View source for Federer-Fleming deformation theorem

The Federer–Fleming deformation theorem can be considered as one of the central results in the theory of integral currents, created by H. Federer and W.H. Fleming at the end of 1950s, [[#References|[a1]]]. The notion of an integral current was introduced to meet all the requirements of the concept of  "k-dimensional domain of integration in Euclidean n-space"  and combines the smoothness properties of differentiable manifolds and the combinatorial structure of polyhedral chains with integer coefficients. The notion of integral current provides a powerful instrument for solving geometrical variational problems like the [[Plateau problem|Plateau problem]].

The deformation theorem shows how one can deform normal and integral currents into similar currents with supports in the appropriate skeletons of a standard cubical cell complex with given cube edge, and gives estimates for the masses of the currents.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100201.png" /> denote the standard cubical complex in the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100202.png" /> generated by the cubes with edge <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100203.png" /> and with centres at the integer points. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100204.png" /> denote the complex obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100205.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100206.png" />-homothety (cf. [[Homothety|Homothety]]). As always, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100207.png" /> denote the space of normal currents in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100208.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f1100209.png" /> be the mass of the current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002010.png" /> (cf. also [[Differential form|Differential form]]; [[Mass and co-mass|Mass and co-mass]]).

The Federer–Fleming deformation theorem reads: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002011.png" /> be an arbitrary positive number. Then any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002012.png" />-dimensional normal current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002014.png" /> can be represented as a sum

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002015.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002016.png" /></td> </tr></table>

with the following properties:

1) The current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002017.png" /> is a polyhedral chain of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002018.png" /> with real coefficients. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002019.png" /> is an integral current, then the coefficients of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002020.png" /> are integers.

2) The supports of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002021.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002022.png" /> lie in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002023.png" />-neighbourhood of the support of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002024.png" />, and the supports of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002025.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002026.png" /> lie in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002027.png" />-neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002028.png" />.

3) For the masses of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002031.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002032.png" /> the following estimates hold:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002033.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002034.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002035.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002036.png" /></td> </tr></table>

4) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002037.png" /> is an integral current, then so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002038.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002039.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002040.png" />.

5) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002041.png" /> is an (integral) Lipschitz chain, so are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002044.png" />.

6) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002045.png" /> is an (integral) Lipschitz chain, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002046.png" />.

As a consequence, one can obtain the following result: For each integral current <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002047.png" /> there exists a sequence of integral Lipschitz chains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002048.png" /> such that

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002049.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002050.png" />.

As other applications of the theorem, one can obtain the theorem about isomorphism between the integer singular homology groups of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002051.png" /> and the homology groups of the integral current chain complex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002052.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002054.png" /> are local Lipschitz neighbourhood retracts in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f110/f110020/f11002055.png" />; the isoperimetric inequalities for integral currents; etc. Some modern applications of the theorem can be found in [[#References|[a2]]], [[#References|[a3]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Federer,   W.H. Fleming,   "Normal and integral currents"  ''Ann. of Math.'' , '''72''' :  3  (1960)  pp. 458–520  {{MR|0123260}} {{ZBL|0187.31301}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  Dao Trong Thi,   A.T. Fomenko,   "Minimal surfaces, stratified multivarifolds and the Plateau problem" , Amer. Math. Soc.  (1991)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.T. Fomenko,   "Variational principles in topology: multidimensional minimal surface theory" , Kluwer Acad. Publ.  (1990)</TD></TR></table>