nLab tangential structure




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


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


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 B O ( 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} }


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_{ k n} \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, Kochman 96, section 1.4)

See also at B-bordism.


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 B S O ( n ) B S O(n) 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.

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} \,.


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


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 March 4, 2024 at 21:31:07. See the history of this page for a list of all contributions to it.