Mazur Manifolds: Exotic 4D Topology

• Mazur manifolds are compact, contractible 4-manifolds built with one 0-handle, one 1-handle, and one 2-handle, producing nontrivial boundaries that distinguish them from the standard 4-ball.
• They bridge classical handlebody constructions with advanced invariants such as Floer and Khovanov homologies, offering a framework to study subtle differences between smooth and topological structures.
• Their applications extend to cork theory, satellite operations, and trisection diagrams, providing key insights into stabilization phenomena, concordance, and surgical techniques in 4-manifold topology.

A Mazur manifold is a compact, contractible 4-manifold admitting a handle decomposition with one 0-handle, a single 1-handle, and a single 2-handle, yet typically with nontrivial boundary and smooth structure. Mazur manifolds are fundamental in the paper of exotic phenomena in four-manifold topology, including corks, trisection theory, Floer and quantum invariants, and the interplay between smooth and topological categories. While originally introduced as "simple" examples of nontrivial contractible manifolds, recent developments reveal a rich structure: infinite families with combinatorial minimality, subtle smooth structures, delicate invariants computable via Floer and Khovanov frameworks, and deep connections to surgery, satellite operations, and concordance theory.

1. Definition, Classical Examples, and Topology

A Mazur manifold M is a compact contractible 4-manifold, not diffeomorphic to the 4-ball, constructed using one 1-handle and one 2-handle in such a way that the boundary ∂M is a homology 3-sphere (often nontrivial). The classical examples $W^{\pm}(l,k)$ due to Mazur, Akbulut, and Kirby are given by attaching a 2-handle along a knot in the boundary of the 1-handle with framing $k$, with specific linking and twist. The handle decomposition encodes both algebraic cancellation (contractibility) and the possibility of exotic smooth structure.

Mazur manifolds are distinguished from the 4-ball by their boundaries and often by their smooth structure: the boundary is not always diffeomorphic to $S^3$, and contractibility does not imply uniqueness of the smooth structure. Simple handle diagrams can realize manifolds with nonstandard differential topology; this "economical" handlebody construction provides a bridge to deeper phenomena, such as corks, exotic pairs, and nontrivial satellite actions.

2. Stable Classification, Tangential Thickness, and Mazur's Theorem

A fundamental aspect of Mazur manifolds is their relation to stabilization under products with Euclidean spaces. Mazur's seminal work established that for sufficiently large $k$, two manifolds $M$ and $N$ satisfy $M \times \mathbb{R}^k \cong N \times \mathbb{R}^k$ (PL, smooth, or topological isomorphism), provided a "tangential homotopy equivalence" exists: a homotopy equivalence $f: M \to N$ such that $f^*(\tau_N) \cong \tau_M$, where $\tau_M$ is the stable tangent bundle.

The normal invariant $n(f) \in [M, G/\mathrm{Top}]$ associated to $f$ must lie in a subgroup $O_k([M, G/\mathrm{Top}])$ for stabilization to produce an isomorphism: $n(f) \in O_k([M, G/\mathrm{Top}]).$ If $k \geq n+2$ for $n$-dimensional base manifolds, Mazur's Theorem 9 ((Kwasik et al., 2011), Section 8.2) asserts that tangential homotopy equivalence implies diffeomorphism after stabilization. This "tangential thickness" is a key concept: Mazur manifolds, as well as other exotic manifolds (such as fake lens spaces), become standard upon stabilization, and many subtle exotic properties disappear for large $k$.

3. Smooth vs. Homeomorphic: Exotic Mazur Manifolds and Diffeomorphism Obstructions

While all Mazur manifolds are contractible and homeomorphic to the 4-ball, smooth structures can be exotic. The construction of pairs of Mazur manifolds that are homeomorphic but not diffeomorphic (Hayden et al., 2019) relies on knot trace techniques: one attaches a 2-handle along a knot $K \subset S^3$ with framing $n$ to build $X_n(K)$, and the smooth structure is distinguished by knot Floer invariants.

