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Complex K-theory of 4-complexes

Jonathan Rosenberg

TL;DR

The paper analyzes $K^0(X)$ for 4-dimensional CW complexes, exploiting the collapse of the Atiyah–Hirzebruch spectral sequence and the surjectivity of Chern class maps to turn the ring $K^0(X)$ into a computable object from $H^{\text{even}}(X;\mathbb Z)$. It proves that all vector bundles over such $X$ are determined by their rank, $c_1$, and $c_2$, via canonical generators $L_x$ and $V_y$, and provides an explicit ring presentation of $K^0(X)$ in terms of these generators and seven relations. The main contribution is an algorithm to compute $K^0(X)$ as a ring from the cup product in $H^{\text{even}}(X;\mathbb Z)$, making the theory practically actionable for 4-dimensional complexes and illustrated by concrete cases like $\\mathbb{RP}^4$.

Abstract

This short note summarizes a number of facts about the ring $K^0(X)$ for $X$ a $4$-dimensional CW-complex. Unusual features of this dimension are that every complex vector bundle is determined up to stable isomorphism by its Chern classes, that every even cohomology class arises as a Chern class of a vector bundle, and that $K^0(X)$ is completely determined as a ring by knowledge of the even-dimensional cohomology ring $H^{\text{even}}(X; \mathbb Z)$. (All of these fail in high dimensions.)

Complex K-theory of 4-complexes

TL;DR

The paper analyzes for 4-dimensional CW complexes, exploiting the collapse of the Atiyah–Hirzebruch spectral sequence and the surjectivity of Chern class maps to turn the ring into a computable object from . It proves that all vector bundles over such are determined by their rank, , and , via canonical generators and , and provides an explicit ring presentation of in terms of these generators and seven relations. The main contribution is an algorithm to compute as a ring from the cup product in , making the theory practically actionable for 4-dimensional complexes and illustrated by concrete cases like .

Abstract

This short note summarizes a number of facts about the ring for a -dimensional CW-complex. Unusual features of this dimension are that every complex vector bundle is determined up to stable isomorphism by its Chern classes, that every even cohomology class arises as a Chern class of a vector bundle, and that is completely determined as a ring by knowledge of the even-dimensional cohomology ring . (All of these fail in high dimensions.)
Paper Structure (2 sections, 4 theorems, 10 equations)

This paper contains 2 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Main Results

Key Result

Theorem 1

Let $X$ be a connected $4$-dimensional CW complex. Then for each $x\in H^2(X;\mathbb Z)$, there is a unique (up to isomorphism) line bundle $L_x$ over $X$ with $c_1(L_x)=x$, and for each $y\in H^4(X;\mathbb Z)$, there is a unique (up to isomorphism) rank-$2$ complex vector bundle $V_y$ over $X$ with

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Remark 2
  • Theorem 3
  • proof
  • Corollary 4
  • proof
  • Remark 5
  • Theorem 6
  • proof
  • ...and 1 more