nLab
tangential structure

Contents

Contents

Idea

Given a smooth manifold XX, by a tangential structure one typically understands (e.g. GMWT 09, Sec. 5) a lift of the classifying map XBGL(n)X \overset{\vdash}{\longrightarrow} B GL(n) of its tangent bundle through any prescribed map BfBGL(n)B \overset{f}{\longrightarrow} B GL(n) into the classifying space of the general linear group, up to homotopy:

B tangentialstructure f X TX BGL(n) \array{ && B \\ & {}^{ \mathllap{ {tangential} \atop {structure} } }\nearrow & \big\downarrow^{ f } \\ X &\underset{ \vdash T X }{\longrightarrow}& B GL(n) }

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 BGL(n)BO(n)B GL(n) \simeq B O(n) to the classifying space of the orthogonal group (the latter being the maximal compact subgroup of GL(n)GL(n)), authors typically consider the equivalent diagram over BO(n)B O(n).

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 bow (for A=BGA = \mathbf{B}G is the classifying stack/delooping of a Lie group GG).

Details

In terms of (B,f)(B,f)-structures

Definition

A (B,f)(B,f)-structure is

  1. for each nn\in \mathbb{N} a pointed CW-complex B nTop CW */B_n \in Top_{CW}^{\ast/}

  2. equipped with a pointed Serre fibration

    B n f n BO(n) \array{ B_n \\ \downarrow^{\mathrlap{f_n}} \\ B O(n) }

    to the classifying space BO(n)B O(n) (def.);

  3. for all n 1n 2n_1 \leq n_2 a pointed continuous function

    ι n 1,n 2:B n 1B n 2\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}

    which is the identity for n 1=n 2n_1 = n_2;

such that for all n 1n 2n_1 \leq n_2 \in \mathbb{N} these squares commute

B n 1 ι n 1,n 2 B n 2 f n 1 f n 2 BO(n 1) BO(n 2), \array{ B_{n_1} &\overset{\iota_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,,

where the bottom map is the canonical one (def.).

The (B,f)(B,f)-structure is multiplicative if it is moreover equipped with a system of maps μ n 1,n 2:B n 1×B n 2B n 1+n 2\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2} which cover the canonical multiplication maps (def.)

B n 1×B n 2 μ n 1,n 2 B n 1+n 2 f n 1×f n 2 f n 1+n 2 BO(n 1)×BO(n 2) BO(n 1+n 2) \array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) }

and which satisfy the evident associativity and unitality, for B 0=*B_0 = \ast the unit, and, finally, which commute with the maps ι\iota in that all n 1,n 2,n 3n_1,n_2, n_3 \in \mathbb{N} these squares commute:

B n 1×B n 2 id×ι n 2,n 2+n 3 B n 1×B n 2+n 3 μ n 1,n 2 μ n 1,n 2+n 3 B n 1+n 2 ι n 1+n 2,n 1+n 2+n 3 B n 1+n 2+n 3 \array{ B_{n_1} \times B_{n_2} &\overset{id \times \iota_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} }

and

B n 1×B n 2 ι n 1,n 1+n 3×id B n 1+n 3×B n 2 μ n 1,n 2 μ n 1+n 3,n 2 B n 1+n 2 ι n 1+n 2,n 1+n 2+n 3 B n 1+n 2+n 3. \array{ B_{n_1} \times B_{n_2} &\overset{\iota_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,.

Similarly, an S 2S^2-(B,f)(B,f)-structure is a compatible system

f 2n:B 2nBO(2n) f_{2n} \colon B_{2n} \longrightarrow B O(2n)

indexed only on the even natural numbers.

Generally, an S kS^k-(B,f)(B,f)-structure for kk \in \mathbb{N}, k1k \geq 1 is a compatible system

f kn:B knBO(kn) f_{k n} \colon B_{ kn} \longrightarrow B O(k n)

for all nn \in \mathbb{N}, hence for all knkk n \in k \mathbb{N}.

(Lashof 63, Stong 68, beginning of chapter II, Kochmann 96, section 1.4)

See also at B-bordism.

Example

Examples of (B,f)(B,f)-structures (def. ) include the following:

  1. B n=BO(n)B_n = B O(n) and f n=idf_n = id is orthogonal structure (or “no structure”);

  2. B n=EO(n)B_n = E O(n) and f nf_n the universal principal bundle-projection is framing-structure;

  3. B n=BSO(n)=EO(n)/SO(n)B_n = B SO(n) = E O(n)/SO(n) the classifying space of the special orthogonal group and f nf_n the canonical projection is orientation structure;

  4. B n=BSpin(n)=EO(n)/Spin(n)B_n = B Spin(n) = E O(n)/Spin(n) the classifying space of the spin group and f nf_n the canonical projection is spin structure.

Examples of S 2S^2-(B,f)(B,f)-structures include

  1. B 2n=BU(n)=EO(2n)/U(n)B_{2n} = B U(n) = E O(2n)/U(n) the classifying space of the unitary group, and f 2nf_{2n} the canonical projection is almost complex structure.
Definition

Given a smooth manifold XX of dimension nn, and given a (B,f)(B,f)-structure as in def. , then a (B,f)(B,f)-structure on the manifold is an equivalence class of the following structure:

  1. an embedding i X:X ki_X \; \colon \; X \hookrightarrow \mathbb{R}^k for some kk \in \mathbb{N};

  2. a homotopy class of a lift g^\hat g of the classifying map gg of the tangent bundle

    B n g^ f n X g BO(n). \array{ && B_{n} \\ &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_n}} \\ X &\overset{g}{\longrightarrow}& B O(n) } \,.

The equivalence relation on such structures is to be that generated by the relation ((i X) 1,g^ 1)((i X) ,g^ 2)((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2) if

  1. k 2k 1k_2 \geq k_1

  2. the second inclusion factors through the first as

    (i X) 2:X(i X) 1 k 1 k 2 (i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2}
  3. the lift of the classifying map factors accordingly (as homotopy classes)

    g^ 2:Xg^ 1B nB n. \hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{n} \longrightarrow B_{n} \,.

Examples

The tangential structures corresponding to lifts through the Whitehead tower of the orthingonal group

References

The concept of tangential structure originates with cobordism theory, originally under the name (B,f)(B,f)-structures:

The terminology “tangential structure” became popular around

Last revised on July 18, 2020 at 04:48:10. See the history of this page for a list of all contributions to it.