,

where q is the quotient homomorphism

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

then for two objects (X1,Y1, f1) and (X2,Y2, f2) of P,

we write(X1,Y1, f1) ≡ (X2,Y2, f2) if there exist U1,U2 ∈ ob(C) such thatY1 + ∂U1

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

then for two objects (X1,Y1, f1) and (X2,Y2, f2) of P, we write

(X1,Y1, f1) ≡ (X2,Y2, f2) if there exist U1,U2 ∈ ob(C) such thatY1 + ∂U1

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

then for two objects (X1,Y1, f1) and (X2,Y2, f2) of P, we write(X1,Y1, f1) ≡ (X2,Y2, f2)

if there exist U1,U2 ∈ ob(C) such thatY1 + ∂U1

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

then for two objects (X1,Y1, f1) and (X2,Y2, f2) of P, we write(X1,Y1, f1) ≡ (X2,Y2, f2) if there exist

U1,U2 ∈ ob(C) such thatY1 + ∂U1

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2,

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

α : S/ ∼→ K (C′cl)/(∂′∗K (C′) + F∗K (Ccl))

∼= Y2 + ∂U2, and

Remark

Definition

The relative cobordism semigroup Ω(F , t, α) is the set of equivalenceclasses under ≡ of objects of P with the operation induced by the sum inP.

Remark

We have homomorphisms

F∗ : Ω(C , ∂, i)→ Ω(C ′, ∂′, i ′) given by Y 7→ FY ,

i : Ω(C ′, ∂′, i ′)→ Ω(F , t, α) given by X 7→ (X , ∅, j), and

∂ : Ω(F , t, α)→ Ω(C , ∂, i) given by (X ,Y , f ) 7→ Y

such that the diagram

Definition

Remark

Definition

Remark

Definition

Remark

Definition

Remark

Definition

Remark

Definition

Remark

Definition

Remark

Ω(C , ∂, i)F∗ // Ω(C ′, ∂′, i ′)

iwwppppppppppp

Ω(F , t, α)

ffNNNNNNNNNN

has period 2, that is iF∗ = ∂i = F∗∂ = 0.

In the following example we consider the geometric case in which αcorresponds to gluing of two manifolds along their common boundary.

Ω(C , ∂, i)F∗ // Ω(C ′, ∂′, i ′)

iwwppppppppppp

Ω(F , t, α)

ffNNNNNNNNNN

Ω(C , ∂, i)F∗ // Ω(C ′, ∂′, i ′)

iwwppppppppppp

Ω(F , t, α)

ffNNNNNNNNNN

has period 2,

that is iF∗ = ∂i = F∗∂ = 0.

Ω(C , ∂, i)F∗ // Ω(C ′, ∂′, i ′)

iwwppppppppppp

Ω(F , t, α)

ffNNNNNNNNNN

has period 2, that is

iF∗ = ∂i = F∗∂ = 0.

Ω(C , ∂, i)F∗ // Ω(C ′, ∂′, i ′)

iwwppppppppppp

Ω(F , t, α)

ffNNNNNNNNNN

Ω(C , ∂, i)F∗ // Ω(C ′, ∂′, i ′)

iwwppppppppppp

Ω(F , t, α)

ffNNNNNNNNNN

Example

Consider an object (X ,Y , f ) of P as a manifold with boundary togetherwith an additional structure on the boundary (e.g. orientation). Suppose

that ((X1,Y1, f1), (X2,Y2, f2)) ∈ S and g : Y1∼= // Y2 is an

isomorphism. Then F (g) : FY1∼= // FY2 is an isomorphism. Let

k : ∂′X1

∼= // ∂′X2 be the isomorphism given by the composition

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

The homomorphism α can be given byα((X1,Y1, f1), (X2,Y2, f2)) = X1 ∪k −X2. The definition of homomorphismα does not depend upon the choice of g .

Example

Consider an object (X ,Y , f ) of P as a manifold with boundary togetherwith an additional structure on the boundary (e.g. orientation).

Suppose

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

Example

isomorphism.

Then F (g) : FY1∼= // FY2 is an isomorphism. Let

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

Example

isomorphism. Then F (g) : FY1∼= // FY2 is an isomorphism.

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

Example

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

Example

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

Example

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

The homomorphism α can be given by

α((X1,Y1, f1), (X2,Y2, f2)) = X1 ∪k −X2. The definition of homomorphismα does not depend upon the choice of g .

Example

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

The homomorphism α can be given byα((X1,Y1, f1), (X2,Y2, f2)) = X1 ∪k −X2.

The definition of homomorphismα does not depend upon the choice of g .

Example

k : ∂′X1

f −12 F (g)f1 : ∂′X1

∼= // FY1

∼= // FY2

∼= // ∂′X2 .

Remark

If C is the subcategory of C′ consisting of initial objects, and F theinclusion. Then β being an epimorphism uniquely determines α. It turnsout that the relative cobordism group in this case can be identified withthe cobordism semigroup Ω(C ′, ∂′, i ′).

Remark

If C is the subcategory of C′ consisting of initial objects, and F theinclusion. Then β being an epimorphism uniquely determines α. It turnsout that the relative cobordism group in this case can be identified withthe cobordism semigroup Ω(C ′, ∂′, i ′).

The End

Thank you!!!

The End

Thank you!!!

M Naeem Ahmad - math.ksu.edunaeem/slideseminar2.pdf · Relative Cobordism M Naeem Ahmad Kansas...

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