MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

2 changed "homologous" to "homotopic"

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and by $f^\ast, g^\ast : \Omega^\bullet N \to \Omega^\bullet M$ the pull-back of differential forms along each map. Note that $\Omega^\bullet$ is a fully-faithful functor from Manifolds to CDGAs.

I have been brought up with two not-obviously-the-same notions of "homotopy" between maps $f,g$:

A geometric homotopy between $f,g$ is a smooth map $H : M \times [0,1] \to N$ such that $H(-,0) = f$ and $H(-,1) = g$.

An algebraic homotopy between $f,g$ is a map $\eta: \Omega^\bullet N \to \Omega^{\bullet - 1} M$ of graded vector spaces such that $f^\ast - g^\ast = \eta \mathrm d_N + \mathrm d_M \eta$.

I believe that the following is true. Any geometric homotopy gives rise to an algebraic homotopy, and two geometric homotopies are homotopic iff the corresponding algebraic homotopies are homologous homotopic. Not every algebraic homotopy comes from a geometric homotopy; rather, it should be required to satisfy some (directly-checkable) condition that says roughly that it's an "antidifferential operator".

Unfortunately, I have been unable to really convince myself of either of the above beliefs. Probably this is textbook material, and so maybe my question is to be pointed to the correct textbook. But really my question is:

How, explicitly, are the above notions of homotopy between maps related? What extra conditions (if any?) should be put on an algebraic homotopy in order for it to be "geometric"?

It is somewhat embarrassing not to know the sharp relationship between the above concepts, but this is one of the many parts of mathematics that I have picked up largely from conversations and working on the examples that come from particular research questions, and not from ever formally learning such material.

1

# How are these algebraic and geometric notions of homotopy of maps between manifolds related?

Let $M$ and $N$ be smooth manifolds, and $f,g: M \to N$ smooth maps. Denote by $(\Omega^\bullet M,\mathrm d_M)$ and $(\Omega^\bullet N, \mathrm d_N)$ the cdgas of de Rham forms in each manifold, and by $f^\ast, g^\ast : \Omega^\bullet N \to \Omega^\bullet M$ the pull-back of differential forms along each map. Note that $\Omega^\bullet$ is a fully-faithful functor from Manifolds to CDGAs.

I have been brought up with two not-obviously-the-same notions of "homotopy" between maps $f,g$:

A geometric homotopy between $f,g$ is a smooth map $H : M \times [0,1] \to N$ such that $H(-,0) = f$ and $H(-,1) = g$.

An algebraic homotopy between $f,g$ is a map $\eta: \Omega^\bullet N \to \Omega^{\bullet - 1} M$ of graded vector spaces such that $f^\ast - g^\ast = \eta \mathrm d_N + \mathrm d_M \eta$.

I believe that the following is true. Any geometric homotopy gives rise to an algebraic homotopy, and two geometric homotopies are homotopic iff the corresponding algebraic homotopies are homologous. Not every algebraic homotopy comes from a geometric homotopy; rather, it should be required to satisfy some (directly-checkable) condition that says roughly that it's an "antidifferential operator".

Unfortunately, I have been unable to really convince myself of either of the above beliefs. Probably this is textbook material, and so maybe my question is to be pointed to the correct textbook. But really my question is:

How, explicitly, are the above notions of homotopy between maps related? What extra conditions (if any?) should be put on an algebraic homotopy in order for it to be "geometric"?

It is somewhat embarrassing not to know the sharp relationship between the above concepts, but this is one of the many parts of mathematics that I have picked up largely from conversations and working on the examples that come from particular research questions, and not from ever formally learning such material.