Attaching a 2-cell

Compare Proposition 1.26 in Hatcher.

Let X be a path-connected space with basepoint x_{0}. Attach a 2-cell to X by means of the attaching map ϕ: S^{1} → X to form the space Y, so that we have the pushout square of spaces

\xymatrix{D^{2} \ar[r]^{Φ} & Y \\ S^{1} \ar[u]_{i}^{∪} \ar[r]^{ϕ} & X \ar[u]_{j}^{∪}}  (1)

Choose a basepoint d_{1} ∈ S^{1} and put x_{1} = f(d_{1}) ∈ X. Write ω_{1}: I → S^{1} for the standard loop in S^{1}, so that π_{1}(S^{1}, d_{1}) = ℤ, generated by [ω_{1}]. Obviously, ω_{1} is nullhomotopic in D^{2}, and hence ϕ ^{∘} ω_{1} is nullhomotopic in Y. Let γ be a path in X from x_{0} to x_{1}.

Theorem 2 The homomorphism j_{∗}: π_{1}(X, x_{0}) → π_{1}(Y, x_{0}) induced by the inclusion j: X ⊂ Y is surjective, and induces the isomorphism of groups π_{1}(X, x_{0})/N ≅ π_{1}(Y, x_{0}), where N is the smallest normal subgroup of π_{1}(X, x_{0}) that contains the element β_{γ}[ϕ ^{∘} ω_{1}].

This follows immediately from the following result, by applying the isomorphisms β_{γ} to replace π_{1}(X, x_{1}) by π_{1}(X, x_{0}) and π_{1}(Y, x_{1}) by π_{1}(Y, x_{0}).

Lemma 3 The square

\xymatrix{π_{1}(D^{2}, d_{1}) \ar[r]^{Φ_{∗}} & π_{1}(Y, x_{1}) \\ π_{1}(S^{1}, d_{1}) \ar[u]_{i_{∗}} \ar[r]^{ϕ_{∗}} & π_{1}(X, x_{1}) \ar[u]_{j_{∗}}}  (4)

is a pushout square of groups.

Proof The given pushout square diagram (1) does not lend itself to direct application of van Kampen’s Theorem. The key idea is to attach a collar.

Define the collar V = {x ∈ D^{2} : ∥x∥ > 1/2} of S^{1} in D^{2}; it contains S^{1} as a deformation retract. Now we expand diagram (1) to the following diagram,

\xymatrix{D^{2} − S^{1} \ar[r]^{⊂} & D^{2} \ar[r]^{Φ} & Y \\ V − S^{1} \ar[u]^{∪} \ar[r]^{⊂} & V \ar[u]^{∪} \ar[r]^{Φ|V} & X ∪_{ϕ} V \\  & S^{1} \ar[u]^{∪} \ar[r]^{ϕ} & X \ar[u]^{∪}}  (5)

which contains five pushout squares, by the stacking properties of pushout squares. (The top left square is a pushout of open subspaces.)

Now we can apply van Kampen, taking A_{1} = Φ(D^{2} − S^{1}) and A_{2} = X ∪_{ϕ} V. We take a new basepoint d_{2} = 3d_{1}/4 in V ⊂ D^{2}, to obtain the pushout square of groups and homomorphisms induced by inclusions, with basepoint y_{2} = Φ(d_{2}),

\xymatrix{π_{1}(A_{1}, y_{2}) \ar[r] & π_{1}(Y, y_{2}) \\ π_{1}(A_{1} ∩ A_{2}, y_{2}) \ar[u] \ar[r] & π_{1}(A_{2}, y_{2}) \ar[u]}

The top left group is trivial, as Φ maps D^{2}−S^{1} homeomorphically to A_{1} and D^{2}−S^{1} is contractible. Further, Φ maps V −S^{1} homeomorphically to A_{1}∩A_{2}, so we may replace the bottom homomorphism by (Φ|V −S^{1})_{∗}: π_{1}(V −S^{1}, d_{2}) → π_{1}(A_{2}, y_{2}). As V −S^{1} ⊂ V is an obvious homotopy equivalence, we may replace this by (Φ|V)_{∗}: π_{1}(V, d_{2}) → π_{1}(A_{2}, y_{2}).

Then we reduce this to the desired homomorphism ϕ_{∗} by the commutative diagram

\xymatrix{π_{1}(V, d_{2}) \ar[d]^{β_{h}} \ar[r]^{(Φ|V)_{∗}} & pi_{1}(A_{2}, y_{2}) \ar[d]^{β_{Φ∘h}} \\ π_{1}(V, d_{1}) \ar[r]^{Φ|V)_{∗}} & π_{1}(A_{2}, y_{2}) \\ π_{1}(S^{1}, d_{1}) \ar[u] \ar[r]^{ϕ_{∗}} & π_{1}(X, x_{1}) \ar[u]}

where h is the obvious path in D^{2} from d_{1} to d_{2}. Since S^{1} is a deformation retract of V, X is a deformation retract of A_{2} and we have the isomorphisms in the lower half of the diagram.