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Computations of Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7

Naoki Kuroda

TL;DR

The paper computes low-dimensional twisted Spin bordism groups $\Omega_*^{\mathrm{Spin}\text{-}G}$ for $G=\mathrm{Sp}(4),\mathrm{SU}(8),\mathrm{Spin}(16)$ in dimensions $\le 7$, using a combination of spectral sequences and obstruction theory. Central methods include the Leray–Serre and Adams spectral sequences, Anderson–Brown–Peterson decomposition to relate Spin bordism to $ko$-homology, and the MTSpin framework to handle Spin-$G$ structures, with explicit $\Z_2$-cohomology computations guiding the Ext calculations. The authors demonstrate no odd torsion, establish 4-equivalences among the three Spin-$G$ bordism theories, and compute the $E_2$-terms that collapse, yielding concrete group structures and manifold representatives up to degree 7. They provide explicit geometric generators in low dimensions, such as $(\mathbb{HP}^1,\mathbb{CP}^2)$ in degree 4 and $\mathbb{HP}^1\times S^1$, $SU(3)/SO(3)$ in degree 5, along with degree-6 constructions like $\mathbb{HP}^1\times S^1\times S^1$ and $\mathbb{CP}^2\times\mathbb{CP}^1$, with detailed discussions of the Spin-$G$-structure realizations. The results advance anomaly analysis in string theory with these symmetry twists and provide a concrete low-dimensional backbone for further higher-dimensional bordism computations.

Abstract

We investigate the structure of Spin-$G$ bordism groups, focusing on the interplay between Spin and additional twisting symmetries such as $Sp(4)$, $SU(8)$ and $Spin(16)$. Using techniques from spectral sequences, obstruction theory, and cohomology operations, we compute explicit generators for the Spin-$G$ bordism groups in dimensions up to 7.

Computations of Spin-Sp(4), Spin-SU(8), and Spin-Spin(16) bordism groups in dimensions up to 7

TL;DR

The paper computes low-dimensional twisted Spin bordism groups for in dimensions , using a combination of spectral sequences and obstruction theory. Central methods include the Leray–Serre and Adams spectral sequences, Anderson–Brown–Peterson decomposition to relate Spin bordism to -homology, and the MTSpin framework to handle Spin- structures, with explicit -cohomology computations guiding the Ext calculations. The authors demonstrate no odd torsion, establish 4-equivalences among the three Spin- bordism theories, and compute the -terms that collapse, yielding concrete group structures and manifold representatives up to degree 7. They provide explicit geometric generators in low dimensions, such as in degree 4 and , in degree 5, along with degree-6 constructions like and , with detailed discussions of the Spin--structure realizations. The results advance anomaly analysis in string theory with these symmetry twists and provide a concrete low-dimensional backbone for further higher-dimensional bordism computations.

Abstract

We investigate the structure of Spin- bordism groups, focusing on the interplay between Spin and additional twisting symmetries such as , and . Using techniques from spectral sequences, obstruction theory, and cohomology operations, we compute explicit generators for the Spin- bordism groups in dimensions up to 7.

Paper Structure

This paper contains 17 sections, 10 theorems, 65 equations, 3 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The relation between Spin-G bordism and quantum theory
  3. Overview of Spin-G bordism groups
  4. Introducing three Spin-G bordism groups
  5. Mathematical Background
  6. The Leray-Serre spectral sequence
  7. The Adams spectral sequence
  8. The Spin-G structure and methods for computing Spin-G bordism groups
  9. The Calculation of the three bordism groups in dimensions up to 7
  10. The relations among the three bordism groups
  11. No odd-prime torsion in the three bordism groups
  12. Computation of cohomology rings with Z2 coefficients
  13. Calculation of the Spin-G bordism in dimensions up to 7
  14. Explicit description of generators by using manifolds
  15. Determining the manifold generator at degree 4
  16. ...and 2 more sections

Key Result

Theorem 1.1

The three Spin-$G$ bordism groups and their respective generators in dimensions up to $7$ are as follows:

Figures (3)

  • Figure 1: The case of Spin-$Sp(4)$
  • Figure 2: The case of Spin-$SU(8)$
  • Figure 3: The case of Spin-$Spin(16)$

Theorems & Definitions (20)

  • Theorem 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 3.3
  • proof
  • Proposition 3.4
  • Lemma 3.5
  • proof
  • ...and 10 more