Real Seiberg-Witten theory - 5

This detailed lecture presents a recent breakthrough by Muazawa regarding exotic embeddings of the real projective plane \(\mathbb{RP}^2\) into the 4-sphere \(S^4\). In particular, it constructs an infinite family of embeddings that are all topologically isotopic but not smoothly isotopic, thereby producing "exotic" smooth structures distinguishable by refined invariants derived from Real Seiberg-Witten theory. This result has deep implications in 4-manifold topology and gauge theory, particularly around involutions and equivariant stable homotopy theoretic invariants. The content is addressed to researchers familiar with gauge theory, 4-manifolds, and equivariant homotopy theory.

The Theorem Setup: Exotic Embeddings of \(\mathbb{RP}^2\) in \(S^4\)

• The speaker considers a standard embedding \(P_+\) of \(\mathbb{RP}^2\) into \(S^4\), arising as the fixed point set of complex conjugation on \(\overline{\mathbb{CP}^2}\). This embedding has positive normal Euler number \(+2\).
• The main theorem states that there exists an infinite family of embeddings \(\{P_n\}_{n=0}^\infty\) of \(\mathbb{RP}^2\) into \(S^4\) such that:
• \(P_0 = P_+\) is the standard embedding.
• For any \(m \neq n\), \(P_m\) and \(P_n\) are topologically isotopic but not smoothly isotopic.
• This provides explicit examples of "exotic" embeddings distinguished by smooth structure, though indistinguishable topologically.
• A corollary, connected to earlier results by Hughes, Kim, and Miller, is the existence of infinitely many exotic involutions on \(\overline{\mathbb{CP}^2}\) (or \(\mathbb{CP}^2\) with reversed orientation). These involutions are related by connected sums with orientable 2-manifolds.