Simplicial Structures in Topology
Simplicial Structures in Topology
Simplicial Structures in Topology
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196 VI Homotopy Groups<br />
and def<strong>in</strong>e the map (called comultiplication <strong>in</strong> S1 )<br />
ν : S 1 → S 1 ∨ S 1 , e 2πti �<br />
(e0,e<br />
↦→<br />
2π2ti ) 0 ≤ t ≤ 1 2<br />
(e2π(2t−1)i ,e0) 1<br />
2 ≤ t ≤ 1;<br />
besides, let us def<strong>in</strong>e the fold<strong>in</strong>g map σ : S 1 ∨ S 1 → S 1 :<br />
(∀0 ≤ t ≤ 1)(e0,e 2tiπ ) ↦→ e 2tiπ , (e 2tiπ ,e0) ↦→ e 2tiπ .<br />
By these def<strong>in</strong>itions, one can easily see that for every po<strong>in</strong>t e 2πti ∈ S 1<br />
( f ∗ g)(e 2πti )=σ( f ∨ g)ν(e 2πti );<br />
moreover, we note that if f ′ ∼ f and g ′ ∼ g, thenσ( f ′ ∨ g ′ )ν is homotopic to<br />
σ( f ∨ g)ν. F<strong>in</strong>ally, for any based space (Y,y0), we def<strong>in</strong>e the function (called multiplication)<br />
[S 1 ,Y ]∗ × [S 1 ,Y ]∗ −→ [S 1 ,Y ]∗<br />
(∀[ f ],[g] ∈ [S 1 ,Y ]∗)[f ] ν<br />
× [g] :=[σ( f ∨ g)ν].<br />
(VI.1.1) Theorem. The set [S 1 ,Y ]∗ with the multiplication ν<br />
× is a group whose unit<br />
element is the homotopy class of the constant function c: S 1 → Y (it takes each<br />
po<strong>in</strong>t of S 1 to y0); besides, the <strong>in</strong>verse of [ f ] is the homotopy class of the based map<br />
h: S 1 → Y def<strong>in</strong>ed, for each e 2πti ∈ S 1 , by the formula h(e 2πti )= f (e 2π(1−t)i ). 1<br />
Proof. We first prove that the function ν : S 1 → S 1 ∨ S 1 is associative (up to homotopy),<br />
<strong>in</strong> other words, that the diagram<br />
ν<br />
S 1 ν ��<br />
��<br />
S 1 × S 1<br />
S 1 ∨ S 1<br />
1 ∨ ν<br />
��<br />
ν ∨ 1 ��<br />
S 1 ∨ S 1 ∨ S 1<br />
commutes up to homotopy (that is to say (1 ∨ ν)ν ∼ (ν ∨ 1)ν). In fact, for every<br />
e2πti ∈ S1 ,<br />
(1 ∨ ν)ν(e 2πti ⎧<br />
⎨ (e0,e0,e<br />
)=<br />
⎩<br />
2π2ti ) 0 ≤ t ≤ 1 2<br />
(e0,e2π2(2t−1)i ,e0) 1 2 ≤ t ≤ 3 4<br />
(e0,e0,e 2π(4t−3)ti ) 3<br />
4 ≤ t ≤ 1<br />
(ν ∨ 1)ν(e 2πti ⎧<br />
⎨ (e0,e0,e<br />
)=<br />
⎩<br />
2π4ti ) 0 ≤ t ≤ 1 4<br />
(e0,e2π(4t−1)i ,e0) 1 4 ≤ t ≤ 1 2<br />
(e2π(2t−1)i ,e0,e0) 1 2 ≤ t ≤ 1<br />
1 Intuitively, the loop h is the loop f , but traveled <strong>in</strong> the opposite direction.