Error in Hatcher on how multiplication is defined in $H^*(X;R)$?

Question

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Should the multiplication $(\sum_i \alpha_i) (\sum_j \beta_j) = \sum_{i,j} \alpha_i\beta_j$ actually be $(\sum_i \alpha_i) \smile (\sum_j \beta_j) = \sum_{i,j} (\alpha_i \smile \beta_j)$?

Answer 1

It's not so much an error as a silent change of notation. If you take a look at the next 10 pages or more, you'll see that Hatcher is omitting the "cup" symbol more and more and more. It kind of makes sense to do this, as long as you keep context in mind, i.e. as long as you can remember that when you write two cohomology classes right next to each other the intention is that you should take their cup product. It's kind of like the ordinary product in the real numbers: after a while, we stop writing $x \times y$, and just write $xy$.