Cover

Title page

Contents

Preface

Equivariant topological complexities

1. Introduction

2. Equivariant Notions

3. Topological complexity

4. Equivariant versions of TC

5. Comments

References

Rational methods applied to sectional category and topological complexity

Introduction

1. Sullivan’s rational homotopy theory

1.1. Sullivan models

1.2. The connection with topology

1.3. Models for homotopy pullbacks

1.4. Models for fibrations

1.5. Models for homotopy pushouts

1.6. Models for cofibrations

1.7. Models for the base point inclusion

1.8. Models for the diagonal map

2. Rational Lusternik-Schnirelmann category

2.1. The mapping theorem for LS category

3. The Whitehead and Ganea characterizations

3.1. The Whitehead characterization

3.2. The Ganea characterization

3.3. Whitehead vs. Ganea

3.4. First algebraic characterizations

4. Rational approximations of sectional category

4.1. Module sectional category

4.2. Poincaré Duality

5. Characterization à la Félix-Halperin

5.1. When 𝑓 admits a homotopy retraction

5.2. Applications to topological complexity

6. A mapping theorem for topological complexity

Acknowledgements

References

Topological complexity of classical configuration spaces and related objects

1. Introduction

2. The plane and the sphere

2.1. Arrangements, I

2.2. Cohomology

2.3. Genus zero

3. Genus one

3.1. Fadell-Neuwirth theorem

3.2. Cohen-Taylor/Totaro spectral sequence

4. Higher genus

4.1. Gröbner bases

4.2. Zero divisors

5. Orbit configuration spaces

5.1. Generalized Fadell-Neuwirth theorem

5.2. Arrangements, II

6. Some discrete groups

6.1. Almost-direct products of free groups

6.2. Fiber-type arrangements

6.3. Subgroup conditions

7. Sins of omission

7.1. Graph configuration spaces

7.2. Unordered configuration spaces

Acknowledgements

References

A topologist’s view of kinematic maps and manipulation complexity

1. Introduction

2. Robot kinematics

3. Topological properties of kinematic maps

4. Overview of topological complexity

5. Complexity of a map

6. Instability of robot manipulation

References

On the cohomology classes of planar polygon spaces

1. Main theorem

2. Proof

References

Sectional category of a class of maps

1. The Ganea point of view

2. The Whitehead point of view

3. The open covering point of view

4. Topological complexity

References

Q-topological complexity

1. Introduction

2. Q-sectional category and Q-topological complexity

3. Some properties of 𝑄secat and 𝑄TC

References

Topological complexity of graphic arrangements

1. Introduction

2. Hyperplane Arrangements

3. Topological Complexity and Motion Planning

4. Large Arrangements

5. Graphic Arrangements

6. Arboricity and a theorem of Nash-Williams

7. Result

8. Applications

References

Hopf invariants, topological complexity, and LS-category of the cofiber of the diagonal map for two-cell complexes

1. Introduction

2. Preliminaries

3. Spheres: The typical example

4. Two-cell complexes in the metastable range with non-trivial Hopf invariant

5. Non-Hopf-sets methods

References

Topological complexity of collision-free multi-tasking motion planning on orientable surfaces

1. Introduction

2. Upper bounds

3. Zero divisors via the Totaro spectral sequence

4. A subquotient of the cohomology of \F(Σ_{𝑔},𝑛)

5. Proof of Theorem 4.2

6. The case 𝑠=2

References

Topological complexity of subgroups of Artin’s braid groups

1. Introduction

2. Topological complexity of aspherical spaces

3. Braid groups and their subgroups

4. Upper bounds

5. Lower bounds

6. Higher topological complexity

References

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