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The geometric loops $t \mapsto (\cos 2 π n t, \sin 2 π n t)$ have degree $n$

Example

For every integer $n$ , the based geometric loop

$γ_{n} (t) = (\cos 2 π n t, \sin 2 π n t)$

has degree $n$ , where degree is transported from the quotient-circle model by the based homeomorphism.

Facts & Assumptions

Given: An integer $n$ and the quotient-circle dictionary.

[L1]

$[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle and sends $[0]$ to $(1, 0)$ ( $[t] \mapsto (\cos 2 π t, \sin 2 π t)$ is a homeomorphism from $R / Z$ to the unit circle).

[L2]

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

[L3]

For every integer $n$ , $ω_{n} (t) = [n t]$ (The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$ ).

Verification

technique · direct

1.1

Under the homeomorphism of [L1], the loop in [L3] has image $t \mapsto (\cos (2 π n t), \sin (2 π n t)) = γ_{n} (t)$ .

L1L3algebra

2.1

The quotient loop has degree $n$ by [L2], so the transported degree of its geometric image $γ_{n}$ is also $n$ . At $n = 0$ this is the constant loop, and the same computation covers every negative $n$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 123 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Allen Hatcher, Algebraic Topology, Ch. 1, Section 1.1 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)
Jonathan Wise, Math 6210 Lecture Notes, Week 3, Section 3.4 (standard reference, not scraped)

The geometric loops t↦(cos⁡2πnt,sin⁡2πnt)t↦(cos2πnt,sin2πnt) have degree nn