Transport of structure

Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them. For example, if ${\displaystyle V}$ and ${\displaystyle W}$ are vector spaces with ${\displaystyle (\cdot ,\cdot )}$ being an inner product on ${\displaystyle W}$, such that there is an isomorphism ${\displaystyle \phi }$ from ${\displaystyle V}$ to ${\displaystyle W}$, then one can define an inner product ${\displaystyle [\cdot ,\cdot ]}$ on ${\displaystyle V}$ by the following rule:

${\displaystyle [v_{1},v_{2}]=(\phi (v_{1}),\phi (v_{2}))}$

Although the equation makes sense even when ${\displaystyle \phi }$ is not an isomorphism, it only defines an inner product on ${\displaystyle V}$ when ${\displaystyle \phi }$ is, since otherwise it will cause ${\displaystyle [\cdot ,\cdot ]}$ to be degenerate. The idea is that ${\displaystyle \phi }$ allows one to consider ${\displaystyle V}$ and ${\displaystyle W}$ as "the same" vector space, and by following this analogy, then one can transport an inner product from one space to the other.

A more elaborated example comes from differential topology, in which the notion of smooth manifold is involved: if ${\displaystyle M}$ is such a manifold, and if ${\displaystyle X}$ is any topological space which is homeomorphic to ${\displaystyle M}$, then one can consider ${\displaystyle X}$ as a smooth manifold as well. That is, given a homeomorphism ${\displaystyle \phi \colon X\to M}$, one can define coordinate charts on ${\displaystyle X}$ by "pulling back" coordinate charts on ${\displaystyle M}$ through ${\displaystyle \phi }$. Recall that a coordinate chart on ${\displaystyle M}$ is an open set ${\displaystyle U}$ together with an injective map

${\displaystyle c\colon U\to \mathbb {R} ^{n}}$

for some natural number ${\displaystyle n}$; to get such a chart on ${\displaystyle X}$, one uses the following rules:

${\displaystyle U'=\phi ^{-1}(U)}$ and ${\displaystyle c'=c\circ \phi }$.

Furthermore, it is required that the charts cover ${\displaystyle M}$ (the fact that the transported charts cover ${\displaystyle X}$ follows immediately from the fact that ${\displaystyle \phi }$ is a bijection). Since ${\displaystyle M}$ is a smooth manifold, if U and V, with their maps ${\displaystyle c\colon U\to \mathbb {R} ^{n}}$ and ${\displaystyle d\colon V\to \mathbb {R} ^{n}}$, are two charts on ${\displaystyle M}$, then the composition, the "transition map"

${\displaystyle d\circ c^{-1}\colon c(U\cap V)\to \mathbb {R} ^{n}}$ (a self-map of ${\displaystyle \mathbb {R} ^{n}}$)

is smooth. To verify this for the transported charts on ${\displaystyle X}$, notice that

${\displaystyle \phi ^{-1}(U)\cap \phi ^{-1}(V)=\phi ^{-1}(U\cap V)}$,

and therefore

${\displaystyle c'(U'\cap V')=(c\circ \phi )(\phi ^{-1}(U\cap V))=c(U\cap V)}$, and

${\displaystyle d'\circ (c')^{-1}=(d\circ \phi )\circ (c\circ \phi )^{-1}=d\circ (\phi \circ \phi ^{-1})\circ c^{-1}=d\circ c^{-1}}$.

Thus the transition map for ${\displaystyle U'}$ and ${\displaystyle V'}$ is the same as that for ${\displaystyle U}$ and ${\displaystyle V}$, hence smooth. That is, ${\displaystyle X}$ is a smooth manifold via transport of structure. This is a special case of transport of structures in general.^[2]

The second example also illustrates why "transport of structure" is not always desirable. Namely, one can take ${\displaystyle M}$ to be the plane, and ${\displaystyle X}$ to be an infinite one-sided cone. By "flattening" the cone, a homeomorphism of ${\displaystyle X}$ and ${\displaystyle M}$ can be obtained, and therefore the structure of a smooth manifold on ${\displaystyle X}$, but the cone is not "naturally" a smooth manifold. That is, one can consider ${\displaystyle X}$ 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 spheres, discovered by John Milnor, which states that there are exactly 28 smooth manifolds which are homeomorphic but not diffeomorphic to ${\displaystyle S^{7}}$, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonical isomorphism between the two objects.

• List of mathematical jargon
• Equivalent definitions of mathematical structures#Transport of structures; isomorphism