Homology of wedge sum is the direct sum of homologies

I want to prove that $H_n(\bigvee_\alpha X_\alpha)\approx\bigoplus_\alpha H_n(X_\alpha)$ for good pairs (Hatcher defines a good pair as a pair $(X,A)$ such that $A\subset X$ and there is a neighborhood of $A$ that deformation retracts onto $A$).

What I tried:

Since $(X_\alpha, x_\alpha)$'s are good pairs, $(\bigsqcup X_\alpha, \{x_\alpha:\alpha\in I\})$ is a good pair, so a theorem (a long exact sequence argument) gives us an isomorphism $$q_*:H_n(\bigsqcup X_\alpha, \{x_\alpha:\alpha\in I\})\to H_n(\bigsqcup X_\alpha/\{x_\alpha:\alpha\in I\},\{x_\alpha:\alpha\in I\}/\{x_\alpha:\alpha\in I\})=H_n(\bigvee X_\alpha, \text{some point})$$ induced by the quotient map $$q:\bigsqcup X_\alpha, \{x_\alpha:\alpha\in I\})\to \bigsqcup X_\alpha/\{x_\alpha:\alpha\in I\},\{x_\alpha:\alpha\in I\}/\{x_\alpha:\alpha\in I\})=(\bigvee X_\alpha, \text{some point})$$

Questions:

Now, do these mean that we have an isomorphism $\phi:H_n(\bigsqcup X_\alpha)=\bigoplus H_n(X_\alpha) \to H_n(\bigvee X_\alpha)$? In general if we have an isomorphism $\theta:H_n(X,Y)\to H_n(A,B)$, then do we also have an isomorphism $\theta':H_n(X)\to H_n(A)$?

• Look at the long exact sequences of the pairs $(X_\alpha, x_\alpha)$. What do they tell you. Hint: The theorem is actually wrong for $n=0$. – Carsten S Oct 8 '13 at 17:42
• I think you mean reduced homology groups $\tilde H_n$ – Stefan Hamcke Oct 8 '13 at 17:43
• And regarding your last question: Surely not, any map $X\to A$ induces an isomorphism $H_n(X, X)\to H_(A, A)$. – Carsten S Oct 8 '13 at 17:43
• @StefanH Actually, yes but they are the same for $n>0$ – Xena Oct 8 '13 at 17:44

Your proof is almost complete, let me suggest you some additional hint to complete it:

• there's an isomorphism $$\tilde H_n(X) \cong H_n(X,x_0)$$ between the reduced homology of a space $X$ and the homology of the pair $(X,x_0)$ where $x_0 \in X$;

• in the category of pairs of topological spaces there's an isomorphism $$H_n\left(\bigsqcup_\alpha (A_\alpha,B_\alpha)\right) \cong \bigoplus_\alpha H_n(A_\alpha, B_\alpha)$$ for a family of pairs $(A_\alpha,B_\alpha)_\alpha$.

Combining these results with what you used should bring you to the solution of the problem.

Hope this helps.

• Of course! I would like to write a complete solution but it took me a while to write this stuff in latex and I have little time but thank you very much – Xena Oct 8 '13 at 18:00
• @Fanni you're welcome – Giorgio Mossa Oct 8 '13 at 18:16
• Where can I find a proof of the second $\bullet$ ? Thanks in advance. – Reinaldo R. Nov 1 '18 at 14:23
• @ReinaldoR. it's a trivial result I believe you can find in every text book on algebraic topology. For instance in Hatcher's Algebraic topology is (a consequence) of proposition 2.6 and the short exact sequence lemma. – Giorgio Mossa Nov 1 '18 at 18:22