Attaching a Cell

Compare Proposition 1.26 in Hatcher.

We determine the effect on the fundamental group of attaching one cell to a general space X. We assume X is path-connected, with basepoint x_{0}. Attaching a 2-cell Attach a 2-cell to X using 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}]. Since ω_{1} is nullhomotopic in D^{2}, ϕ ^{∘} ω_{1} is nullhomotopic in Y. Take a path γ 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}] = β_{γ}(ϕ_{∗}[ω_{1}]). Lemma 3 In the commutative diagram

\xymatrix{π_{1}(D^{2}, d_{1}) \ar[r]^{Φ_{∗}} & π_{1}(Y, x_{1}) \ar[r]^{≅}_{β_{γ}} & π_{1}(Y, x_{0}) \\ π_{1}(S^{1}, d_{1}) \ar[u]_{i_{∗}} \ar[r]^{ϕ_{∗}} & π_{1}(X, x_{1}) \ar[u]_{j_{∗}} \ar[r]^{≅}_{β_{γ}} & π_{1}(X, x_{0}) \ar[u]_{j_{∗}}}  (4)

the left square, hence the whole rectangle, are pushout squares of groups. Proof of Theorem 2 Given homomorphisms ψ_{1}: π_{1}(D^{2}, d_{1}) → G and ψ_{2}: π_{1}(X, x_{0}) → G such that ψ_{1}^{∘}i_{∗} = ψ_{2}^{∘}β_{γ}^{∘}ϕ_{∗}, where G is any group, we seek a unique homomorphism ψ: π_{1}(Y, x_{0}) → G such that ψ^{∘} β_{γ} ^{∘} Φ_{∗} = ψ_{1} and ψ^{∘} j_{∗} = ψ_{2}. As D^{2} is contractible, the first requirement is trivial and Ker ψ_{2} must contain the element β_{γ}ϕ_{∗}[ω_{1}], and hence the whole of N. We have enough to identify π_{1}(Y, x_{0}) with π_{1}(X, x_{0})/N. Proof of Lemma 3 The given pushout square (1) does not lend itself to direct application of van Kampen’s Theorem. The key idea is to attach a collar to X.

Define the collar V = {x ∈ D^{2} : ∥x∥ > 1/2} of S^{1} in D^{2}; it contains S^{1} as a deformation retract. We attach V to X using the same attaching map ϕ: S^{1} → X to produce the subspace X ∪_{ϕ} V of Y. 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 \ar[u]^{∪} \\  & S^{1} \ar[u]^{∪}_{i} \ar[r]^{ϕ} & X \ar[u]^{∪}_{j}}  (5)

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

We need a new basepoint d_{2} = 3d_{1}/4 ∈ V ⊂ D^{2} that does not lie in S^{1}. Now we can apply van Kampen, taking A_{1} = Y − X and A_{2} = X ∪_{ϕ} V with basepoint y_{2} = Φ(d_{2}), to obtain the pushout square of groups and homomorphisms induced by inclusion

\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 the subspace 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 in turn replace this by (Φ|V)_{∗}: π_{1}(V, d_{2}) → π_{1}(A_{2}, y_{2}).

Finally, we reduce to the desired homomorphism ϕ_{∗} by the commutative diagram

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

where h denotes a 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 two lower isomorphisms.

Attaching an n-cell We now consider attaching an n-cell to X to form Y, where n > 2, using an attaching map ϕ: S^{n−1} → X. Then diagram (4) becomes

\xymatrix{π_{1}(D^{n}, d_{1}) \ar[r]^{Φ_{∗}} & π_{1}(Y, x_{1}) \ar[r]^{≅}_{β_{γ}} & π_{1}(Y, x_{0}) \\ π_{1}(S^{n−1}, d_{1}) \ar[u]_{i_{∗}} \ar[r]^{ϕ_{∗}} & π_{1}(X, x_{1}) \ar[u]_{j_{∗}} \ar[r]^{≅}_{β_{γ}} & π_{1}(X, x_{0}) \ar[u]_{j_{∗}}}

However, since π_{1}(S^{n−1}, d_{1}) is now trivial, i_{∗} and hence j_{∗} are isomorphisms.

Proposition 6 Form the space Y from X by attaching an n-cell using an attaching map ϕ: S^{n−1} → X, where n > 2. Then the inclusion map j: X ⊂ Y induces an isomorphism j_{∗}: π_{1}(X, x_{0}) ≅ π_{1}(Y, x_{0}).