nLab zero-section into Thom space of universal line bundle is weak equivalence

Idea

The Thom space of the universal complex line bundle is weakly homotopy equivalent to the base space of the line bundle, hence to the classifying space $B U(1)$ for the circle group, and this is equivalence is exhibited by the zero section of the universal line bundle, followed by its inclusion into its Thom space (Adams 74, Part I, Example 2.1, see Prop. below).

This statement plays a key role in the discussion of complex oriented cohomology, as it implies that any choice of universal first Conner-Floyd Chern class $c^E_1$ is equivalent to a choice of universal Thom class on the universal complex line bundle, and hence induces a Thom class, hence a “fiberwise complex orientation” on any complex line bundle. (See at Conner-Floyd E-Chern classes are E-Thom classes for more on this.)

In this context the statement is naturally stated in the form

$$B \mathrm{U}(1) \overset{\simeq}{\longrightarrow} M \mathrm{U}(1) \,,$$

where on the right the Thom space is thought of as the component space in degree 2 (complex degree 1) of the universal unitary Thom spectrum MU.

The analogous statement is true also for the universal real- and quaternionic line bundles, and it implies the analogous consequence for quaternionic oriented cohomology theory, etc., notably

$$B Sp(1) \overset{\simeq}{\longrightarrow} M Sp(1) \,,$$

where on the right the Thom space is thought of as the component space in degree 4 (quaternionic degree 1) of the universal quaternionic unitary Thom spectrum MSp.

Preliminaries

The universal line bundle

Let $\mathbb{K} \,\in\, \{\mathbb{R}, \mathbb{C}, \mathbb{H}\}$ be the real numbers or complex numbers or quaternions. Write

(1)$$S(\mathbb{K}) \;\coloneqq\; \big\{ q \in \mathbb{K} \;\big\vert\; q q^\ast =1 \big\} \;\; \in \; Groups$$

for its multiplicative group of unit-norm elements. Specifically this is the cyclic group of order 2, the circle group or the quaternionic unitary group/SU(2):

$$S(\mathbb{R}) \,\simeq\, \mathbb{Z}/2 \,, \phantom{AA} S(\mathbb{C}) \,\simeq\, \mathrm{U}(1) \,, \phantom{AA} S(\mathbb{H}) \,\simeq\, Sp(1) \,\simeq\, SU(2)$$

By either left or right multiplication in $\mathbb{K}$ this group acts on $\mathbb{K}$, $\mathbb{R}$-linearly, making $\mathbb{K}$ a linear representation

(2)$$\mathbb{K} \,\in\, S(\mathbb{K}) Representations_{\mathbb{R}} \,.$$

Hence with

$$E \big( S(\mathbb{K}) \big) \overset{\;\;\;}{\longrightarrow} B \big( S(\mathbb{K}) \big)$$

denoting the $S(\mathbb{K})$-universal principal bundle over the classifying space for the group (1), the real vector bundle underlying the universal K-line bundle is the corresponding associated bundle via the above action (2):

(3)$$\array{ E \big( S(\mathbb{K}) \big) \underset{ S(\mathbb{K}) }{\times} \mathbb{K} \\ \big\downarrow \\ B \big( S(\mathbb{K}) \big) }$$

Thom spaces

The Thom space of a real topological vector bundle $\mathcal{V}_X$ over some base space $X$ is the homotopy cofiber of its associated spherical fibration:

(4)$$S_X(\mathcal{V}_X) \overset{ p_{S(\mathcal{V}_X)} }{\longrightarrow} X \overset{ hocofib }{\longrightarrow} Th(X) \,.$$

When the topological space $X$ has the structure of a CW-complex then a cofibration which models the homotopy type of $p_{S(\mathcal{V}_X)}$ in the classical model structure on topological spaces is the inclusion of the unit sphere bundle into the unit disk bundle $D_X(\mathcal{V}_X)$ of $\mathcal{V}_X$ (with respect to any choice of fiberwise metric) since this is then a relative cell complex-inclusion. Therefore the homotopy cofiber (4) is then represented by the 1-category theoretic cofiber

