Weaker Freudenthal

[Ali Caglayan]

Can we prove that the map A -> loop susp A is (k+1)-connected for a k-connected A, without having to invoke Freudenthal?

[Ali Caglayan]

Here is an observation I had made:

We have that A is k-connected, we can show that susp A is (k+1)-connected and then show that loop susp A is k-connected. This gives us a k-connected map A -> 1 and a k-connected map loop susp A -> 1. This gives us a diagram which commutes with A and loop susp A in the top corners and 1 in the bottom. The LHS composition is homotopic to the RHS composition hence naming eta : A -> loop susp A, we have eta o unitmap being k-connected hence eta must also be k-connected. This isn't quite there.

Now I was hoping to use the fact that loop spaces are pullbacks hence there are maps coming out of 1s hence (k+1)-connectedness appears, but I couldn't get it to work.