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Contents
Context
Manifolds and cobordisms
Contents
Idea
Given a smooth manifold , by a tangential structure one typically understands (e.g. GMWT 09, Sec. 5) a lift of the classifying map of its tangent bundle through any prescribed map into the classifying space of the general linear group, up to homotopy:
Since this is all considered (only) for homotopy types of topological spaces (e.g. via the classical model structure on topological spaces) and there is a weak homotopy equivalence to the classifying space of the orthogonal group (the latter being the maximal compact subgroup of ), authors typically consider the equivalent diagram over .
Beware that the same kind of lift but understood in differentiable classifying stacks instead of just classifying spaces is a G-structure as commonly understood now (for , the classifying stack/delooping of a Lie group ).
Details
In terms of -structures
Definition
A -structure is
-
for each a pointed CW-complex
-
equipped with a pointed Serre fibration
to the classifying space (def.);
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for all a pointed continuous function
which is the identity for ;
such that for all these squares commute
where the bottom map is the canonical one (def.).
The -structure is multiplicative if it is moreover equipped with a system of maps which cover the canonical multiplication maps (def.)
and which satisfy the evident associativity and unitality, for the unit, and, finally, which commute with the maps in that all these squares commute:
and
Similarly, an --structure is a compatible system
indexed only on the even natural numbers.
Generally, an --structure for , is a compatible system
for all , hence for all .
(Lashof 63, Stong 68, beginning of chapter II, Kochman 96, section 1.4)
See also at B-bordism.
Example
Examples of -structures (def. ) include the following:
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and is orthogonal structure (or “no structure”);
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and the universal principal bundle-projection is framing-structure;
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the classifying space of the special orthogonal group and the canonical projection is orientation structure;
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the classifying space of the spin group and the canonical projection is spin structure.
Examples of --structures include
- the classifying space of the unitary group, and the canonical projection is almost complex structure.
Definition
Given a smooth manifold of dimension , and given a -structure as in def. , then a -structure on the manifold is an equivalence class of the following structure:
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an embedding for some ;
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a homotopy class of a lift of the classifying map of the tangent bundle
The equivalence relation on such structures is to be that generated by the relation if
-
-
the second inclusion factors through the first as
-
the lift of the classifying map factors accordingly (as homotopy classes)
Examples
The tangential structures corresponding to lifts through the Whitehead tower of the orthogonal group
References
The concept of tangential structure originates with cobordism theory, originally under the name -structures:
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Richard Lashof, Poincaré duality and cobordism, Trans. AMS 109 (1963), 257-277 (doi:10.1090/S0002-9947-1963-0156357-4)
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Robert Stong, beginning of chapter II of: Notes on Cobordism theory, 1968,
reprinted as: Princeton Legacy Library, Princeton University Press 2016 (ISBN:9780691649016, toc pdf)
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Stanley Kochman, section 1.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
The terminology “tangential structure” became popular around