Scientists / Open questions

Michael H. Freedman

Low-dimensional topology; topological quantum computing

1986 Fields Medal

20 questions

Weak Poincaré transversality for atomic four-dimensional surgery targets (Freedman's homotopy-theoretic reformulation of the four-dimensional topological surgery conjecture)
Minimal projective-plane cellulations for single-qubit topological quantum codes: optimality of nine edges and classification of the minimal codes
Decidability of periodic 2-SAT ( $HN_2$ ) on finitely generated word-hyperbolic groups and effective bounds on periodicity of solutions
Combining fault tolerance and strictly unitary simulation in a universal model of quantum computation based on topological modular functors
The quantum separation axioms of Cui-Freedman-Wang: the BQP-NDQC1 relationship and DQC1-hardness of trace-closure Jones approximation for $r > 6$
Existence and classification of topological arbiters: the Bing-doubling axiom and 4-dimensional surgery, arbiters on $\mathbb{CP}^2$ , and multi-arbiters
Explicit non-poly-local unitaries: exhibit a unitary family beyond polynomial-size quantum circuits, and locate the ancilla/dilation boundary (from Freedman, Poly-Locality in Quantum Computing, 2002)
Optimal ε–δ–dimension constants in the Freedman–Wang controlled cancellation theorem for controlled modules
Are the Borromean rings A-B-slice? (the A-B slice reformulation of the 4-dimensional topological surgery conjecture)
Open directions in the Freedman–Press truncation of wavelet matrices: boundary-wavelet optimization, sharp $\epsilon(\delta)$ control bounds, and higher-dimensional extension
Reducing universal 4-dimensional surgery models to $\pi_1$ -null kernels and free-slicing the links $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}$
Freedman-He asymptotic crossing number conjecture: is the asymptotic crossing number of every knot equal to its crossing number?
The good groups question in 4-manifold topology: the $\pi_1$ -null disk lemma for free and amenable fundamental groups
Curve-to-word conversion complexity: expressing Dehn twists about geometrically given curves as bounded-length words in mapping-class-group generators
A genuine A-B slice solution for the generalized Borromean rings: combining Freedman–Krushkal's link-homotopy+ stabilization with Whitehead doubling
Explicit diffeomorphism types, multiplicities, and distinguishing invariants in the De Michelis-Freedman family of exotic $\mathbb{R}^4$ 's
Freely slicing the Whitehead double of the Borromean rings: the three-or-more-component case of the Topological Whitehead Double Conjecture
Injectivity of one-generator one-relator quotients of nonzero exponent sum over an arbitrary group (the strong form of the Kervaire conjecture), as posed in Freedman's 1978 remarks on first-degree equations in groups
Are all good boundary links freely slice? The link-slice core of the 4-dimensional topological surgery conjecture
Sharp effective bounds for Kneser-Haken finiteness in bounded 3-manifolds and the handlebody threshold of the Freedman-Freedman periodic-curve example

Back to open problems

Weak Poincaré transversality for atomic four-dimensional surgery targets (Freedman's homotopy-theoretic reformulation of the four-dimensional topological surgery conjecture)

Minimal projective-plane cellulations for single-qubit topological quantum codes: optimality of nine edges and classification of the minimal codes

Decidability of periodic 2-SAT (HN2HN_2HN2​) on finitely generated word-hyperbolic groups and effective bounds on periodicity of solutions

Combining fault tolerance and strictly unitary simulation in a universal model of quantum computation based on topological modular functors

The quantum separation axioms of Cui-Freedman-Wang: the BQP-NDQC1 relationship and DQC1-hardness of trace-closure Jones approximation for r>6r > 6r>6

Existence and classification of topological arbiters: the Bing-doubling axiom and 4-dimensional surgery, arbiters on CP2\mathbb{CP}^2CP2, and multi-arbiters

Explicit non-poly-local unitaries: exhibit a unitary family beyond polynomial-size quantum circuits, and locate the ancilla/dilation boundary (from Freedman, Poly-Locality in Quantum Computing, 2002)

Optimal ε–δ–dimension constants in the Freedman–Wang controlled cancellation theorem for controlled modules

Are the Borromean rings A-B-slice? (the A-B slice reformulation of the 4-dimensional topological surgery conjecture)

Open directions in the Freedman–Press truncation of wavelet matrices: boundary-wavelet optimization, sharp ϵ(δ)\epsilon(\delta)ϵ(δ) control bounds, and higher-dimensional extension

Reducing universal 4-dimensional surgery models to π1\pi_1π1​-null kernels and free-slicing the links {Wh(Bing(Wh))}\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}{Wh(Bing(Wh))}

Freedman-He asymptotic crossing number conjecture: is the asymptotic crossing number of every knot equal to its crossing number?

The good groups question in 4-manifold topology: the π1\pi_1π1​-null disk lemma for free and amenable fundamental groups

Curve-to-word conversion complexity: expressing Dehn twists about geometrically given curves as bounded-length words in mapping-class-group generators

A genuine A-B slice solution for the generalized Borromean rings: combining Freedman–Krushkal's link-homotopy+ stabilization with Whitehead doubling

Explicit diffeomorphism types, multiplicities, and distinguishing invariants in the De Michelis-Freedman family of exotic R4\mathbb{R}^4R4's

Freely slicing the Whitehead double of the Borromean rings: the three-or-more-component case of the Topological Whitehead Double Conjecture

Injectivity of one-generator one-relator quotients of nonzero exponent sum over an arbitrary group (the strong form of the Kervaire conjecture), as posed in Freedman's 1978 remarks on first-degree equations in groups

Are all good boundary links freely slice? The link-slice core of the 4-dimensional topological surgery conjecture

Sharp effective bounds for Kneser-Haken finiteness in bounded 3-manifolds and the handlebody threshold of the Freedman-Freedman periodic-curve example

Decidability of periodic 2-SAT ( $HN_2$ ) on finitely generated word-hyperbolic groups and effective bounds on periodicity of solutions

The quantum separation axioms of Cui-Freedman-Wang: the BQP-NDQC1 relationship and DQC1-hardness of trace-closure Jones approximation for $r > 6$

Existence and classification of topological arbiters: the Bing-doubling axiom and 4-dimensional surgery, arbiters on $\mathbb{CP}^2$ , and multi-arbiters

Open directions in the Freedman–Press truncation of wavelet matrices: boundary-wavelet optimization, sharp $\epsilon(\delta)$ control bounds, and higher-dimensional extension

Reducing universal 4-dimensional surgery models to $\pi_1$ -null kernels and free-slicing the links $\{\mathrm{Wh}(\mathrm{Bing}(\mathrm{Wh}))\}$

The good groups question in 4-manifold topology: the $\pi_1$ -null disk lemma for free and amenable fundamental groups

Explicit diffeomorphism types, multiplicities, and distinguishing invariants in the De Michelis-Freedman family of exotic $\mathbb{R}^4$ 's