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i.e. the following diagram commutes:
B
φ
B ′
φ B
M ∐ φ ′ B
N
(c) For any object M ∈ Cob(n), the identity map is represented by the product bordism
B = M × [0, 1], i.e. the so-called cylinder over M.
(d) Composition of morphisms in Cob(n) is given by gluing bordisms together.: given
objects M, M ′ , M ′′ ∈ Cob(n), and bordisms B : M → M ′ and B ′ : M ′ → M ′′ , the
composition is defined to be the morphism represented by the manifold B ∐ M ′ B ′ .
(To get a smooth structure on this manifold, choices like collars are necessary. They
lead to diffeomorphic glued bordisms, however.)
2. For each n, the category Cob(n) can be endowed with the structure of a tensor category.
The tensor product
⊗ : Cob(n) × Cob(n) → Cob(n)
is given by disjoint union. The unit object of Cob(n) is the empty set, regarded as a
smooth manifold of dimension n − 1.
Example 2.5.16.
The objects of Cob(1) are finitely many oriented points. Thus objects are finite unions of ( •, +)
and (•, −).
The morphisms are oriented one-dimensional manifolds, possibly with boundary, i.e. unions
of intervals and circles.
An isomorphism class of objects is characterized by the numbers (n + , n − ) of points with
positive and negative orientation. Sometimes, one also considers another equivalence relation
on objects: two d − 1-dimensional closed manifolds M and N are called cobordant, if there is
a cobordism B : N → N. Since there is a cobordism (•, +) ∐ (•, −) → ∅, the objects (n + , n − )
and (n ′ +, n ′ −) are cobordant, if and only if n + − n − = n ′ + − n ′ −.
One can also define a category of unoriented cobordisms. In this case, simple objects are
disjoint unions of points, isomorphism classes are in bijection to the number of points. Since a
pair of points can annihilate, there are only two cobordism classes, depending on whether the
number of points is odd or even.
Observation 2.5.17.
Let M be a closed oriented n − 1-dimensional manifold. Then the oriented n-dimensional
manifold B := M × [0, 1], the cylinder over M, has boundary M ∐ M. The manifold B can be
considered as a cobordism in six different ways, corresponding to decomposition of its boundary:
• As a bordism M → M. This represents the identity on M.
• As a bordism M → M. This represents the identity on M.
• As a morphism d M : M ∐ M → ∅ or, alternatively, as a morphism ˜d M : M ∐ M → ∅.
• As a morphism ˜b M : ∅ → M ∐ M or, alternatively, as a morphism b M : ∅ → M ∐ M.
One checks that the axioms of a left and a right duality hold. We conclude that the category
Cob(n) is rigid.
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