*This is a follow-up question on a [previous question](https://mathoverflow.net/questions/387491/nonvanishing-section-of-infinite-dimensional-tautological-bundle/387495#387495) of mine, which ended up to be trivial, because I overlooked the obvious problem with Hilbert space bundles, which I fix here.*




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Let us write $E$ for the space of complex-valued sequences $\{\dots, a\_{-2}, a\_{-1}, a\_1, a\_2, \dots\}$, labeled by $(\mathbb{Z}\setminus\{0\})$, such that only finitely many $a\_n$ are non-zero. For a subset $I \subset \mathbb{Z}$, write
$$E\_I = \mathrm{span}\{ e^i ~|~ i \in I\} \subseteq E,$$
where $e^i$, $i \in \mathbb{Z}$ are the canonical basis vectors of $E$.


Let us call a vector subspace $W \subset E$ *half-dimensional* if there exists $N \in \mathbb{N}$, such that
$$ W = W^\prime \oplus E\_{[N, \infty)},$$
where $W^\prime$ is a half-dimensional subspace of the (finite-dimensional) space $E\_{[-N, N]}$. Consider the following space
$$\mathrm{Gr}\_\infty = \{ W \subset E ~|~ W \text{ is half-dimensional} \}.$$
This has a natural direct limit topology:
$ \mathrm{Gr}\_\infty = \lim\_{n \to \infty} \mathrm{Gr}\_n(E\_{[-n, n]}),$
where the map $\mathrm{Gr}\_n(E\_{[-n, n]}) \to \mathrm{Gr}\_{n+1}(E\_{[-(n+1), n+1]})$ is given by sending $W \to W \oplus \mathbb{C} e^{n+1}$.


There is a tautological bundle $\tau$ over $\mathrm{Gr}\_\infty$, with fibers $\tau(W) = W$.


**Q: Does this bundle have a non-vanishing section?**


In the previous version of the question, the answer was trivially yes: the bundle was a bundle of Hilbert spaces, which has contractible automorphism group. Here, we have a bundle with typical fiber $\mathbb{C}^\infty$, the automorphism group of which is $U\_\infty = \lim\_{n \to \infty} U\_n$, which has lots of trivial topology.



Nonvanishing section of infinite-dimensional tautological bundle II
$\DeclareMathOperator{\Gr}{Gr}\newcommand{\C}{\mathbb{C}}\DeclareMathOperator\*{\colim}{colim}$The total space of your bundle is the colimit of the spaces $\tau\_n\times\C^{[N,\infty)}$, where $\tau\_n\to \Gr\_n(E\_{[-n,n]})$ is the tautological bundle over $\Gr\_n(E\_{[-n,n]})$ and the transition functions are the product of the bundle map $\tau\_n\oplus \C\to \tau\_{n+1}$, which exhibits the left-hand side as the pullback of $\tau\_{n+1}$ along the inclusion $\Gr\_n(E\_{[-n,n]})\hookrightarrow \Gr\_{n+1}(E\_{[-(n+1),n+1]})$, and the identity of $\C^{[N+1,\infty)}$.
In particular, it receives a map from $\colim\_n \tau\_n$, and the induced map from the complements of zero sections $\colim\_n \tau\_n^\times$ lands in the complement of the zero section. Now the map $\tau\_n^\times\to \Gr\_n(E\_{[-n,n]})$ is a fiber bundle, in particular a Serre fibration, with fiber $S^{2n-1}$. Since a filtered colimit of Serre fibrations is a Serre fibration, the map $\colim\_n \tau\_n^\times\to \Gr\_\infty$ is a Serre fibration with fiber $\colim\_n S^{2n-1}\simeq \*$, i.e. an acyclic Serre fibration. Since the base $\Gr\_\infty$ has the homotopy type of a CW complex, there is a section $\Gr\_\infty\to\colim\_n\tau\_n^\times$, which then maps to a nonzero section of your bundle.


It is probably also possible to explicitly construct such a section by using that the shift map on sequences is homotopic to the identity.


