Serre's 1951 problem on fibrations of Euclidean space with connected fibres: the triviality question

In his 1951 doctoral thesis that introduced the Leray–Serre spectral sequence, Jean-Pierre Serre proved that a locally trivial fibration with total space $E=\mathbb{R}^n$ and connected fibre has acyclic fibre and acyclic base, and that the fibration is trivial whenever the base is a locally finite polyhedron; he then recorded his expectation — which he found difficult to establish — that there are no other connected-fibre fibrations of $\mathbb{R}^n$ beyond the direct-product decompositions $\mathbb{R}^n=\mathbb{R}^p\times\mathbb{R}^q$. Three companion questions raised in the same thesis (whether the trivial-local-system hypothesis in the multiplicativity law $\chi(E)=\chi(B)\chi(F)$ for the Euler–Poincaré characteristic is superfluous; the determination of the exponents $k$ in $p$-primary torsion summands of homotopy groups of spheres; and the extension of the spectral-sequence methods to free loop spaces and closed geodesics) have been absorbed by later literature. The surviving open core is the triviality question for locally trivial fibrations of Euclidean space with connected fibres, with no hypotheses on base or fibre beyond connectedness: whether every such fibration is trivial. The literal 'only $\mathbb{R}^p\times\mathbb{R}^q$' clause of Serre's expectation is contradicted by classical manifold factors such as the Whitehead manifold $W$ with $W\times\mathbb{R}\cong\mathbb{R}^4$, but all known examples are trivial bundles, and no nontrivial example of such a fibration is recorded in the cited literature.

Background

A fibration is a continuous map $p:E\to B$ from a total space onto a base; the special case relevant here is a locally trivial fibration (fibre bundle): every point of $B$ has an open neighbourhood $U$ such that $p^{-1}(U)$ is homeomorphic to $U\times F$ over $U$, where $F=p^{-1}(b)$ is the fibre. The fibration is trivial when $E\cong B\times F$ compatibly with the projection. A space is acyclic if its reduced singular homology with integer coefficients vanishes, i.e. it has the homology of a point; the Euler–Poincaré characteristic $\chi$ is the alternating sum of the ranks of the homology groups, and it is the subject of the classical law $\chi(E)=\chi(B)\chi(F)$ for fibrations.

Serre's thesis (Annals of Mathematics, 1951) constructed the singular-homology spectral sequence of a fibration — the Leray–Serre spectral sequence, a computational machine whose input is the homology of the base with coefficients twisted by the monodromy action of $\pi_1(B)$ on $H_*(F;k)$ (a local system) and whose output converges to $H_*(E;k)$. Chapter I, no. 3, 'Fibrations des espaces euclidiens', contains Proposition 4: if $E=\mathbb{R}^n$ is the total space of a locally trivial fibration with connected fibre $F$ and base $B$, then $F$ and $B$ are acyclic (and $B$ is simply connected, by the long exact homotopy sequence of the fibration, a step Serre records in his proof). A corollary, via a classical theorem of Feldbau (a fibre bundle over a base that is a contractible locally finite polyhedron is trivial) recorded in the thesis, gives triviality $E=B\times F$ whenever $B$ is a locally finite polyhedron (a space triangulated so that each compact region meets finitely many simplices); Serre adds that the same conclusion holds if instead the fibre $F$ is a locally finite polyhedron.

At the time, the nonexistence of fibrations of $\mathbb{R}^n$ by compact non-point fibres (a conjecture of Montgomery and Samelson) had just been established by A. Shapiro and independently by Borel and Serre, whose method also excludes connected-fibre fibrations of $\mathbb{R}^n$ over compact non-point bases; related factorizations of manifolds were studied by Young, whose paper Serre cites in a footnote for 'more details on this question'. Serre then recorded the remark at the heart of this problem: it seems probable that the restrictive hypotheses on $B$ and $F$ can be dispensed with entirely, and that there are no connected-fibre fibrations of $\mathbb{R}^n$ other than the direct-product decompositions $\mathbb{R}^n=\mathbb{R}^p\times\mathbb{R}^q$ ($p+q=n$), 'but that seems difficult to establish'.

The geometric topology of 1955–1965 complicates the literal conjecture. Whitehead constructed a contractible open 3-manifold $W$ that is not homeomorphic to $\mathbb{R}^3$ (it fails the property, enjoyed by $\mathbb{R}^n$, that complements of sufficiently large compact subsets are simply connected — 'simple connectivity at infinity'), and Glimm proved $W\times\mathbb{R}\cong\mathbb{R}^4$ (a result first obtained by A. Shapiro); McMillan and Andrews–Rubin produced further examples of spaces, not homeomorphic to Euclidean spaces, whose product with a line is Euclidean — the theory of manifold factors of Euclidean space. Transported along such a homeomorphism, the projection $W\times\mathbb{R}\to\mathbb{R}$ becomes a locally trivial fibration of $\mathbb{R}^4$ with connected non-Euclidean fibre, so the strict reading of Serre's expectation (the fibre must be some $\mathbb{R}^q$) fails; but all known examples are trivial bundles, leaving open the deeper part of his question — whether any nontrivial connected-fibre fibration of Euclidean space exists at all.

