Algebra + Homotopy = ??

Explaining why A_\infty algebras

There is a comprehensive review article on this recently. Check out what it's all about.

http://math.unice.fr/~brunov/publications/Algebra+Homotopy=Operad.pdf

Have a dga A with a homotopy to another complex H. From the dga structure of A, you can get all the higher operations needed on H side.

For example



where is the map H to A, then use the multiplication on A and then take it back with p.

I refuse to write the formulas for any of the higher products. Instead I'll give you a pretty picture.

That is the three product on the H side.

That is the nth higher product in terms of all sorts of degenerations of planar binary rooted trees with n leafs.

There are also the horrible relations that describe all sorts of degenerations of the trees.

So we have gotten an  algebra from a dga. This construction is in fact functorial and dga's are a full subcategory of  algebras.

We could do algebraic topology with this by using the example of a chain complex being homotopic to the homology complex with 0 differential. The higher products we got by abstract nonsense become the familiar Massey products of say the Boromean rings or other such configurations where linking numbers just don't cut it.

We have gotten  , next time we continue with operads.