Homotopies and fixed endpoints

Try

Two routes in the plane can bend into one another while their starting and finishing points stay fixed. Take α(t)=(t,t) and β(t)=(t,t²), 0≤t≤1. The formula H(t,s)=(t,(1−s)t+st²) continuously changes the first route into the second. The parameter t locates a point along a route; s chooses which intermediate route to inspect. They have different roles.

Predict

Predict what happens if the space has a forbidden point. Straight-line interpolation may pass through that point even when both original paths avoid it. Also predict whether a deformation that moves the endpoints is enough to compare routes with prescribed endpoints. It is not: endpoint conditions are part of the problem, not an optional feature of a drawing.

Experiment

For the fixed example, inspect the midpoint of each intermediate route.

Deformation s	H(1/2,s)
0	(1/2,1/2)
1/2	(1/2,3/8)
1	(1/2,1/4)

At t=0 every row starts at (0,0), and at t=1 every row ends at (1,1). Draw the parameter square: its bottom and top are the two paths; its vertical edges are constant endpoint tracks. This boundary bookkeeping will later make lifting and group operations precise.

Understand

For continuous maps f,g:X→Y, a homotopy is a jointly continuous H:X×[0,1]→Y with H(x,0)=f(x) and H(x,1)=g(x). It is relative to a subset A⊆X if H(a,s)=f(a)=g(a) for every a in A and every s. For paths, take X=[0,1] and A consisting of its two endpoints. Endpoint homotopy therefore keeps the starting and ending points fixed through the whole deformation.

The relation is reflexive using H(x,s)=f(x), symmetric by reversing s, and transitive by running two homotopies in succession. In the transitive construction, use the first at time 2s for s≤1/2 and the second at time 2s−1 for s≥1/2. Their values agree on the shared seam because the first ends at the map where the second begins. The pasting lemma for the two closed parameter pieces proves joint continuity. The same construction preserves any relative subset.

If Y is convex in a real vector space, straight-line interpolation stays in Y and proves that any two maps are homotopic. For paths with common endpoints it is automatically relative to those endpoints. Convexity is a sufficient geometric condition, not a definition of homotopy. In a punctured plane, opposite semicircle midpoints interpolate through the missing origin, so this particular construction fails.

A contraction is a homotopy from the identity map of a space to a constant map. This is stronger than path connectedness: it selects paths to a common point continuously as the starting point varies. A finite collection of sketches cannot verify that joint continuity requirement.

Apply

Repair the statement “Every path can be shrunk by shortening its parameter interval.” Shortening a path generally changes its endpoint. For a loop, shrinking the parameter interval can also move the basepoint unless the formula is designed to preserve it. State the four sides of the parameter square before claiming an endpoint homotopy. Later the circle will provide an obstruction to shrinking certain loops with endpoints fixed.

Continue

This bridge assumes the point-set topology course and elementary group operations. Its theorem scope is checked against Hatcher’s official Chapter 1 outline. The explanations and practice here are original; homology and higher homotopy lie beyond this bridge. Next, endpoint-homotopy classes turn loop concatenation into a group.

Continue to Build the fundamental group.