The knot Floer homology concordance invariant $\nu(K)$ is shown to be an invariant of the smooth structure of $X_n(K)$: if Mazur manifolds $X_n(K)$ and $X_n(K')$ are diffeomorphic, then $\nu(K) = \nu(K')$, providing a diffeomorphism obstruction. In contrast, concordance invariants $\tau$ and $\epsilon$ do not pass to 4-manifold invariants; they may differ even when two traces are diffeomorphic. This suggests that $\nu$ detects subtler aspects of the smooth structure than $\tau$ or $\epsilon$, especially within contractible settings.

A key corollary is that integer homology 3-spheres arise as boundaries of Mazur manifolds that admit two distinct $S^1 \times S^2$ surgeries, answering a longstanding question from Kirby's list.

4. Floer and Quantum Invariants: Heegaard Floer and Khovanov Homology

Heegaard Floer Theory

For boundaries $Y$ of Mazur type manifolds, Heegaard Floer homology takes the form

$$HF^+(Y) = \mathcal{T}^+_{(d)} \oplus HF^{\mathrm{red}}(Y),$$

with correction term $d(Y)=0$ and $HF^{\mathrm{red}}(Y)$ a finitely generated abelian group. The Casson invariant $\lambda(Y)$ satisfies

$\chi(HF^{\mathrm{red}}(Y)) = \lambda(Y),$

and for specific families of Mazur type manifolds $W_n(k)$,

$$\mathrm{rank}\, HF^{\mathrm{red}}(-\partial W_n(k)) = -\lambda(\partial W_n(k)) = \frac{n(n+1)(n+2)}{3}$$

(Akbulut et al., 2012). This cubic growth in rank demonstrates the deep Floer-theoretic complexity of boundaries arising from simple contractible core constructions.

Khovanov Homology

Recent work shows that Khovanov homology can distinguish exotic Mazur manifolds (Nahm, 12 Oct 2025). By analyzing cobordism maps associated to smoothly embedded disks (or their blowups) in $D^4$ or $(\mathbb{C}P^2)^\circ$, one constructs pairs of disks $\Sigma_k, \Sigma_k'$ in $D^4$ bounded by the same knot $J_k$ with different cobordism maps: for some $\phi \in Kh(J_k)$,

$$Kh(m(\Sigma_k))(\phi) = \pm1,\quad Kh(m(\Sigma_k'))(\phi) = 0.$$

Consequently, their exteriors are homeomorphic but not diffeomorphic: Khovanov homology is sensitive to smooth structure, providing a combinatorial (rather than gauge-theoretic) invariant for detecting exotic smooth structures in Mazur manifolds.

5. Satellite Operations, Generalized Mazur Patterns, and Concordance

Mazur manifolds are closely related to satellite knots and pattern operations, notably the Mazur pattern and its generalizations. A satellite operator defined by $(K, \eta)$ acts on any knot $J$: $P_{K, \eta}(J).$ The Mazur pattern is conjectured to act trivially on the topological concordance group, as recently evidenced by vanishing classical obstructions (Casson-Gordon and $\rho$-invariants) when curves $\eta_0$ and $\eta_1$ are freely homotopic in the complement of a concordance (Manchester, 2022). Thus, while the Mazur pattern does not act trivially on smooth concordance (with explicitly nontrivial examples), it is topologically indistinguishable from the identity operator modulo $(1)$-solvable knots.

Generalized Mazur patterns $Q_n^{i,j}$, obtained by adding twists and winding number parameters, produce satellites with explicit formulas for $\tau$ and $\epsilon$ invariants in terms of companion data (Bodish, 14 May 2024). Remarkably, these invariants never take the value $-1$ for satellites using the generalized Mazur pattern, implying that the satellite operator cannot be surjective on the (smooth or rational) concordance group. Floer thinness typically fails for these satellites unless specific conditions are met (e.g., the companion is the unknot and the twist is $-1$), adding further depth to the algebraic structure of Mazur manifolds constructed by such operations.

