Smooth homotopy - Imported Knowlpedia Content

Definition

Let $M$ and $N$ be smooth manifolds without boundary, and let $f_0,f_1:M\to N$ be smooth maps. A smooth homotopy from $f_0$ to $f_1$ is a smooth map in the manifold-with-boundary sense

F:M\times[0,1]\longrightarrow N

such that $F(x,0)=f_0(x)$ and $F(x,1)=f_1(x)$ for every $x\in M$ . Here the product $M\times[0,1]$ has its standard smooth manifold-with-boundary structure, and smoothness includes smooth extension in the interval direction at the endpoints. Smooth homotopy is an equivalence relation on smooth maps and is the differential-category refinement of continuous homotopy.

Homotopy formula for differential forms

Write $F_t(x)=F(x,t)$ . Integration in the interval direction defines an operator $K:\Omega^k(N)\to\Omega^{k-1}(M)$ satisfying

F_1^*-F_0^*=dK+Kd.

Thus the two pullbacks of de Rham complexes are cochain-homotopic and induce the same map on de Rham cohomology. This is the smooth homotopy-invariance mechanism used throughout de Rham theory Tu, chapter on de Rham theory.

Examples and distinctions

If $f:M\to N$ is constant along a smooth path in $N$ , that path gives a smooth homotopy between the corresponding constant maps. A smooth isotopy is stronger than a smooth homotopy: it requires each time slice to remain in a specified class, commonly diffeomorphisms or embeddings. A general smooth homotopy imposes no such condition on the slices.

Conventions and scope

Some authors require a homotopy to be constant for $t$ near $0$ and $1$ , especially when homotopies will be concatenated smoothly. That is an additional collar convention, not part of the basic definition. If $M$ itself has boundary, $M\times[0,1]$ naturally has corners, so the ambient smooth category must be enlarged or the definition phrased using smooth extension on a neighborhood.

References

Loring W. Tu, An Introduction to Manifolds, 2nd ed., Springer, 2011. DOI record. Relevant: smooth homotopy and the homotopy formula in de Rham theory.
Raoul Bott and Loring W. Tu, Differential Forms in Algebraic Topology, Springer, 1982. DOI record. Relevant: Chapter I, homotopy operators and de Rham theory.