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π
1
(X, x)
e = [c
x
] [f]
−1
= [f
−1
].
X
φ: (X, x
0
) → (Y, y
0
)
PT op π
1
(φ): π
1
(X, x
0
) → π
1
(Y, y
0
)
π
1
PT op → Gr
f : I → X, f(0) = x
0
= f(1)
e
f : (S
1
, s
0
) → (X, x
0
)
π
1
(X × Y, (x
0
, y
0
))
∼
=
π
1
(X, x
0
) × π
1
(Y, y
0
),
π
1
PT op Gr
X Y
π
1
((X, x
0
) ∨ (Y, y
0
))
∼
=
π
1
(X, x
0
) ∗ π
1
(Y, y
0
),
π
1
Gr
PT op
