This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975. They focus on the uses of homotopy theory in physics, and especially on the calculation of homotopy groups. The next chapter summarizes briefly the ways in which homotopy theory can be applied to quantum field theory.

Elements of Homotopy Theory

Cambridge University Press v. Georgia State University

This paper presents an opportunity for the uncertainty that has plagued the novel's criticism to appear as absences in the body of historical knowledge, particularly regarding the notion of life after death. Taking appearance (eg. proof of existence), as opposed to disappearance, as a universally accepted value allows this analysis to interrogate the novel's logic in relation to a variety of conventional systems whose very existence depends on the reproduction of their systems. The ineffectuality of Foucauldian disciplinary institutions in the novel establishes the threat of nonexistence. A significant relationship to Dante's Inferno is rendered, lending the appearance of language an 'enchanted' value through allusions to Dante's intentional invocation of Augustinian corporeal vision. The novel's metalanguage appears enchanted by the body of historical knowledge, particularly as the product of capitalism, discipline and Judeo-Christianity, and programmed by literary precursors William S. Burroughs, Gertrude Stein and Ernest Hemingway. Foregrounded by this complex network, an analysis of the novel’s first chapter demonstrates how an attention to appearance brings the language to life and draws the narrator, equally invested in appearance, into its realm of representation.

Will To Appear

We consider two families X_n of varieties on which the symmetric group S_n acts: the configuration space of n points in C and the space of n linearly independent lines in C^n. Given an irreducible S_n-representation V, one can ask how the multiplicity of V in the cohomology groups H*(X_n;Q) varies with n. We explain how the Grothendieck-Lefschetz Fixed Point Theorem converts a formula for this multiplicity to a formula for the number of polynomials over F_q (or maximal tori in GL_n(F_q), respectively) with specified properties related to V. In particular, we explain how representation stability in cohomology, in the sense of [CF, arXiv:1008.1368] and [CEF, arXiv:1204.4533], corresponds to asymptotic stability of various point counts as n goes to infinity.

/pdf/representation-stability-in-cohomology-and-asymptotics-for-1dz3xjyvwy.pdf

Representation stability in cohomology and asymptotics for families of varieties over finite fields

Foundational essays on topological manifolds, smoothings and triangulations

We define bounded generation for $E_n$-algebras in chain complexes and prove that this property is equivalent to homological stability for $n \ge 2$. Using this we prove a local-to-global principle for homological stability, which says that if an $E_n$-algebra A has homological stability (or equivalently the topological chiral homology $\int _{\mathbb {R}^n} A$ has homology stability), then so has the topological chiral homology $\int _M A$ of any connected non-compact manifold M. Using scanning, we reformulate the local-to-global homological stability principle so that it applies to compact manifolds. We also give several applications of our results.

$$E_n$$-cell attachments and a local-to-global principle for homological stability

We prove that in dimensions not equal to 4, 5, or 7, the homology and homotopy groups of the classifying space of the topological group of diffeomorphisms of a disk fixing the boundary are finitely generated in each degree. The proof uses homological stability, embedding calculus and the arithmeticity of mapping class groups. From this we deduce similar results for the homeomorphisms of $R^n$ and various types of automorphisms of 2-connected manifolds.

https://msp.org/gt/2019/23-5/gt-v23-n5-p03-s.pdf

Some finiteness results for groups of automorphisms of manifolds

Homological stability for topological chiral homology of completions

We prove a new kind of stabilisation result, "secondary homological stability", for the homology of mapping class groups of orientable surfaces with one boundary component. These results are obtained by constructing CW approximations to the classifying spaces of these groups, in the category of $E_2$-algebras, which have no $E_2$-cells below a certain vanishing line.

$E_2$-cells and mapping class groups

We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^g S^n \times S^n$ relative to a disc in a stable range, for $2n \geq 6$. Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite dimensional in a stable range.

/pdf/on-the-cohomology-of-torelli-groups-br1bo0pwys.pdf

Alexander Kupers

Papers

$$E_n$$-cell attachments and a local-to-global principle for homological stability

Some finiteness results for groups of automorphisms of manifolds

Homological stability for topological chiral homology of completions

$E_2$-cells and mapping class groups

On the cohomology of Torelli groups