Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0211216

Quadratic functions in geometry, topology,and M-theory

M. J. Hopkins·I. M. Singer
View original source ↗Read source document ↗Export metadata ↓

Abstract

We describe an interpretation of the Kervaire invariant of a Riemannian manifold of dimension $4k+2$ in terms of a holomorphic line bundle on the abelian variety $H^{2k+1}(M)\otimes R/Z$. Our results are inspired by work of Witten on the fivebrane partition function in $M$-theory (hep-th/9610234, hep-th/9609122). Our construction requires a refinement of the algebraic topology of smooth manifolds better suited to the needs of mathematical physics, and is based on our theory of "differential functions." These differential functions generalize the differential characters of Cheeger-Simons, and the bulk of this paper is devoted to their study.

Explore related subjects

math.AThep-thmath-phmath.MP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. J. Hopkins, I. M. Singer. 2005-08-24. Quadratic functions in geometry, topology,and M-theory. https://arxiv.org/abs/math/0211216

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Adams filtrations and secondary coherence for fibers of Steenrod operations

Motivated by the Bruner--Rognes two-extension theorem and open questions regarding the secondary cohomology modeling of fibers of Steenrod operations, we develop a general framework to compute their Adams filtrations and delineate the underlying algebraic obstructions. Let $f:Y\to\prodΣ^{d_λ}H\mathbb F_2$ be a locally finite family of positive-degree stable operations, where $Y=H\mathbb F_2$ or $H\mathbb Z$. We determine the Adams filtration of its fiber from the minimal generators of $\operatorname{im}(f^*)$ as a left Steenrod module. An explicit target automorphism separates a minimal subfamily, and combining this with the two-extension theorem and an acyclic cancellation gives $E_3=E_\infty$. A terminating Adem recursion realizes this splitting for the full positive even-square family. Separately, we identify two fundamental algebraic obstructions to directly constructing the secondary cohomology of these fibers: an augmentation discrepancy when the primary row has a nonzero kernel, and a coherence obstruction class in the third cohomology of a Hom complex. These results demonstrate why the secondary Ext groups of these fibers must be derived topologically via the Baues--Jibladze comparison theorem, rather than through a direct algebraic resolution.

math.AT↗
arXiv2026

Discrete homotopy groups of cubical sets

We extend the notion of discrete homotopy groups of graphs to arbitrary cubical sets, and show that the discrete homotopy groups of quasisymmetric cubical sets are naturally isomorphic to the homotopy groups of their geometric realizations. Here, quasisymmetric cubical sets are cubical sets equipped with coordinate permutation symmetries that are compatible with faces and degeneracies, but not necessarily with connections. We give a purely combinatorial construction of the left adjoint of the forgetful functor from the category of quasisymmetric cubical sets to the category of cubical sets, and prove that the unit of this adjunction is a natural weak equivalence. As a consequence, we obtain a purely combinatorial description of the homotopy groups of the geometric realizations of arbitrary cubical sets. As an application, we establish the discrete Hurewicz map from the discrete homotopy groups of a cubical set to the homology groups, and prove a Hurewicz theorem with respect to the discrete Hurewicz map whenever the cubical set is quasisymmetric. Applied to the 1-nerve of a directed graph, it also yields a Hurewicz theorem relating the homotopy groups of Li, Wu, Yau, and Zhang to singular cubical homology.

math.AT↗
arXiv2026

Monoidal and symmetrized parametrized topological complexity

We introduce and study monoidal and symmetrized versions of parametrized topological complexity. First, we develop parametrized analogues of the monoidal topological complexity theories of Iwase--Sakai, Aguilar-Guzmán--González (based on the Fadell--Husseini approach), and Dranishnikov. We investigate the parametrized Iwase--Sakai conjecture and provide sufficient conditions under which it holds. We then introduce symmetrized parametrized topological complexity and combine it with the monoidal perspective to define monoidal symmetrized parametrized topological complexity, showing that the resulting notions agree. Finally, we compute these invariants for Fadell--Neuwirth fibrations.

math.AT↗