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The special unitary groups $SU(2n)$ as framed manifolds

Haruo Minami

TL;DR

The paper investigates how twisting the left-invariant framing on framed Lie groups alters the complex Adams $e$-invariant, showing that for $G=SU(2n)$ a specific twist by the real representation $\rho$ yields a nonzero value, namely $e_\mathbb{C}([SU(2n), \mathscr{L}^{(n-1)\rho}]) = (-1)^{n-1} \frac{B_{n^2}}{2n^2}$, while the untwisted case vanishes. It develops a tensor-product decomposition of a canonical complex line bundle $E$ over $G/S$ arising from a circle subgroup, and demonstrates that the induced stable framing corresponds to $\mathscr{L}^{(n-1)\rho}$, enabling an explicit computation of the $e$-invariant via known formulas (LS). The method extends to the quotient $SU(2n+1)/C$ with induced framing, obtaining $e_\mathbb{C}([SU(2n+1)/C, \mathscr{L}_C]) = -\frac{B_{n^2+n}}{2(n^2+n)}$. Collectively, the results show that suitable framing twists convert zero $e$-invariants into generators of the image of $e_\mathbb{C}$, and provide explicit representatives tied to Bernoulli numbers, revealing a robust link between representation-theoretic twists and bordism invariants.

Abstract

Let $[SU(2n), \mathscr{L}]$ denote the bordism class of $SU(2n)$ $(n\ge 2)$ equipped with its left invariant framing $\mathscr{L}$. Then it is well known that $e_\mathbb{C}([SU(2n), \mathscr{L}])=0$ where $e_\mathbb{C}$ denotes the complex Adams $e$-invariant. In this note we show that replacing $\mathscr{L}$ by the framing obtained by twisting it by a specific map the zero value of $e_\mathbb{C}([SU(2n), \mathscr{L}])$ can be transformed into a generator of $\mathrm{Im} \, e_\mathbb{C}$ which is isomorphic to a cyclic group. In addition we show that the same procedure affords an analogous result for a quotient of $SU(2n+1)$ by a circle subgroup which inherits a canonical framing from $SU(2n+1)$ in the usual way. .

The special unitary groups $SU(2n)$ as framed manifolds

TL;DR

The paper investigates how twisting the left-invariant framing on framed Lie groups alters the complex Adams -invariant, showing that for a specific twist by the real representation yields a nonzero value, namely , while the untwisted case vanishes. It develops a tensor-product decomposition of a canonical complex line bundle over arising from a circle subgroup, and demonstrates that the induced stable framing corresponds to , enabling an explicit computation of the -invariant via known formulas (LS). The method extends to the quotient with induced framing, obtaining . Collectively, the results show that suitable framing twists convert zero -invariants into generators of the image of , and provide explicit representatives tied to Bernoulli numbers, revealing a robust link between representation-theoretic twists and bordism invariants.

Abstract

Let denote the bordism class of equipped with its left invariant framing . Then it is well known that where denotes the complex Adams -invariant. In this note we show that replacing by the framing obtained by twisting it by a specific map the zero value of can be transformed into a generator of which is isomorphic to a cyclic group. In addition we show that the same procedure affords an analogous result for a quotient of by a circle subgroup which inherits a canonical framing from in the usual way. .
Paper Structure (3 sections, 5 theorems, 41 equations)

This paper contains 3 sections, 5 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Tensor product decomposition of $E$
  3. Proofs of Theorem and Proposition

Key Result

Theorem

Let $\rho : SU(2n)\to GL(4n, \mathbb{R})$$(n\ge 2)$ be the standard real representation of $SU(2n)$. Then we have

Theorems & Definitions (11)

  • Theorem
  • Proposition
  • Lemma 1
  • proof
  • Lemma 2: cf. LS, §2, Example 3
  • proof
  • Lemma 3
  • proof
  • proof : Proof of Theorem
  • proof : Proof of Proposition
  • ...and 1 more