TITLE: Homology of sphere-complements
QUESTION [4 upvotes]: I have to solve the following questions:
"For a subset $X \subset S^n$ determine the homology group $H_i(S^n - X)$, where
(a) $X \cong S^l \vee S^k$
(b) $X \cong S^l \sqcup S^k$ (disjoint union)
"

The second one I could solve by myself: I used de Morgan's rule, the Proposition 2B.1. from Hatcher's Algebraic topology (Brouwer, Jordan) and a long exact Mayer-Vietoris sequence (if someone wants to see it she/he may ask for it). But I'm not able to see what I have to do in the first case (a)...could perhaps someone give me a hint? Thank you very much!

REPLY [5 votes]: The argument you've outlined for (b) works for (a) after a shift in perspective: 
Suppose $L$ and $K$ are closed subspaces of $S^n$. Let $A=S^n- L$ and $B=S^n -K$. Then we have $A \cap B=S^n -(L \cup K)$, and the Mayer-Vietoris sequence reads
$$H_{i+1}(A \cup B)\to H_i(S^n-(L\cup K)) \to H_i(S^n - L) \oplus H_i(S^n-K) \to H_i(A \cup B).$$ 
In part (b), you had $L=S^l$ and $K=S^k$ disjoint. This means $A \cup B=S^n$, so you get isomorphisms in the middle of the above sequence for all $i \leq n-2$. But in part (a), $L=S^l$ and $K=S^k$ intersect at a single point -- the wedge point. So we have 
$$A \cup B = (S^n - L) \cup (S^n - K) = S^n - (L \cap K)=S^n -\{pt\} \cong \mathbb{R}^n.$$
Leveraging the boring homology of $\mathbb{R}^n$ and the exactness above, we get isomorphisms
 $$H_i(S^n-(L \cup K)) \cong H_i(S^n - L) \oplus H_i(S^n - K)$$
for all $i\geq 0$. (The $i=0$ case can be argued using reduced homology, for example.) Now use the result from Hatcher again, just as in part (b).
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