I am studying algebraic topology and I am a beginner in it. I want to prove that if $ (X, x_0) \sim (Y, y_0) $ and $(Z, z_0) \sim (W , w_0) $ then $ X\vee Z \sim Y\vee W .$ X,Y, Z, W are top spaces

I have just an intuition that wedge product seems something like glueing two spaces together in a point. I don't know how can I start to think about it and solve it. Any help would be great thanks.

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    $\begingroup$ This is most easily done by mapping properties of wedges and quotients. $\endgroup$ – Randall Jun 21 '19 at 11:40

Generally speaking, (particularly if you're new to algebraic topology), a good approach can be we actually write down the maps which give you a homotopy equivalence of spaces. I will give you a few hints to get you going.


  • You start by assuming that $(X, x_0) \sim (Y, y_0)$ and $(Z, z_0) \sim (W , w_0)$. This is extremely important. That tells you that you have some maps back and forth which satisfy a certain property to do with their composition being homotopy equivalent to the identity. Keep these maps, and the homotopies which give you your homotopy equivalences, in mind.

  • You can use these maps back and forth to build yourself some maps back and forth on the wedges of spaces $X\vee Z \leftrightarrow Y\vee W$.

  • You then want to show that the composition of these maps is homotopic to the identity on both spaces - the definition of the spaces being homotopy equivalent. To do that, you're going to want to use the homotopies from earlier, which give you that your spaces are homotopy equivalent.

The above is the bare bones of what I imagine is the argument you'll want to run. Give it a shot, and then if you want more details please comment and I can try to say a little more.


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