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Relative Cobordism
M Naeem Ahmad
Kansas State University
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 2 / 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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 2 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 3 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 3 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 3 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 3 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 3 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 3 / 13
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)) $$HHHH
HHHH
H
t(A)// F (∂A)
F (i(A))vvvv
vvvv
v
F (A)
is commutative.
Notation
The subcategory of closed objects of C will be denoted by Ccl.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 4 / 13
The Category P
Let (C, ∂, i) and (C′, ∂′, i ′) be two cobordism categories, F : C → C′ an
additive functor, and t : ∂′F∼= // F∂ a natural equivalence of additive
functors such that the diagram
∂′F (A)
i ′(F (A)) $$HHHH
HHHH
H
t(A)// F (∂A)
F (i(A))vvvv
vvvv
v
F (A)
is commutative.
Notation
The subcategory of closed objects of C will be denoted by Ccl.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 4 / 13
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)) $$HHHH
HHHH
H
t(A)// F (∂A)
F (i(A))vvvv
vvvv
v
F (A)
is commutative.
Notation
The subcategory of closed objects of C will be denoted by Ccl.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 4 / 13
The Category P
Let (C, ∂, i) and (C′, ∂′, i ′) be two cobordism categories, F : C → C′ an
additive functor, and t : ∂′F∼= // F∂ a natural equivalence of additive
functors such that the diagram
∂′F (A)
i ′(F (A)) $$HHHH
HHHH
H
t(A)// F (∂A)
F (i(A))vvvv
vvvv
v
F (A)
is commutative.
Notation
The subcategory of closed objects of C will be denoted by Ccl.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 4 / 13
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)) $$HHHH
HHHH
H
t(A)// F (∂A)
F (i(A))vvvv
vvvv
v
F (A)
is commutative.
Notation
The subcategory of closed objects of C will be denoted by Ccl.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 4 / 13
The Category P
Let (C, ∂, i) and (C′, ∂′, i ′) be two cobordism categories, F : C → C′ an
additive functor, and t : ∂′F∼= // F∂ a natural equivalence of additive
functors such that the diagram
∂′F (A)
i ′(F (A)) $$HHHH
HHHH
H
t(A)// F (∂A)
F (i(A))vvvv
vvvv
v
F (A)
is commutative.
Notation
The subcategory of closed objects of C will be denoted by Ccl.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 4 / 13
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
FY1
Fψ
∂′X2f2 // FY2
commutes.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 5 / 13
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
FY1
Fψ
∂′X2f2 // FY2
commutes.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 5 / 13
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
FY1
Fψ
∂′X2f2 // FY2
commutes.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 5 / 13
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
FY1
Fψ
∂′X2f2 // FY2
commutes.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 5 / 13
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
FY1
Fψ
∂′X2f2 // FY2
commutes.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 5 / 13
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
FY1
Fψ
∂′X2f2 // FY2
commutes.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 5 / 13
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).
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 6 / 13
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).
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 6 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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 setof 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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 7 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 8 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 9 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 10 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 10 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 10 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 10 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 10 / 13
Relative Cobordism Semigroup
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.
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 10 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 by
α((X1,Y1, f1), (X2,Y2, f2)) = X1 ∪k −X2. The definition of homomorphismα does not depend upon the choice of g .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 .
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 11 / 13
Relative Cobordism Semigroup
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 ′).
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 12 / 13
Relative Cobordism Semigroup
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 ′).
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 12 / 13
The End
Thank you!!!
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 13 / 13
The End
Thank you!!!
M Naeem Ahmad (KSU) Relative Cobordism September 24, 2010 13 / 13