Aspherical 4-manifolds with positive Euler characteristic and their geography

TL;DR

The paper constructs closed oriented aspherical 4-manifolds with prescribed Euler characteristic and signature, showing that for every $n$ there is an $X_n$ with $χ(X_n)=σ(X_n)=n$ and thereby proving sharpness of the inequality $χ(X)≥|σ(X)|$ in dimension four while disproving the real Bogomolov–Miyaoka–Yau bound in this setting. The strategy glue together four types of aspherical building blocks along π1-injective boundaries: a cusp-derived complex-hyperbolic core contributing $(χ,σ)=(1,1)$, torus-bundle and semi-bundle pieces with $χ=0$, and a torus-trick construction; additivity of Euler characteristic and signature under gluings drives the construction. A detailed analysis of torus bundles and semi-bundles via the Meyer function and Wall’s non-additivity yields explicit fillability results for monodromies in $SL_2(Z)$ and its derived subgroup, enabling the realization of all $(χ,σ)=(n,n)$. The paper also addresses questions about simplicial volume and filling properties for amenable 3-manifolds, proving a virtual small-filling theorem and showing that the overall manifolds $X_n$ have positive simplicial volume, while the building-block fillings can have vanishing volume, highlighting a nuanced interplay between topology and geometric invariants.

Abstract

We present an explicit construction of closed oriented aspherical smooth 4-manifolds with $χ= σ= n$ for every positive integer $n$. This proves a conjecture of Edmonds by providing a closed oriented aspherical 4-manifold with Euler characteristic 1, and it shows that the real analogue of the Bogomolov-Miyaoka-Yau inequality fails for aspherical 4-manifolds. By the Hitchin-Thorpe inequality, these manifolds do not admit Einstein metrics. As a further consequence of our construction, we show that every closed aspherical 3-manifold with amenable fundamental group is virtually the $π_1$-injective boundary of an aspherical 4-manifold with vanishing Euler characteristic and vanishing simplicial volume, thereby answering questions of Edmonds and of Löh-Moraschini-Raptis up to finite covers.

Figures (3)

• Figure 1: Gluing scheme illustrating Lemma \ref{['lem:double_covering_trick']}.
• Figure 2: The boundary component $T(\varphi\tau\varphi^{-1}\tau)$ of $W$ can be described as follows. We glue $T_1$ to $T_-\times \{1\}$ via $\tau$, and to $T_+\times \{0\}$ via $\varphi$. Likewise, we glue $T_2$ to $T_-\times \{0\}$ via $\varphi^{-1}$, and to $T_+ \times \{0\}$ via $\tau$.
• Figure 3: Gluing scheme illustrating Lemma \ref{['lem:torus_trick_semi_bundles']}

Theorems & Definitions (27)

• Conjecture 1
• Theorem 1
• Corollary 2
• proof
• Theorem 3
• Remark 2.1
• Lemma 2.2: Edm20
• Lemma 2.3
• proof
• Corollary 2.4