Real Monopole Floer Homology
- Real monopole Floer homology is a refinement of monopole Floer theory that uses conjugation symmetry and real spin^c structures to capture deeper geometric information.
- It integrates involutive and Pin(2)-equivariant methods, employing mapping cone constructions and Morse–Bott techniques to reveal new invariants.
- The framework bridges 3-manifold topology and link theory, enabling explicit computations via skein exact triangles and spectral sequence analyses.
Searching arXiv for recent and foundational papers on real, involutive, and -equivariant monopole Floer homology. Real monopole Floer homology is a conjugation-sensitive refinement of monopole Floer theory in which Seiberg–Witten configurations are endowed with an anti-linear symmetry and the resulting Floer package is built from the corresponding invariant sector. In the literature, the phrase refers to several closely related constructions rather than a single universal formalism. One strand treats self-conjugate structures on closed $3$-manifolds and produces involutive or -equivariant monopole Floer groups from the conjugation symmetry on the configuration space (Lin, 2016, Lin, 2014). Another, more recent strand studies links via branched double covers equipped with the deck involution and defines genuinely real groups $\widehat{\HMR}$, $\widecheck{\HMR}$, $\overline{\HMR}$, and a based-link mapping-cone refinement (Li, 2023, Li, 2024). Across these variants, the defining feature is that ordinary monopole Floer theory is augmented by data invisible to the purely -equivariant theory, typically coming from conjugation, quaternionic symmetry, or real structures.
1. Definition and geometric input
The geometric source of real monopole Floer theory is the conjugation symmetry of Seiberg–Witten theory. In the involutive formulation, the basic map on configurations is
0
where 1 is a connection, 2 is the conjugate connection, and 3 uses the quaternionic structure on the spinor representation (Lin, 2016). When 4, this becomes an involutive symmetry of a single Floer complex up to continuation, and the corresponding theory is a 5-equivariant enhancement of ordinary monopole Floer homology (Lin, 2016).
In the 6-equivariant formulation, one works with a closed oriented 7-manifold 8 equipped with a self-conjugate spin9 structure $3$0. The extra symmetry arises because the spinor bundle is quaternionic, and the family of Dirac operators over the torus of reducibles becomes equivariant under the involution induced by conjugation (Lin, 2017). The relevant symmetry group is
$3$1
and the bar version $3$2 is a module over
$3$3
with $3$4 (Lin, 2017).
The recent link-theoretic theory is formulated directly for an involutive $3$5-manifold. If $3$6 is a closed oriented $3$7-manifold with an orientation-preserving involution $3$8, a real spin$3$9 structure on 0 is a spin1 structure 2 together with an anti-linear involution
3
covering 4, compatible with the Hermitian metric and Clifford multiplication (Li, 2023). For a link 5, the main example is the branched double cover
6
with deck involution 7. In this setting, the real Floer groups are defined on the 8-invariant configuration space and its blown-up counterpart (Li, 2023). This construction is then refined for based links 9 by the tilde theory $\widehat{\HMR}$0 (Li, 2024).
A common misconception is to identify “real” with real coefficients or characteristic-zero de Rham models. That interpretation is explicitly excluded in the integral computation of $\widehat{\HMR}$1, which is not a Real, involutive, quaternionic, or $\widehat{\HMR}$2-equivariant theory (Lin et al., 2021). The same distinction applies to several boundary or Fueter-type monopole theories, which are monopole-based but not real in the conjugation-equivariant sense (Esfahani, 2023, Wang, 2020).
2. Ordinary monopole Floer theory as the underlying model
Real refinements are built on the blow-up formalism of ordinary monopole Floer homology. For a closed oriented $\widehat{\HMR}$3-manifold $\widehat{\HMR}$4 with spin$\widehat{\HMR}$5 structure $\widehat{\HMR}$6, the configuration space is
$\widehat{\HMR}$7
with Seiberg–Witten vector field
$\widehat{\HMR}$8
the $\widehat{\HMR}$9-gradient of the Chern–Simons–Dirac functional (Lidman et al., 2016). Because the gauge action is not free at reducibles, one passes to the blown-up configuration space
$\widecheck{\HMR}$0
whose quotient is a manifold with boundary (Lidman et al., 2016).
The resulting chain complex has three types of generators: irreducibles $\widecheck{\HMR}$1, boundary-stable reducibles $\widecheck{\HMR}$2, and boundary-unstable reducibles $\widecheck{\HMR}$3. The “to” complex is
$\widecheck{\HMR}$4
with differential
$\widecheck{\HMR}$5
(Lidman et al., 2016). The $\widecheck{\HMR}$6-action is defined by counting cut-down trajectories and lowers degree by $\widecheck{\HMR}$7 (Lidman et al., 2016).