6. Stein Fillings, Planar Lefschetz Fibrations, Corks, and Shadow Complexity

Mazur type manifolds frequently admit Stein structures, especially those constructed using positive allowable Lefschetz fibrations (PALFs) with planar fibers (e.g., 4-holed spheres) (Oba, 2014). The resulting contact structures on the boundary are supported by planar open books, with minimal support genus and explicit monodromy constructed from Dehn twists. Stein fillable contact homology 3-spheres with nontrivial Casson invariants arise via this mechanism.

Many corks—contractible 4-manifolds with nontrivial boundary involution—are of Mazur type and thus play a central role in changing smooth structures on closed 4-manifolds. The theory of Turaev's shadows provides a combinatorial framework for producing infinite families of Mazur manifolds and corks with shadow complexity one (Naoe, 2015). Specifically, acyclic special polyhedra with a single true vertex encode the topology and smooth structure efficiently, and all corks of minimal shadow complexity arise from two basic families $A(m,n)$ and $\widetilde{A}(m, n-\tfrac{1}{2})$, connecting directly to classical Mazur examples.

7. Trisection Theory, Stabilization, and Standard Diagrams

The double of a Mazur type manifold, $D W^{-}(0,n+2)$, is always diffeomorphic to $S^4$ (Isoshima, 26 Mar 2024). Explicit trisection diagrams for such doubles can be constructed via two distinct methods: doubling relative trisection diagrams and algorithms converting Kirby diagrams. Both approaches require handle slides, Dehn twists, and destabilizations, ultimately simplifying the presentations to the standard genus-$0$ trisection diagram for $S^4$. This supports the conjecture that every trisection of $S^4$ is standard and shows that even complicated Mazur type diagrams can be reduced to canonical forms. The interplay between diagrammatic techniques for relative trisections and Kirby diagrams provides new tools for examining corks, twists, and other cut-and-paste operations in 4-manifold topology.

8. Boundary L-spaces, Fiberedness, and Floer-theoretic Rigidity

If a Mazur-type 4-manifold has boundary an irreducible L-space (with minimal Heegaard Floer homology), the manifold must be diffeomorphic to $B^4$ and the boundary to $S^3$ (Conway et al., 2018). This establishes a strong rigidity: no exotic smooth structures survive when the boundary is a Floer-theoretically simple L-space. The proof uses handle techniques and contact-geometric arguments (including monodromy and right-veering conditions), and rederives Gabai's Property R as a corollary. The result supports broader conjectures that $S^3$ is the only irreducible homology sphere with L-space Floer homology.

Fiberedness of satellites with generalized Mazur patterns depends intricately on the pattern parameters and companion properties: if $i=0$ and winding and twist constraints are met, the satellite is fibered if the companion is fibered (Bodish, 14 May 2024). Floer thinness tends to fail for such satellites, except in special cases, indicating additional topological complexity.

9. Connections, Open Problems, and Future Directions

The paper of Mazur manifolds unlocks central problems at the interface of 3- and 4-manifold topology, the smooth vs. topological categories, and quantum topology. New invariants (Floer, Khovanov, Casson-Gordon, $\rho$) reveal fine distinctions in smooth structure. Combinatorial models with minimal shadow complexity offer efficient constructions, while trisection theory provides universal decompositions. Rigidity results for boundaries, explicit algorithms for invariants, and the power of satellite operations underscore the far-reaching consequences of Mazur's classical construction. Current open questions include:

• To what extent do quantum (e.g., Khovanov) invariants distinguish general exotic 4-manifolds?
• Can combinatorial methods be systematically extended to describe all contractible 4-manifolds of Mazur type?
• Are all trisections of $S^4$ reducible to standard diagrams?
• How do satellite patterns beyond Mazur's interact with Floer-theoretic and quantum invariants, especially outside the L-space setting?

Mazur manifolds thus remain a central testbed for deep phenomena in low-dimensional topology, with rich interplay between geometric, algebraic, and quantum invariants and ongoing implications for concordance, exotic structures, and diagrammatic topology.