Lift homotopies and detect injectivity

Try

A path lift records how a route travels through sheets. If one route is continuously deformed into another while both endpoints stay fixed, its lifted endpoint should not jump between discrete sheets. To prove this, lifting paths one at a time is insufficient unless their dependence on the deformation parameter is also continuous. We therefore lift the whole parameter square.

Predict

Predict the role of the right edge of an endpoint homotopy square. Every point on that edge projects to the same base endpoint. If the lifted square is continuous, its right edge is a continuous path in a discrete fiber and must be constant. The existence of a jointly continuous lift is the missing bridge between the two observations.

Experiment

Draw a square grid in the homotopy parameters t and s. Each cell will map into one evenly covered neighborhood.

Known data	Next construction	Agreement check
Bottom edge lift	First row of cells	Bottom and shared vertical edges
Completed row	Row immediately above	Shared horizontal edge
All rows	Whole square lift	Finite pasting

Color cells by their chosen sheet, remembering that sheet labels belong to local neighborhoods and need not be globally comparable. At a shared edge, equality is proved by lift uniqueness, not by matching arbitrary color names.

Understand

Let H:[0,1]²→B be continuous and suppose its bottom path has a specified lift to E. The inverse images of evenly covered neighborhoods cover the compact square. Choose a Lebesgue number and a sufficiently fine finite rectangular grid, so each closed cell maps inside one such neighborhood.

For the first cell, its bottom-edge lift stays in one sheet because that edge is connected. Use that sheet’s inverse map composed with H to lift the cell. For the next cell in the row, the already lifted common corner selects a sheet over its neighborhood. Its prescribed bottom and left edges agree with the new formula: each is a path lift of the same base edge with the same initial point, so path uniqueness applies. Continue across the row, then upward through successive rows. Every common edge agrees, and the finite pasting lemma gives a continuous lift of the square. The same local agreement argument proves uniqueness once one starting value is fixed on the connected square.

For an endpoint homotopy, the left and right base edges are constant. Their lifts lie in discrete fibers, so each lifted edge is constant. Thus the two endpoint-homotopic paths, lifted from the same point, have the same endpoint. Notice that a free homotopy can move the basepoint and then its lifted boundary track need not stay in one fiber.

A covering induces an injective homomorphism on fundamental groups. If a loop upstairs projects to the identity class, choose a based null homotopy of its projection. Lift the square beginning with the original loop. The top edge projects to a constant path and therefore is constant. Both side edges remain at the lifted basepoint. The lifted square is a based null homotopy upstairs, proving that the kernel is trivial.

Apply

Apply injectivity to the exponential covering R→S¹. The real line has trivial fundamental group, so injectivity alone does not compute the circle group; it only says no nontrivial upstairs loop is killed. The stronger computation comes from classifying which downstairs loops lift to closed paths and recording their integer endpoint displacement.

Continue

This lesson proves the square version of homotopy lifting needed here. General parameter-space lifting statements require their own formulation, though related local arguments extend the construction. Next, combine path lifting, fixed endpoints and straight-line homotopies upstairs to compute π₁(S¹) exactly.

Continue to Compute the fundamental group of the circle.