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arxiv: 2606.28596 · v1 · pith:7ZGD5ZY2new · submitted 2026-06-26 · 🧮 math.GT

Multi-framed real monopole Floer theory

Jiakai Li This is my paper

Pith reviewed 2026-06-30 00:54 UTC · model grok-4.3

classification 🧮 math.GT
keywords real monopole Floer homologyframed Floer homologythree-manifolds with involutionsSeiberg-Witten invariantsfour-manifolds with involutionsmultiple basepointsrelative gradingsmulti-framing
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The pith

The paper constructs framed real monopole Floer homology for three-manifolds with involutions marked by multiple basepoints and proposes Z-valued invariants for four-manifolds with involutions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This work develops a version of monopole Floer homology adapted to three-manifolds that carry involutions together with multiple basepoints at which framing data is recorded. The resulting homology groups carry relative gradings that depend on the framing information, and a sufficient condition is supplied for the existence of relative mod-two gradings. Assuming the four-manifolds are orientable and orientations have been chosen, the paper also supplies a definition of integer-valued framed real Seiberg-Witten invariants for four-manifolds equipped with involutions and marked by circles. The constructions extend classical Floer and Seiberg-Witten theories into the setting of real structures and multi-point framings.

Core claim

The central claim is that there exists a framed real monopole Floer homology for three-manifolds with involutions, marked with multiple basepoints, whose relative gradings depend on the framing information, with a sufficient condition given for the existence of relative mod two gradings; furthermore, assuming orientability and choices of orientations, Z-valued framed real Seiberg-Witten invariants are defined for four-manifolds with involutions marked with circles.

What carries the argument

Framed real monopole Floer homology, which incorporates framing data at multiple basepoints to define the homology groups for three-manifolds equipped with involutions.

If this is right

  • The homology groups are equipped with relative gradings determined by the chosen framings at the basepoints.
  • A sufficient condition on the framing data guarantees the existence of relative mod-two gradings.
  • Z-valued invariants for four-manifolds with involutions are obtained once orientability and orientation choices are fixed.
  • The invariants are marked by circles on the four-manifolds and are framed real versions of Seiberg-Witten invariants.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The dependence of gradings on framing data implies that distinct framings can produce different homology groups for the same underlying manifold and involution.
  • The multi-basepoint marking suggests the theory may be compatible with operations that add or remove basepoints while preserving the involution.
  • The construction supplies a new source of invariants that could be compared with other real or equivariant Floer theories once explicit examples are computed.

Load-bearing premise

The definition of the Z-valued invariants requires the assumption of orientability of the four-manifolds together with choices of orientations.

What would settle it

An explicit calculation on a concrete four-manifold with involution showing that the proposed invariants change with different orientation choices or fail to take integer values would falsify the definition.

Figures

Figures reproduced from arXiv: 2606.28596 by Jiakai Li.

Figure 1
Figure 1. Figure 1: Table of notations in Morse and monopole theories Example 2.2 (Tt ). Suppose 0 ≤ r ≤ t. This example models the case of a pointed real 3-manifold that admits an invariant metric of positive scalar curvature. The prototype is the double branched cover #t (S 1 × S 2 ) of the (t + 1)-component unlink Ut+1, and there are (r + 1) basepoints on (r + 1) components of Ut+1, cf § 3. Let B = Bred = Tt ∼= (R/Z) t ; t… view at source ↗
read the original abstract

This paper constructs a framed real monopole Floer homology for three-manifolds with involutions, marked with multiple basepoints. The relative gradings of these Floer homologies depend on the framing information and the paper gives a sufficient condition for the existence of relative mod two gradings. Assuming orientability and choices of orientations, this paper also proposes a definition of $\mathbf{Z}$-valued framed real Seiberg--Witten invariants for 4-manifolds with involutions, marked with circles.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. This paper constructs a framed real monopole Floer homology for three-manifolds with involutions, marked with multiple basepoints. The relative gradings of these Floer homologies depend on the framing information and the paper gives a sufficient condition for the existence of relative mod two gradings. Assuming orientability and choices of orientations, this paper also proposes a definition of Z-valued framed real Seiberg-Witten invariants for 4-manifolds with involutions, marked with circles.

Significance. If the constructions and definitions are rigorously established, the work would extend real monopole Floer theory to multi-basepoint and multi-framed settings for 3-manifolds with involutions, while also supplying Z-valued invariants for 4-manifolds with involutions and circle markings. Such invariants could strengthen existing mod-2 versions and find applications in equivariant 3- and 4-manifold topology.

minor comments (1)
  1. The abstract uses boldface Z for the integers; consistent notation should be checked throughout the manuscript.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the summary of our manuscript, which accurately reflects its content and potential applications. No specific major comments appear in the report, so we offer no point-by-point responses. We maintain that the constructions and definitions are rigorously established in the paper as written.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper is a construction of framed real monopole Floer homology for 3-manifolds with involutions and multiple basepoints, together with a proposed definition of Z-valued invariants for 4-manifolds under explicit orientability assumptions. No equations, self-referential definitions, fitted parameters renamed as predictions, or load-bearing self-citations appear in the abstract or description. The central content consists of new definitions and sufficient conditions whose validity is independent of the target objects by construction, making the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; the ledger is therefore minimal and records only the orientability assumption explicitly stated for the four-manifold invariants.

axioms (1)
  • domain assumption Orientability of the four-manifolds together with choices of orientations are required for the Z-valued invariants.
    Explicitly stated in the final sentence of the abstract as a prerequisite for the definition.

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Reference graph

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119 extracted references · 70 canonical work pages · 1 internal anchor

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