[Paper Review] A classification of 5-dimensional manifolds, souls of codimension two and non-diffeomorphic pairs

Sadeeb Ottenburger|arXiv (Cornell University)|Mar 1, 2011

Homotopy and Cohomology in Algebraic Topology7 references3 citations

TL;DR

This paper classifies 5-dimensional manifolds fibered over $S^2 \times S^2$ with fundamental group $\mathbb{Z}/r$ for $r > 1$ coprime to six, and constructs infinite families of complete metrics of nonnegative sectional curvature on certain total spaces of complex line bundles over lens spaces. The key result is the existence of manifolds admitting infinitely many such metrics with pairwise non-homeomorphic souls of codimension two, resolving a problem on non-diffeomorphic soul pairs under curvature constraints.

ABSTRACT

Let T(γ) be the total space of the canonical line bundle γover CP^1 and r an integer which is greater than one and coprime to six. We prove that L_r^3 imes T(γ) admits an infinite sequence of metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls, where L_r^3 is the standard 3-dimensional lens space with fundamental group isomorphic to Z/r. We classify the total spaces of S^1-fibre bundles over S^2 imes S^2 with fundamental group isomorphic to Z/r up to diffeomorphism and use these results to give examples of manifolds N which admit two complete metrics of nonnegative sectional curvature with souls S and S' of codimension two such that S and S' are diffeomorphic whereas the pairs (N,S) and (N,S') are not diffeomorphic. This solves a problem posed by I. Belegradek, S. Kwasik and R. Schultz.

Motivation & Objective

• To classify total spaces of $S^1$-fiber bundles over $S^2 \times S^2$ with fundamental group $\mathbb{Z}/r$ up to diffeomorphism for $r > 1$ coprime to six.
• To construct examples of 5-manifolds admitting infinitely many complete metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls of codimension two.
• To resolve a problem posed by Belegradek, Kwasik, and Schultz on the existence of non-diffeomorphic soul pairs under nonnegative curvature.
• To establish a classification theorem for 5-manifolds homotopy equivalent to $L^{a,b}$ with $\pi_1 \cong \mathbb{Z}/r$, using $\rho$-invariants and Reidemeister torsion.

Proposed method

• Use surgery theory and classification results in the simple structure set to classify 5-manifolds with $\pi_1 \cong \mathbb{Z}/r$ up to diffeomorphism.
• Apply results from Grove and Ziller on the existence of nonnegative curvature metrics on total spaces of complex line bundles over $L^{a,b}$ with primitive first Chern class.
• Employ the $\rho$-invariant and Reidemeister torsion $\Delta(N)$ as invariants to distinguish diffeomorphism types of manifolds in $\mathcal{L}$.
• Construct self-homotopy equivalences with non-trivial normal invariants to show that certain cohomological isomorphisms cannot be induced by diffeomorphisms.
• Use normal $s$-cobordism arguments and parameter congruences to derive contradictions when assuming diffeomorphisms exist under specific algebraic conditions.
• Leverage the injectivity of the $\rho$-invariant for $r^2 q \neq 0$ to rule out certain diffeomorphisms and prove non-diffeomorphic soul pairs.

Experimental results

Research questions

• RQ1 Do there exist 5-dimensional manifolds admitting infinitely many complete metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls of codimension two?
• RQ2 Can two complete metrics of nonnegative curvature on the same manifold have diffeomorphic souls but non-diffeomorphic soul pairs?
• RQ3 Under what algebraic conditions on the fundamental group $\mathbb{Z}/r$ does the existence of a unit $s$ with $s^2 = -1$ or $s^3 = 1$ affect the diffeomorphism type of soul pairs?
• RQ4 Is the moduli space of nonnegative curvature metrics on $L_r^3 \times T(\gamma)$ infinite-dimensional in the sense of having infinitely many path components?
• RQ5 Does the existence of a metric of nonnegative curvature on a manifold $M$ imply the existence of such a metric on any simply homotopy equivalent manifold $N$?

Key findings

• The manifold $L_r^3 \times T(\gamma)$, where $L_r^3$ is the lens space with $\pi_1 \cong \mathbb{Z}/r$, admits an infinite sequence of complete metrics of nonnegative sectional curvature with pairwise non-homeomorphic souls.
• The moduli space $\mathfrak{M}^{c}_{\text{sec} \geq 0}(L_r^3 \times T(\gamma))$ has infinitely many path components, implying the souls cannot be connected by a continuous path of metrics.
• For $r = 5$, the existence of a unit $s = 2$ with $s^2 = -1$ in $\mathbb{Z}/5$ leads to examples of manifolds with non-diffeomorphic soul pairs, solving Problem 4.9 in [BKS1-09].
• The classification theorem (Theorem 4) states that two 5-manifolds $N$ and $N'$ with $\pi_1 \cong \mathbb{Z}/r$ are diffeomorphic if and only if there exists an orientation-preserving homotopy equivalence satisfying $\rho(h_*(g),N) = \rho(g,N')$ and $\Delta(N) \sim \Delta(N')$.
• When $r$ has no non-trivial unit with $s^3 = 1$ and $q$ is not divisible by $r$, the pair $(N^{r,qr}_x, S)$ is diffeomorphic if and only if $S$ and $S'$ are diffeomorphic, showing rigidity in this case.
• The existence of a self-homotopy equivalence with non-trivial normal invariant that cannot be realized by a diffeomorphism implies that certain cohomological isomorphisms are not induced by diffeomorphisms, leading to non-diffeomorphic soul pairs.