This ordinary theory admits a spectrum-level interpretation. For rational homology spheres,
$\widecheck{\HMR}$8
as absolutely graded $\widecheck{\HMR}$9-modules (Lidman et al., 2016). This is structurally important for real refinements: it provides a template in which enhanced symmetry should be encoded by a more equivariant Floer spectrum, with the homology theory obtained by applying the corresponding equivariant functor (Lidman et al., 2016). This suggests that real monopole Floer homology is best viewed not merely as an involution on $\overline{\HMR}$0, but as an equivariant shadow of a stronger spectrum-level object.
3. Involutive and $\overline{\HMR}$1-equivariant constructions
The simplest real enhancement is involutive monopole Floer homology. For a self-conjugate spin$\overline{\HMR}$2 structure, the conjugation map induces a chain endomorphism $\overline{\HMR}$3, and the involutive chain complex is defined as the mapping cone
$\overline{\HMR}$4
over $\overline{\HMR}$5, with $\overline{\HMR}$6 of degree $\overline{\HMR}$7 (Lin, 2016). The resulting homology $\overline{\HMR}$8 fits into an exact triangle relating it to the ordinary based tilde theory $\overline{\HMR}$9 (Lin, 2016). For non-self-conjugate spin0 structures, the involution exchanges 1 and 2, and the involutive theory reduces to
3
with trivial 4-action (Lin, 2016).
The fuller equivariant refinement is 5-monopole Floer homology. In the Morse–Bott approach, one generalizes Kronheimer–Mrowka’s construction to gradient flows with Morse–Bott singularities and then restricts to the special case of self-conjugate spin6 structures, where the conjugation symmetry produces the 7-equivariant package (Lin, 2014). The resulting groups
8
are modules over
9
and fit into a long exact sequence analogous to the usual monopole Floer triangle (Lin, 2014). The Morse–Bott formalism is essential because 0-equivariant perturbations do not make all critical points Morse nondegenerate; instead, one obtains Bott families, especially reducible 1-families over fixed reducibles (Lin, 2014).
The bar version 2 admits a concrete topological classification in terms of ordinary cohomological data plus Rokhlin data. For a closed oriented 3-manifold with self-conjugate spin4 structure 5, 6 is determined, up to overall grading shift, by the triple cup product 7 and the Rokhlin map 8 on the set of spin structures inducing 9 (Lin, 2017). The fixed points of the involution on the torus of flat connections are precisely the spin connections, and there are exactly 0 of them (Lin, 2017). The extra equivariant information is torsion in 1, identified by mod 2 spectral flow and hence by Rokhlin invariants (Lin, 2017).
A further structural development is the 3-package for 4-monopole Floer chains. The chain complex of 5 carries a partially defined 6-algebra structure, and for every 7 the chain complex is a partially defined 8-module over it (Lin, 2016). The connected sum formula is then expressed by a quasi-isomorphism
9
and the associated Eilenberg–Moore spectral sequence has
00
(Lin, 2016).
4. Real monopole Floer homology for links and branched double covers
The link-theoretic theory 01 is a real version of monopole Floer homology built from the fixed-point locus of an involution. For a closed oriented 02-manifold 03 with orientation-preserving involution 04, a real spin05 structure 06 determines the real Seiberg–Witten configuration space
07
where 08 is modeled on the 09-anti-invariant imaginary-valued 10-forms, and 11 consists of the 12-invariant spinors (Li, 2023). The corresponding blown-up quotient 13 is a manifold with boundary, and the real chain complexes are defined exactly in the same three-flavor pattern as in ordinary monopole Floer homology (Li, 2023).
For a link 14, one writes
15
since every spin16 structure on the branched double cover supports a unique compatible real structure up to equivalence (Li, 2023). For links with 17 components, the completed groups are modules over
18
with each 19 of degree 20 and 21 (Li, 2023).
The based-link refinement is the tilde real monopole Floer homology
22
defined as the mapping cone of the degree 23 operator
24
(Li, 2024). On the chain level,
25
and its homology is 26 (Li, 2024). This yields a long exact sequence
27
(Li, 2024).
