Let's have an algebraic topology thread. Yesterday I learned

Maybe there is a glueing of the K(Gi,nj)'s which does this, I'd ask on SE

>>7767928
I know category theory and need to learn some basic algebraic topology for a homotopy type theory course. What's a good introductory resource?

>>7768076
OP here. I like Tom Dieck's "Algebraic Topology" and Davis & Kirk's "Lecture Notes in Algebraic Topology". Bredon's "Topology and Geometry" also contains many chapters on algebraic topology, but I only briefly looked into that book.

>>7767928
What you're looking for is called a product.

>>7768076
Hatcher is the standard reference. His book is certainly the most well written and provides a lot of geometric intuition. On the down side he avoids using category theory as much as he can, but if you'll be taking a homotopy theory class then it's no loss.

>>7770028
K(G,n) ^ K(G,m) is m+n-1-connected (first nontrivial homotopy group is the n+m th), where ^ is the smash product.

>>7770095
Anyway OP, you can construct your space by playing with bouquets of spheres, similar to the standard CW complex construction of the K(G,n).
If you take a sphere for each generator of G (m of them, say), you can form K(G,n) by taking the wedge sum of the spheres, giving you [math]\bigvee^{m} S^{n}[/math], using attaching maps to impose relations, then adding cells in higher dimension to kill the higher homotopy.
Also, look up the Borel construction.

>>7770095
Wrong product, just use the cartesian one.

>>7770167
Thanks bro.

>>7770172
Interesting. I'm not sure though whether the higher homotopy groups (n>1) still satisfy pi_n(XxY) = pi_n(X) x pi_n(Y). Do they?

>>7770263
Yes, they do. Just apply the universal property for products to the relevant hom space and then look at path components on both sides of the homeomorphism.