(5)$$S_X(\mathcal{V}_X) \overset{ i_{S_X(\mathcal{V}_X)} }{\longrightarrow} D_X(\mathcal{V}_X) \overset{ cofib }{\longrightarrow} Th(X) \,.$$

Finally, since the zero section of the unit disk bundle is manifestly the weak homotopy equivalence that exibits this cofibrant resolution, we may call

(6)$$0_X \;\colon\; X \underoverset {\simeq} { 0_{D_X(\mathcal{V}_X)} } {\longrightarrow} D(\mathcal{V}_X) \overset {cofib\big( i_{S_X(\mathcal{V}_X)} \big)} {\longrightarrow} Th(X)$$

the zero-section into the Thom spaces.

Statement

Proposition

The zero-section (6) into the Thom space of the universal $\mathbb{K}$-line bundle (3)

$$\array{ Th \Big( E (S(\mathbb{K})) \underset{ S(\mathbb{K}) }{\times} \mathbb{K} \Big) \\ {}^{{}_{\mathllap{ 0_{ B \big( S(\mathbb{K}) \big) } }}} \big\uparrow {}^{{}_{ \mathrlap{\simeq}} } \\ B \big( S(\mathbb{K}) \big) }$$

is a weak homotopy equivalence.

Proof

The point is that for the universal line bundle, the associated sphere bundle is homotopy equivalent to the universal principal bundle and hence weakly contractible.

One way to see it is to unwind the definition of the unit sphere bundle in the universal line bundle as follows:

(7)$$S_{B \big(S(\mathbb{K})\big)} \Big( E\big(S(\mathbb{K})\big) I \underset{S(\mathbb{K})}{\times} \mathbb{K} \Big) \;=\; \Big( E\big(S(\mathbb{K})\big) \underset{S(\mathbb{K})}{\times} S(\mathbb{K}) \Big) \;=\; E\big(S(\mathbb{K})\big) \;\simeq\; \ast \,.$$

Another way to see the same is to observe that the sphere bundle associated to the universal line bundle is the sequential colimit over the tautological principal bundles over the finite-dimensional complex projective space, which themselves are n-spheres (see there). With this, the statement (7) follows from the fact that the infinite-dimensional sphere is weakly contractible (see there):

$$S_{\mathbb{K}P^\infty} \big( \mathcal{L}^\ast_{\mathbb{K}P^\infty} \big) \;\simeq\; \underset{ \underset{n}{\longrightarrow} }{\lim} \; \big( S_{\mathbb{K}P^n} \left( \mathcal{L}^\ast_{\mathbb{K}P^n} \right) \big) \;\simeq\; \underset{ \underset{n}{\longrightarrow} }{\lim} \; \left( S^{ (n+1) \cdot dim_{{}_{\mathbb{R}}}(\mathbb{K}) - 1 } \right) \;\simeq\; S^\infty \;\simeq\; \ast \,.$$

In any case, this means that we have the following solid commuting diagram, where the solid vertical morphisms are all weak homotopy equivalences:

(Here the left vertical map picks any point of the sphere bundle. There is then a unique horizontal map on the left to make the left square commute.)

Now, since the classifying space $B(S(\mathbb{K}))$ does have the structure of a CW-complex (given, for instance, by its realization as infinite real/complex/quaternionic projective space $\mathbb{K}P^\infty$, via the cell structure of projective space), the bottom cofiber here represents, as in (5), the defining homotopy cofiber.

Since homotopy cofibers are preserved, up to weak equivalence, by weak equivalences of their diagrams (by this Prop.), it follows that the dashed vertical morphism is a weak equivalence in the classical model structure on topological spaces, hence a weak homotopy equivalence.

This is the statement that was to be shown. Or more explicitly: By two-out-of-three also the composite vertical morphism on the right is a weak homotopy equivalence, which is the desired morphism in the form (6).

Related statements

• cell structure of projective space

• Conner-Floyd E-Chern classes are E-Thom classes

See also:

• complex oriented cohomology, universal complex orientation on MU

• quaternionic oriented cohomology

References

• Frank Adams, Part I, Example 2.1 in: Stable homotopy and generalised homology, 1974

• Stanley Kochman, Lemma 2.6.5 in: Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996