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A loop that traverses the circle once and then pauses is homotopic to the standard loop

Example

Define $u : I \to R$ by

$u (t) = {\begin{cases} 2 t, & 0 \leq t \leq 1 / 2, \\ 1, & 1 / 2 \leq t \leq 1, \end{cases}$

and put $α = p \circ u$ . Then $α$ traverses the quotient circle once during the first half of the parameter interval and remains at $[0]$ during the second half. It is path-homotopic to $ω_{1}$ .

Facts & Assumptions

Given: The displayed function $u$ and the loop $α = p \circ u$ .

[L1]

For every integer $n$ , define ${\tilde{ω}}_{n} (t) = n t$ and $ω_{n} = p \circ {\tilde{ω}}_{n}$ (The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$ ).

[L3]

Functions continuous on each member of a finite closed cover, and agreeing where the pieces meet, paste to a continuous function (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous).

[L4]

Degree is the endpoint of the unique lift beginning at zero (The degree of a based circle loop).

[L5]

$\deg (ω_{n}) = n$ for every integer $n$ ( $\deg (ω_{n}) = n$ for every integer $n$ ).

[L6]

Straight-line interpolation between two continuous real-valued maps is a continuous homotopy (For continuous maps into a convex subset of $R^{n}$ , the straight-line formula defines a continuous homotopy).

[L7]

The quotient projection is continuous, $p (0) = [0]$ , and $p (1) = [0]$ (The circle as $S^{1} = R / Z$ with basepoint $[0]$ ).

[L8]

Constant functions, the identity, finite sums, and scalar multiples are continuous on real intervals (Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function).

Verification

technique · direct

1.1

The two formulas for $u$ agree at $t = 1 / 2$ , where both equal $1$ , and each piece is continuous by [L8], so [L3] makes $u$ continuous. It has $u (0) = 0$ and $u (1) = 1$ , hence [L7] makes $α = p \circ u$ a based loop. Since $u$ starts at zero and projects to $α$ , it is the defining lift and [L4] gives $\deg (α) = u (1) = 1$ .

L3L4L7L8

2.1

By [L1], the standard loop $ω_{1}$ is the projection of $v (t) = t$ , and [L5] gives $\deg (ω_{1}) = 1 = \deg (α)$ .

step 1.1L1L5

3.1

The formula $K (t, s) = (1 - s) u (t) + s t$ is a continuous homotopy from $u$ to $v$ by [L6]. Since $u (0) = v (0) = 0$ and $u (1) = v (1) = 1$ , it fixes both endpoints for every $s$ . Postcomposing with $p$ gives the explicit path homotopy $H (t, s) = p (K (t, s))$ from $α$ to $ω_{1}$ , relative to $t = 0, 1$ .

step 1.1step 2.1L6L7∎

Depends on

Used by

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