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Geometric Constructions of Mod $p$ Cohomology Operations

Herng Yi Cheng

TL;DR

This work develops explicit geometric representations of mod $p$ cohomology operations, including Steenrod powers and Bockstein homomorphisms, by embedding cohomology into Brown representations via spaces of mod $p$ cycles. Central to the approach are the cyclic product map $ extit{cyc}$, gluings, and an inductive-limit topology that together yield Brown representatives for generators, products, and Steenrod operations, all realized as continuous maps between cycle spaces equivalent to Eilenberg–MacLane spaces. Key contributions include a geometric construction of $ extit{cyc}$, a novel Almgren Isomorphism for mod $p$ cycles via gluings, and explicit Brown representatives for the Steenrod powers $P^i$ and $eta P^i$ for all primes $p$, leveraging equivariant cohomology and lens-space Thom complexes. The results open avenues for quantitative homotopy theory and potentially a quantitative Adams spectral sequence, linking geometric measure theory with stable homotopy theoretic information and giving a concrete geometric framework for mod $p$ cohomology operations.

Abstract

The Brown Representability Theorem implies that cohomology operations can be represented by continuous maps between Eilenberg-Maclane spaces. These Eilenberg-Maclane spaces have explicit geometric models as spaces of cycles on round spheres and spaces of relative cycles on unit disks, due to the Almgren Isomorphism Theorem. A. Nabutovsky asked what maps between spaces of cycles represent the Steenrod squares. In this work we answer this question by constructing maps with explicit formulas from spaces of cycles on spheres to spaces of relative cycles on disks that represent all Steenrod squares, as well as all Steenrod powers and Bockstein homomorphisms on mod $p$ cohomology, for all primes $p$.

Geometric Constructions of Mod $p$ Cohomology Operations

TL;DR

This work develops explicit geometric representations of mod cohomology operations, including Steenrod powers and Bockstein homomorphisms, by embedding cohomology into Brown representations via spaces of mod cycles. Central to the approach are the cyclic product map , gluings, and an inductive-limit topology that together yield Brown representatives for generators, products, and Steenrod operations, all realized as continuous maps between cycle spaces equivalent to Eilenberg–MacLane spaces. Key contributions include a geometric construction of , a novel Almgren Isomorphism for mod cycles via gluings, and explicit Brown representatives for the Steenrod powers and for all primes , leveraging equivariant cohomology and lens-space Thom complexes. The results open avenues for quantitative homotopy theory and potentially a quantitative Adams spectral sequence, linking geometric measure theory with stable homotopy theoretic information and giving a concrete geometric framework for mod cohomology operations.

Abstract

The Brown Representability Theorem implies that cohomology operations can be represented by continuous maps between Eilenberg-Maclane spaces. These Eilenberg-Maclane spaces have explicit geometric models as spaces of cycles on round spheres and spaces of relative cycles on unit disks, due to the Almgren Isomorphism Theorem. A. Nabutovsky asked what maps between spaces of cycles represent the Steenrod squares. In this work we answer this question by constructing maps with explicit formulas from spaces of cycles on spheres to spaces of relative cycles on disks that represent all Steenrod squares, as well as all Steenrod powers and Bockstein homomorphisms on mod cohomology, for all primes .
Paper Structure (35 sections, 69 theorems, 94 equations, 4 figures)

This paper contains 35 sections, 69 theorems, 94 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Brown representability and the main theorems
  3. Main technical challenges
  4. Proving \ref{['thm:SteenrodPowers']} using gluing
  5. Proving the Almgren Isomorphism Theorem using gluing
  6. Earlier geometric representations of cohomology classes and operations
  7. Potential applications to quantitative homotopy theory
  8. Organization of Content
  9. Definitions
  10. Mod $p$ relative integral currents and the flat metric
  11. The inductive limit topology
  12. Concentration of mass
  13. Piecewise Smooth Families of Cycles
  14. The Piecewise Smooth Approximation Theorem
  15. Gluings
  16. ...and 20 more sections

Key Result

Theorem 1.2

For each $n \geq 1$, the Bockstein homomorphism $\beta : H^n(-) \to H^{n+1}(-)$ has a Brown representative where each $x_i \in \mathbb{S}^{n}$ and the fraction denotes the barycenter of $x_{i_1}, \dotsc, x_{i_p}$, considered as points in $\mathbb{R}^{n+1}$. In particular, when $p = 2$, $b(x_1 + \dotsb + x_k)$ is the sum of the midpoints of every unordered pair $\{x_i,x_j\}$.