The real theory also has an absolute grading in the torsion case. If 28, 29, and 30 is torsion, the paper defines
31
for a critical point 32, where 33 is a cobordism from the unknot, 34 is its branched double cover, and
35
(Li, 2023). This grading leads to a real Frøyshov invariant 36, defined from the lowest absolute grading in the image of
37
(Li, 2023).
5. Algebraic structures, exact triangles, and computational consequences
A recurrent structural theme is that the real refinements introduce new coefficient algebras and exact sequences. In ordinary monopole Floer theory the dominant formal variable is 38 of degree 39; in involutive theory one adds 40 with 41 (Lin, 2016); in 42-theory the ring becomes
43
with 44, 45 (Lin, 2016, Lin, 2014); in the link-theoretic real theory one instead obtains the 46-module structure with 47 (Li, 2023).
The 48-package has a Gysin exact triangle relating 49 to ordinary 50, with 51 acting on 52 and 53 acting on 54 as 55 (Lin, 2018). This triangle is a basic computational tool because many Massey products in 56 can be identified in terms of the usual 57-action on 58 (Lin, 2018). The paper gives explicit formulas such as
59
under the stated annihilation hypotheses (Lin, 2018).
These higher products are not formal artifacts. The 60-algebra underlying 61-monopole Floer homology is non-formal, and a basic witness is the four-fold Massey product
62
(Lin, 2018). This non-formality is already visible for 63 and controls higher differentials and extension problems in the connected-sum Eilenberg–Moore spectral sequence (Lin, 2016, Lin, 2018). A plausible implication is that the “real” information carried by 64-theory is encoded not only in module structure but also in higher coherences.
For the bar version 65, the spectral sequence description in terms of the Rokhlin map is particularly concrete. If 66 is a basis of 67 and 68 denotes the spin structure corresponding to a subset 69, then there is a spectral sequence converging to 70 with
71
and the differential 72 is nonzero exactly when 73 for 74 and 75, in which case it is, up to grading shift, multiplication by 76 (Lin, 2017).
6. Computations, relationships, and scope
Several families admit explicit calculations. For involutive monopole Floer homology,
77
up to the stated grading normalization, and
78
with 79 in degree 80 (Lin, 2016). For 81-theory, if 82, then up to grading shift
83
while for 84 there are exactly two possibilities depending on whether the two spin structures have the same Rokhlin invariant or not (Lin, 2017). For 85, seven spin structures have Rokhlin invariant 86 and one has Rokhlin invariant 87, and the corresponding 88 can be written explicitly in terms of the cubic Rokhlin map (Lin, 2017).
Mapping tori provide a different bridge between ordinary and real refinements. For an automorphism 89 of a compact Riemann surface 90 with quotient 91, 92-invariant theta characteristics 93 correspond naturally to self-conjugate spin94 structures 95 on the mapping torus 96 (Lin, 2022). The monopole Floer homology of 97 is explicitly determined by the eigenvalues of the lifted action of 98 on 99, and the paper remarks that the chain-level computation respects the extra symmetry and can be used to compute $3$00 as well (Lin, 2022). This suggests a close relationship between algebraic geometry of theta characteristics and the self-conjugate sectors in real or $3$01-equivariant Floer theory.
The recent link-theoretic real theory adds two major computational results. First, $3$02 satisfies an unoriented skein exact triangle for skein triples $3$03 (Li, 2023, Li, 2024). Second, it is the target of a spectral sequence
$3$04
where $3$05 is the reduced Khovanov homology of the mirror (Li, 2024). The $3$06-page is assembled from unlink resolutions, and for the $3$07-component unlink one has
$3$08
(Li, 2024). The Euler characteristic of $3$09 is equal to Miyazawa’s invariant: $3$10 for nonzero determinant links (Li, 2024). For the pretzel knot $3$11, the paper computes
$3$12
while
$3$13
showing that the real theory is not merely ordinary monopole Floer homology of the branched double cover (Li, 2024).
The scope of the subject is therefore twofold. On one side, “real monopole Floer homology” denotes involutive and $3$14-equivariant refinements of monopole Floer theory for self-conjugate spin$3$15 structures, governed by conjugation, Rokhlin invariants, and higher algebra (Lin, 2016, Lin, 2017, Lin, 2016). On the other, it denotes a link-theoretic theory built from branched double covers with deck involution and real spin$3$16 structures, endowed with skein exact triangles, Frøyshov-type invariants, and a Khovanov spectral sequence (Li, 2023, Li, 2024). What unifies these constructions is the principle that anti-linear symmetry in Seiberg–Witten theory yields Floer-theoretic information inaccessible to the ordinary $3$17-equivariant package.