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I have a question regarding notation in the proof leading up to Mayer-Vietoris sequence. Suppose the manifold $M=U\cup V$. And $i_{1}:U \to M$ and $i_{2}:V \to M$. Then we have a map:

$\alpha_{k}:H_{dR}^{k}\left ( M \right )\rightarrow H_{dR}^{k}\left ( U \right ) \oplus H_{dR}^{k}\left ( V \right )$

$\left [ \omega \right ] \mapsto \left (i_{1}^{*}\left [ \omega \right ] ,i_{2}^{*}\left [ \omega \right ] \right )$

My question is this: why do we have a $\oplus $ here? To me the result looks more like a Cartesian product. For instance, I would have written $\alpha_{k}:H_{dR}^{k}\left ( M \right )\rightarrow H_{dR}^{k}\left ( U \right ) \times H_{dR}^{k}\left ( V \right )$ in place of $\alpha_{k}:H_{dR}^{k}\left ( M \right )\rightarrow H_{dR}^{k}\left ( U \right ) \oplus H_{dR}^{k}\left ( V \right )$

So why do we use $\oplus$ and in what sense do we have addition between $i_{1}^{*}\left [ \omega \right ]$ and $i_{2}^{*}\left [ \omega \right ]$ ?

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This may just be gibberish, but this is how I understand it. Remember, the cartesian product and direct sum are the same in the finite case, so you are not incorrect. It is just more common to use $\oplus$ when dealing with algebraic objects to denote that there is some underlying structure there. If you simply write $X \times Y$, one would think that you are simply wanting to look at all tuples $(x,y)$. On the other hand, $X \oplus Y$ suggests that you also have an operation defined on $X \times Y$, hence the extra structure.

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    Thanks for your answer, I had wondered if it might be as you said. In my question, I suppose I am just thinking of the meaning in mathematical analysis where the $\oplus$ means addition of two spaces that are orthogonal. i.e. where we are indeed adding the elements in both orthogonal spaces together. But I suspect your answer is the correct one. Thanks :) – 2017-02-27