Figures (4)

  • Figure 1: (Suppose that $p = 2$.) (a) A Brown representative $a : \mathbb{S}^{1} \to \mathcal{Z}_{1}^{\mathrm{}}(\mathbb{S}^{2})$ of the generator of $H^1(\mathbb{S}^{1})$, where $\mathbb{S}^{1}$ is parametrized as $[-1,1]$. (b) $\mathbb{RP}^2$ can be visualized as the upper hemisphere of $S^2$, with antipodal points on the equator identified. Each point on the upper hemisphere corresponds to a line $\ell$ through the origin. (c) A Brown representative of the generator of $H^1(\mathbb{RP}2)$ sends $\ell \in \mathbb{RP}^2$ to $\ell^\perp \cap \mathbb{D}^3 \in \mathcal{Z}_{2}^{\mathrm{}}(\mathbb{D}^3, \partial\mathbb{D}^3)$.
  • Figure 2: Brown representatives $b : \mathcal{Z}_{0}^{\mathrm{}}(\mathbb{S}^{1}) \to \mathcal{Z}_{0}^{\mathrm{}}(\mathbb{D}^2, \partial\mathbb{D}^2)$ of the Bockstein homomorphism $\beta : H^1(-) \to H^2(-)$ for $p = 2$ in (a)--(b) and $p = 3$ in (c)--(d). (a) The mod 2 0-cycle $x_1 + \dotsb + x_4$ shown as red points. (b) $b(x_1 + \dotsb + x_4)$ shown as red points, where $(ij)$ denotes the midpoint $\frac{1}{2}(x_1 + x_j)$. (c) The mod 3 0-cycle $x_1 + x_2 + x_3$ shown as red points. (b) $b(x_1 + x_2 + x_3)$ shown as red points, where $(ijk)$ denotes the barycenter $\frac{1}{3}(x_1 + x_j + x_k)$. The point $(123) = (321)$ has multiplicity 2.
  • Figure 3: A Brown representative $\mathit{cyc}$ for the total Steenrod power, when $p = 2$, $k = 0$, and $n = 1$. The input 0-cycle $x_1 + \dotsb + x_4$, made of the vertices of a square, is shown as red points in (a). The output $\mathit{cyc}(x_1 + \dotsb + x_4)$ is a sum of pairs $(\ell_{ij}, \frac{1}{2}(x_i + x_j))$, where $\ell_{ij} = \mathop{\mathrm{span}}\nolimits\{x_i - x_j\}$ is a line in $\mathbb{R}^2$ shown in (b), and $\frac{1}{2}(x_i + x_j)$ is a point in $\mathbb{D}^2$ that is shown in red and labeled as $(ij)$ in (c).
  • Figure 4: A Brown representative $\mathit{sq}^i$ for $\mathit{Sq}^i$, when $p = 2$, $k = 1$, $n = 2$, and $i \geq 0$. If the input is a planar cycle $V \cap \mathbb{S}^{2}$ as shown in (a), then the output is a union, over lines $\ell \subset \mathbb{R}^3$ through the origin and parallel to $V$, of a Cartesian product of copies of $\ell^\perp$ and a copy of $V$ (and then intersected with $\mathbb{D}^{(n+1)(k+i)}$), as shown in (b).

Theorems & Definitions (147)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7
  • Lemma 2.1
  • Lemma 2.1
  • Definition 2.2: Mass concentration profile
  • ...and 137 more