Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/9910026

Representations of the homotopy surface category of a simply connected space

M. Brightwell·P. Turner
View original source ↗Read source document ↗Export metadata ↓

Abstract

We introduce the homotopy surface category of a space which generalizes the 1+1-dimensional cobordism category of circles and surfaces to the situation where one introduces a background space. We explain how for a simply connected background space, monoidal functors from this category to vector spaces can be interpreted in terms of Frobenius algebras with additional structure.

Explore related subjects

math.ATmath.QA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

M. Brightwell, P. Turner. 1999-10-05. Representations of the homotopy surface category of a simply connected space. https://arxiv.org/abs/math/9910026

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

KEEP EXPLORING

Related papers

arXiv2026

Higher path groupoids and the holonomy of formal power series connections

Building on ideas of Kohno, we develop a framework for the construction of higher holonomy functors via the transport of formal power series connections. Using these techniques, we obtain functors from the path groupoid, the path 2-groupoid, and the path 3-groupoid of a manifold. We show that the resulting higher holonomy functors can detect non-trivial classes in higher homotopy groups. As an application, we construct a Gray functor from the path 3-groupoid of the configuration space of $m$ points in $\mathbb{R}^n$ for $n\geqslant 4$ and compute its restriction to 3-tracks with constant boundary when $n=4$.

math.AT↗
arXiv2026

Multicomplexes in Action: Strong Formality of Compact Hyperkähler Manifolds

We show that all compact hyperkähler manifolds are bigraded strongly formal in the sense of Milivojević and Stelzig, proving that in the context of pluripotential homotopy theory they play the same role as compact Kähler manifolds in rational homotopy theory. Indeed, the main result arises from general algebraic considerations on categories of objects with more differentials, that of $n$-complexes and commutative $n$-differential graded algebras. For them we discuss a standard model category structure, introduce a generalization of the $\partial\bar{\partial}$-Lemma and establish an algebraic formality theorem.

math.AT↗
arXiv2026

On bordism classes of symmetric products of surfaces

We explicitly determine the complex bordism class of the $n$-th symmetric product of a complex curve of genus $g$. In particular, we provide an explicit formula expressing this class as an integral linear combination of bordism classes of products of complex projective spaces. Similarly, we determine the unoriented bordism class of the $n$-th symmetric product of a non-orientable surface of genus $g$ (i.e., $\#_{i=1}^{g+1}\mathbb R P^2$). It turns out that these classes are also expressed as a $\mathbb Z_2$-linear combination of bordism classes of products of even-dimensional real projective spaces. We also give an affirmative answer to a non-orientable version of the Blagojević--Grujić--Živaljević conjecture stating that the topological type of any symmetric product of a noncompact surface of finite type determines the topological type of the surface itself.

math.AT↗