The thesis also raised three companion questions nearby. (i) Serre proved $\chi(E)=\chi(B)\chi(F)$ under hypothesis (a), triviality of the local systems $H_i(F;k)$ over $B$, and hypothesis (b), finiteness of the Betti numbers; he noted (a) is superfluous when $B$ is a finite polyhedron and asked whether it is superfluous in general. (ii) His field-coefficient method detected $p$-torsion in the homotopy groups of spheres $\pi_{n+2p-3}(S^n)$ only up to an undetermined exponent $k$ in $\mathbb{Z}/p^k$-summands, integral computations being 'incomparably more difficult'. (iii) He called it highly desirable to apply his methods to the free loop space of a compact Riemannian manifold $X$ (the space of continuous maps $S^1\to X$, intimately related to closed geodesics) and to Lusternik–Schnirelmann category (a homotopy invariant measuring how many contractible open sets are needed to cover a space). These three questions have since been absorbed by later mathematics; the problem stated below is the surviving open question of the thesis.

Problem Statement

Source-faithful formulation (Serre 1951, Chapter I, no. 3, Proposition 4, its Corollary, and Remarques 1; the same remark appears in Serre's Séminaire Henri Cartan 1950–51 exposé no 10, no. 3). Let $p:\mathbb{R}^n\to B$ be a locally trivial fibration (fibre bundle) of the Euclidean space $\mathbb{R}^n$ onto a base $B$, with fibre $F=p^{-1}(b)$, and assume only that $F$ is connected — no regularity hypothesis on $B$ or $F$ beyond what follows from being the base and fibre of a locally trivial fibration of Euclidean space is imposed (in particular no paracompactness, metrizability, local compactness, or polyhedron/absolute-neighbourhood-retract (ANR) assumption; Serre observes in his proof of Proposition 4 that local contractibility and covering dimension at most $n$ do follow). Proposition 4 of the source then gives that $B$ and $F$ are acyclic, i.e. their reduced singular homology with integer coefficients vanishes. Moreover $B$ is simply connected, by the long exact homotopy sequence of the fibration (a step Serre records in his proof), and $F$ is simply connected as well: $\pi_1(F)$ is a homomorphic image of $\pi_2(B)$ by exactness, and $\pi_2(B)\cong H_2(B)=0$ by the Hurewicz theorem since $B$ is simply connected and acyclic.

Question (triviality). Is every such fibration trivial, i.e. is there a homeomorphism $\mathbb{R}^n\cong B\times F$ over $B$ (commuting with the projections to $B$)? Equivalently: does there exist a nontrivial locally trivial fibration of some Euclidean space $\mathbb{R}^n$ with connected fibre? Serre: 'il paraît probable que l'on peut se passer complètement d'hypothèses restrictives sur $B$ ou $F$; sans doute n'existe-t-il pas d'autres fibrations de $\mathbb{R}^n$, à fibres connexes, que la décomposition en produit direct: $\mathbb{R}^n=\mathbb{R}^p\times\mathbb{R}^q$ ($p+q=n$), mais cela paraît difficile à établir.'

References

1. Jean-Pierre Serre, 'Homologie singulière des espaces fibrés. Applications', Annals of Mathematics (2) 54 (1951), no. 3, 425–505. DOI: 10.2307/1969485, URL: https://doi.org/10.2307/1969485 (doctoral thesis)
2. Armand Borel and Jean-Pierre Serre, 'Impossibilité de fibrer un espace euclidien par des fibres compactes', Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences Paris 230 (1950), 2258–2260. Zbl: 0037.26304, URL: https://zbmath.org/?q=an:0037.26304
3. Arnold Shapiro, 'Cohomologie dans les espaces fibrés', Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences Paris 231 (1950), 206–207. Zbl: 0038.36502, URL: https://zbmath.org/?q=an:0038.36502
4. Gail S. Young, 'On the factors and fiberings of manifolds', Proceedings of the American Mathematical Society 1 (1950), no. 2, 215–223. DOI: 10.1090/s0002-9939-1950-0034582-7, URL: https://doi.org/10.1090/s0002-9939-1950-0034582-7 (the paper cited by Serre's footnote 3)
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18. Moira Chas and Dennis Sullivan, 'Closed string operators in topology leading to Lie bialgebras and higher string algebra', arXiv: math/0212358 (2002); published in: O. A. Laudal et al. (eds.), The Legacy of Niels Henrik Abel, Springer, Berlin, 2004, pp. 771–784. DOI: 10.1007/978-3-642-18908-1_27, URL: https://arxiv.org/abs/math/0212358
19. Detlef Gromoll and Gerard Walschap, 'The metric fibrations of Euclidean space', Journal of Differential Geometry 57 (2001), no. 2, 233–238. DOI: 10.4310/jdg/1090348110, URL: https://doi.org/10.4310/jdg/1090348110 (nearest modern namesake of Serre's question, but a different problem: metric fibrations in the sense of Riemannian geometry)
20. MathOverflow question 424165, 'How to prove the product of Whitehead manifold and $\mathbb{R}$ is homeomorphic to $\mathbb{R}^4$?' (asked by user 'Math Diego', 2022; one accepted answer), URL: https://mathoverflow.net/q/424165
21. Jean-Pierre Serre, 'La suite spectrale des espaces fibrés. Applications', Séminaire Henri Cartan, tome 3 (1950–1951), exp. no 10, URL: https://www.numdam.org/item/SHC_1950-1951__3__A10_0.pdf (section 3 states, as an unresolved conjecture, the same fibration-of-Euclidean-space remark as the thesis Remarques 1)