1 Introduction: topological and algebro-geometric motivations  
1.1 Open contractible manifolds  
1.2 Contractible algebraic varieties  
2 A user’s guide to $\mathbb{A}^{1}$-homotopy theory  
2.1 Brief topological motivation  
2.2 Homotopy functors in algebraic geometry  
2.3 The unstable $\mathbb{A}^{1}$-homotopy category: construction  
2.4 The unstable $\mathbb{A}^{1}$-homotopy category: basic properties  
2.5 A snapshot of the stable motivic homotopy category  
3 Concrete $\mathbb{A}^{1}$-weak equivalences  
3.1 Constructing $\mathbb{A}^{1}$-weak equivalences of smooth schemes  
3.2 $\mathbb{A}^{1}$-weak equivalences vs. weak equivalences  
3.3 Cancellation questions and $\mathbb{A}^{1}$-weak equivalences  
3.4 Danielewski surfaces and generalizations  
3.5 Building quasi-affine $\mathbb{A}^{1}$-contractible varieties  
4 Further computations in $\mathbb{A}^{1}$-homotopy theory  
4.1 $\mathbb{A}^{1}$-homotopy sheaves  
4.2 $\mathbb{A}^{1}$-connectedness and geometry  
4.3 $\mathbb{A}^{1}$-homotopy sheaves spheres and Brouwer degree  
4.4 $\mathbb{A}^{1}$-homotopy at infinity  
5 Cancellation questions and $\mathbb{A}^{1}$-contractibility  
5.1 The biregular cancellation problem  
5.2 $\mathbb{A}^{1}$-contractibility vs topological contractibility  
5.3 Cancellation problems and the Russell cubic  
5.4 $\mathbb{A}^{1}$-contractibility of the Koras–Russell threefold  
5.5 Koras–Russell fiber bundles