Transport of structure
Encyclopedia
In mathematics
, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists. Definitions by transport of structure are regarded as canonical.
Since mathematically structures are often defined in reference to an underlying spaces, many examples of transport of structure involve spaces and mappings between them. For example, if V and W are vector space
s, and if
is an isomorphism
, and if
is an inner product on 
, then we can define an inner product 
on V by
.
Although the equation makes sense even when
is not an isomorphism, it only defines an inner product on V when 
is, since otherwise it will cause 
to be degenerate. The idea is that 
allows us to consider V and W as "the same" vector space, and if we follow this analogy, we can transport an inner product from one to the other.
A more involved example comes from differential topology
, in which we have the notion of a smooth manifold. If M is such a manifold, and if X is any topological space which is homeomorphic to M, we can consider X as a smooth manifold as well. That is, let
be a homeomorphism; we must define coordinate charts on X, which we will do by "pulling back" coordinate charts on M through 
. Recall that a coordinate chart on 
is an open set U together with an injective map
for some n; to get such a chart on X, we let
and 
.
Furthermore, it is required that the charts cover M, we must check that the transported charts cover X, which follows immediately from the fact that
is a bijection
. Finally, since M is a smooth manifold, we have that if U and V, with their maps
and 
,
are two charts on M, then the composition, the "transition map"
(a self-map of 
)
is smooth. We must check this for our transported charts on X. We have
,
and therefore
, and
.
Therefore the transition map for
and 
is the same as that for U and V, hence smooth. Therefore X is a smooth manifold via transport of structure.
Although the second example involved considerably more checking, the principle was the same, and any experienced mathematician would have no difficulty performing the necessary verifications. Therefore when such an operation is indicated, it is invoked merely as "transport of structure" and the details left to the reader, if desired.
The second example also illustrates why "transport of structure" is not always desirable. Namely, we can take M to be the plane, and we can take X to be an infinite one-sided cone. By "flattening" the cone we achieve a homeomorphism of X and M, and therefore the structure of a smooth manifold on X, but the cone is not "naturally" a smooth manifold. That is, we can consider X as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of exotic sphere
s, discovered by Milnor, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition not diffeomorphic) to
, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonical
isomorphism between the two objects.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists. Definitions by transport of structure are regarded as canonical.
Since mathematically structures are often defined in reference to an underlying spaces, many examples of transport of structure involve spaces and mappings between them. For example, if V and W are vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s, and if
is an isomorphismIsomorphism
In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations. If there exists an isomorphism between two structures, the two structures are said to be isomorphic. In a certain sense, isomorphic structures are...
, and if
is an inner product on , then we can define an inner product on V by.Although the equation makes sense even when
is not an isomorphism, it only defines an inner product on V when is, since otherwise it will cause to be degenerate. The idea is that allows us to consider V and W as "the same" vector space, and if we follow this analogy, we can transport an inner product from one to the other.A more involved example comes from differential topology
Differential topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.- Description :...
, in which we have the notion of a smooth manifold. If M is such a manifold, and if X is any topological space which is homeomorphic to M, we can consider X as a smooth manifold as well. That is, let
be a homeomorphism; we must define coordinate charts on X, which we will do by "pulling back" coordinate charts on M through . Recall that a coordinate chart on is an open set U together with an injective mapfor some n; to get such a chart on X, we let
and .Furthermore, it is required that the charts cover M, we must check that the transported charts cover X, which follows immediately from the fact that
is a bijectionBijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
. Finally, since M is a smooth manifold, we have that if U and V, with their maps
and ,are two charts on M, then the composition, the "transition map"
(a self-map of )is smooth. We must check this for our transported charts on X. We have
,and therefore
, and.Therefore the transition map for
and is the same as that for U and V, hence smooth. Therefore X is a smooth manifold via transport of structure.Although the second example involved considerably more checking, the principle was the same, and any experienced mathematician would have no difficulty performing the necessary verifications. Therefore when such an operation is indicated, it is invoked merely as "transport of structure" and the details left to the reader, if desired.
The second example also illustrates why "transport of structure" is not always desirable. Namely, we can take M to be the plane, and we can take X to be an infinite one-sided cone. By "flattening" the cone we achieve a homeomorphism of X and M, and therefore the structure of a smooth manifold on X, but the cone is not "naturally" a smooth manifold. That is, we can consider X as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of exotic sphere
Exotic sphere
In differential topology, a mathematical discipline, an exotic sphere is a differentiable manifold M that is homeomorphic but not diffeomorphic to the standard Euclidean n-sphere...
s, discovered by Milnor, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition not diffeomorphic) to
, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonicalCanonical
Canonical is an adjective derived from canon. Canon comes from the greek word κανών kanon, "rule" or "measuring stick" , and is used in various meanings....
isomorphism between the two objects.