nLab tangential structure

Idea

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

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

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

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 $A = \mathbf{B}G$, the classifying stack/delooping of a Lie group $G$).

Details

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

Definition

A $(B,f)$-structure is

1. for each $n\in \mathbb{N}$ a pointed CW-complex $B_n \in Top_{CW}^{\ast/}$

2. equipped with a pointed Serre fibration

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

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

3. for all $n_1 \leq n_2$ a pointed continuous function

   $\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$

   which is the identity for $n_1 = n_2$;

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

$$\array{ B_{n_1} &\overset{\iota_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,,$$

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

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

$$\array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) }$$

and which satisfy the evident associativity and unitality, for $B_0 = \ast$ the unit, and, finally, which commute with the maps $\iota$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:

$$\array{ B_{n_1} \times B_{n_2} &\overset{id \times \iota_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} }$$

and

$$\array{ B_{n_1} \times B_{n_2} &\overset{\iota_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,.$$

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

$$f_{2n} \colon B_{2n} \longrightarrow B O(2n)$$

indexed only on the even natural numbers.

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

$$f_{k n} \colon B_{ k n} \longrightarrow B O(k n)$$

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

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

See also at B-bordism.

Example

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

1. $B_n = B O(n)$ and $f_n = id$ is orthogonal structure (or “no structure”);

2. $B_n = E O(n)$ and $f_n$ the universal principal bundle-projection is framing-structure;

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

4. $B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the spin group and $f_n$ the canonical projection is spin structure.

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

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

Definition

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

1. an embedding $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;

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

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

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

1. $k_2 \geq k_1$

2. the second inclusion factors through the first as

   $$(i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2}$$

3. the lift of the classifying map factors accordingly (as homotopy classes)

   $$\hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{n} \longrightarrow B_{n} \,.$$

Examples

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

• orientation

• spin structure

• string structure

• fivebrane structure

• …

• framing

References

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

• Richard Lashof, Poincaré duality and cobordism, Trans. AMS 109 (1963), 257-277 (doi:10.1090/S0002-9947-1963-0156357-4)

• 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)

• Stanley Kochman, section 1.4 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

The terminology “tangential structure” became popular around

• Søren Galatius, Ib Madsen, Ulrike Tillmann,Michael Weiss, Section 5 of: The homotopy type of the cobordism category, Acta Math. 202 (2009), no. 2, 195–239 (arXiv:math/0605249)