Trends in Differential Geometry and Algebraic Topology

1. Introduction

With this Editorial, we present a Special Issue of Axioms entitled “Trends in Differential Geometry and Algebraic Topology.” This Special Issue provides a platform for recent research at the intersection of differential geometry and algebraic topology, two areas whose interaction continues to generate deep results across pure mathematics and mathematical physics.

The scope of this Special Issue includes, but is not limited to, Riemannian geometry; harmonic map theory; the geometry of moduli spaces; generalized cohomology theories and their equivariants, differentials, and twisted refinements; homotopy theory and covering space theory; higher category theory; and applications of geometric and topological methods to mathematical physics. In practice, the published contributions also reflect connections to neighboring fields such as combinatorial geometry, general topology, homological algebra, and algebraic geometry arising from string compactification. We believe this breadth illustrates the vitality of the subject and the diversity of techniques it inspires.

2. Overview of Published Papers

This Special Issue contains eleven papers that were accepted for publication after a rigorous peer-review process. The contributions reflect the breadth and vitality of current research in differential geometry and algebraic topology, ranging from classical curvature problems and harmonic map theory to modern topics such as differential cohomology, string compactification, and equivariant homotopy theory. We briefly summarize the content of each paper below.

The authors of the first contribution 1 address the maximum number of halving lines and the rectilinear crossing number for finite point sets in the plane. They establish recursive inequalities relating these quantities for sets of n points to those for $n+1$ or $n+2$ points, showing that adding two suitably placed points increases the halving line count by at least five. Applying these bounds to known optimal configurations for 37, 61, 97, and 99 points, the authors improve the best known lower bounds on halving lines for 35, 59, 95, and 97 points, raising them to 137, 286, 539, and 553, respectively. Under a conjecture linking crossing number minimizers to halving line maximizers, these also tighten the best known upper bounds on the rectilinear crossing number.

The authors of the second contribution 2 study continuous multi-utility representations for closed preorders on topological spaces. They introduce net-compact topologies that generalize sequential compactness: every net indexed by a directed set admitting some convergent net itself has an accumulation point. The central result establishes that, for a net-compact Hausdorff space, every closed preorder admits a continuous multi-utility representation if and only if the space is normal.

The authors of the third contribution 3 establish new connections among F-harmonic maps, f-harmonic maps, Riemannian submersions, and de Rham cohomology. For horizontally conformal submersions $u:M\to N$ ($dimM>dimN=n$) from a closed manifold M that are F-harmonic or f-harmonic under appropriate conditions on their dilation $\lambda$, the authors prove that if the zeros of a certain expression involving $F^{\prime }$ and $F^{″}$ are isolated and the n-th de Rham cohomology of M vanishes, then the horizontal distribution is never integrable. Complementarily, the pullback of the volume form of N is a harmonic n-form on M if and only if the horizontal distribution is completely integrable. The proofs use the Hodge theorem together with the characterization of minimal fibers via the relevant harmonicity conditions.

The authors of the fourth contribution 4 investigate superstring compactification on Calabi–Yau manifolds, extending the framework to include non-algebraic spaces. For Calabi–Yau hypersurfaces in Hirzebruch scrolls, when the twist parameter is large, the anticanonical hypersurfaces necessarily factorize (Tyurin degeneration). The paper resolves this by allowing Laurent deformations via an “intrinsic limit” procedure. The central combinatorial tool is the “transpolar” operation, a local dual of the standard polar operation, which maps the Newton polytope to the fan of the ambient toric variety. When the ambient space is non-Fano, the transpolar image is a “flip-folded” multitope encoding a torus manifold rather than an algebraic variety. The transposition mirror construction of Berglund–Hübsch extends to this setting, producing mirror pairs where one side lives in a conventional toric variety and the other in a generalized torus manifold. Explicit Calabi–Yau threefolds with $h^{1,1}=2$ and $h^{2,1}=86$ are constructed. A regularizing alternative via a fractional change in variables converting Laurent-deformed systems into polynomial systems on a different variety is also proposed.

The authors of the fifth contribution 5, motivated by Hypothesis H (that M-brane charges are classified by twisted equivariant four-cohomotopy), compute explicit models for quasi-elliptic cohomology applied to four-spheres acted on by finite subgroups of $\mathrm{Spin}\left(3\right)$ and $\mathrm{Spin}\left(4\right)$. Both complex and real versions are computed; the real version uses Freed–Moore K-theory, with the decomposition into fixed and free conjugacy classes yielding equivariant KR-theory and complex K-theory factors, respectively. For each finite subgroup G acting on $S^4$, explicit ring-theoretic descriptions of the form $K{R}^{*}\left(pt\right)[x,q^{\pm }]/\langle x^n-q^k\rangle$ are given, with exponents determined by centralizer data. For product subgroups $H\times K\subset \mathrm{Spin}\left(4\right)$, a product formula reduces the computation to tensor products, and an explicit computation for $E_6\times E_7$ is carried out.

The authors of the sixth contribution 6 investigate when filtered products of copies of an injective module remain injective. The authors introduce $(\aleph ,M)$-Noetherian rings, generalizing classical ℵ-Noetherianity. The central result: for an injective R-module M, every filtered product of copies of M with respect to any filter $\mathcal{F}$ with $\mathrm{cpl}\left(\mathcal{F}\right)\ge \aleph$ is injective if and only if R is $(\aleph ,M)$-Noetherian. This property is shown to be stable under submodules, quotients, finite direct sums, finitely generated modules, quotient rings, and finite ring extensions. The theory is then extended to torsion-theoretic settings: given a hereditary torsion theory $\tau$ with an injective cogenerator E, the $(\aleph ,\tau )$-ascending chain condition is equivalent to R being $(\aleph ,E)$-Noetherian, and under this condition filtered products of $\tau$-torsion-free injective modules remain injective and filtered products of $\tau$-closed modules remain $\tau$-closed.

