Open Problems | The Portal

Sphere Eversion

A sphere can be turned inside out through a smooth regular homotopy, allowing only self-intersections and no creases. The process is counterintuitive because local smoothness is preserved while global orientation reverses.

The statement is often written as:

f : S² \to ℝ³, with a homotopy fₜ where f₀ = f and f₁ = -f

This is a landmark example showing that topology and intuition can diverge dramatically.

Boy's Surface

Boy's surface is an immersion of the real projective plane in three dimensions. It realizes a non-orientable object through self-intersection structure rather than pinch-point singularities.

The 3-fold symmetry seen in visualizations reflects the deep geometric structure of the immersion and makes it one of topology's most iconic surfaces.

Hopf Fibration

The Hopf fibration decomposes $S³$ into linked circular fibers over points of $S²$ . Under stereographic projection, these fibers become linked circles in $ℝ³$ .

S¹ ↪ S³ \to S²

It is locally a product but globally non-trivial, making it a canonical example of a non-trivial fiber bundle.

π₁ of SO(3)

The rotation group $SO(3)$ has fundamental group $π₁(SO(3)) ≅ ℤ/2ℤ$ . A 360-degree turn is non-trivial; a 720-degree turn is contractible.

The belt trick and plate trick make this topological fact tangible by showing how entanglement introduced by one full turn can be undone by a second.