The authors of the seventh contribution 7 generalize classical homotopy and covering theory by replacing continuous maps with irresolute maps (preimages of semi-open sets are semi-open). The authors define irresolute paths and irresolute homotopy, construct the irresolute fundamental group ${\pi }_1^{\mathrm{ir}}(T,p_0)$, and introduce irresolute covering maps. They prove that for every subgroup G of ${\pi }_1^{\mathrm{ir}}(T,p_0)$, there exists an irresolute covering space with characteristic subgroup G. The main lifting theorem shows that if $(G,\ast ,\tau )$ is an irresolute topological group whose underlying space is semi-connected, locally irresolute path-connected, and semi-locally simply connected, then for any irresolute covering map $c_i:\tilde{G}\to G$, the group operations lift to $\tilde{G}$, making it an irresolute topological group and $c_i$ a group morphism.

The eighth contribution 8 determines the fixed-point locus of $\mathrm{\Phi }\left(E\right)=\sigma \left({\tau }^{*}E\right)$ on the moduli space of principal $E_6$-bundles over a compact Riemann surface X of genus $g\ge 2$ with involution $\tau$, where $\sigma$ represents the outer involution of $E_6$. Since the fixed-point subgroup of $\sigma$ is either $F_4$ or $\mathrm{PSp}(8,\mathbb{C})$, both connected, the topological obstruction in $H^2(X/\tau ,{\pi }_0\left(E_6^{\sigma }\right))$ always vanishes. The fixed-point subvariety is isomorphic to the moduli space of principal H-bundles over $X/\tau$, consisting of $2^{g-k+1}$ components when $\tau$ has $2k$ fixed points, each of dimension $26(g-k+1)$ for $H=F_4$ and $18(g-k+1)$ for $H=\mathrm{PSp}(8,\mathbb{C})$. The semi-stability of a fixed $E_6$-bundle implies semi-stability of the corresponding H-bundle. The second Chern class of a fixed $E_6$-bundle equals the pullback of the second Chern class of the associated H-bundle when $H=F_4$, and twice that pullback when $H=\mathrm{PSp}(8,\mathbb{C})$. The $F_4$ fixed points correspond bijectively to octonionic structures on $X/\tau$, and an explicit construction of such structures via direct sums of line bundles satisfying a degree-zero condition is provided.

The authors of the ninth contribution 9 investigate the curvature geometry of Kähler golden manifolds, which arise from Kähler manifolds by setting $\mathrm{\Phi }={\displaystyle \frac{\epsilon \sqrt{5}}{2}}J+{\displaystyle \frac{1}{2}}I$. It is shown that, under mild dimension and genericity assumptions, a Kähler golden manifold with constant sectional curvature must be flat. Two curvature invariants are introduced: the $\mathrm{\Phi }$-holomorphic sectional curvature and the $\mathrm{\Phi }$-holomorphic bi-sectional curvature. The $\mathrm{\Phi }$-holomorphic sectional curvature coincides with the holomorphic sectional curvature of the underlying Kähler manifold, so a Kähler golden manifold is a complex golden space form if and only if the underlying Kähler manifold is a complex space form. An explicit formula for the bi-sectional curvature in terms of the golden and ordinary sectional curvatures is established.

The authors of the tenth contribution 10 develop a method for extending a digraph Brown functor—a contravariant functor from the homotopy category of finite directed graphs to abelian groups satisfying triviality, additivity, and Mayer–Vietoris axioms—to arbitrary directed graphs. The extension is constructed via the inverse limit over all finite subdigraphs, and is shown to be functorial, additivity-preserving, and independent of the exhausting family.An alternative description via the Yoneda lemma shows that the extended functor at Y is naturally isomorphic to the set of natural transformations from $[-,Y]$ (restricted to finite digraphs) to the original functor, connecting the construction to classical representability theory.

The central result of the eleventh contribution 11 is that in the Bunke–Schick model of differential K-theory, the differential form component is redundant: every class can be represented by a geometric family alone. The proof uses the relation between Bunke and Bismut–Cheeger $\eta$-forms, the Venice lemma of Simons and Sullivan (expressing any odd form as a Chern–Simons form of connections on a trivial bundle), and a relation between the Bismut–Cheeger $\eta$-form and the Chern–Simons form to absorb the remaining form into a geometric family via the kernel bundle construction.

3. Conclusions

A total of eleven papers were published in this Special Issue, “Trends in Differential Geometry and Algebraic Topology.” The contributions cover a wide range of subjects in differential geometry, algebraic topology, and their interactions with algebra and mathematical physics. Topics include combinatorial bounds on halving lines and crossing numbers, continuous multi-utility representations on topological spaces, harmonicity conditions for Riemannian submersions and their cohomological consequences, Calabi–Yau compactification beyond the algebraic setting, equivariant quasi-elliptic cohomology of four-spheres, filtered products of injective modules and generalized Noetherian conditions, irresolute homotopy and covering theory, fixed-point loci on moduli spaces of principal $E_6$-bundles, curvature invariants of Kähler golden manifolds, the extension of digraph Brown functors to arbitrary directed graphs, and a simplification of the Bunke–Schick model of differential K-theory. We hope that the results presented here will inspire further investigation and fruitful interactions across these areas.