Almgren isomorphism theorem is a result in
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfac ...
and
algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
about the topology of the space of
flat cycles in a Riemannian manifold.
The theorem plays a fundamental role in the
Almgren–Pitts min-max theory In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces.
The theory started with the efforts for generalizing George David Birkhoff's m ...
as it establishes existence of topologically non-trivial families of cycles, which were used by
Frederick J. Almgren Jr.,
Jon T. Pitts and others to prove existence of (possibly singular)
minimal submanifolds in every Riemannian manifold. In the special case of the space of null-homologous codimension 1 cycles with mod 2 coefficients on a closed Riemannian manifold Almgren isomorphism theorem implies that it is weakly homotopy equivalent to
the infinite
real projective space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space.
Basic properti ...
.
Statement of the theorem
Let M be a
Riemannian manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...
. Almgren isomorphism theorem asserts that the m-th
homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homo ...
of the space of
flat k-dimensional cycles in M is isomorphic to the (m+k)-th dimensional
homology group of M. This result is a generalization of the
Dold–Thom theorem
In algebraic topology, the Dold-Thom theorem states that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as its reduced homology groups. The most common version of its proof consists of showing that the ...
, which can be thought of as the k=0 case of Almgren's (
1962a (ver. PhD thesis),
1962b (ver. Topology (Elsevier)) theorem.
The isomorphism is defined as follows. Let G be an abelian group and
denote the space of flat cycles with coefficients in group G. To each family of cycles
we associate an (m+k)-cycle C as follows. Fix a fine
triangulation
In trigonometry and geometry, triangulation is the process of determining the location of a point by forming triangles to the point from known points.
Applications
In surveying
Specifically in surveying, triangulation involves only angle m ...
T of
. To each vertex v in the 0-skeletion of T we associate a cycle f(v). To each edge E in the 1-skeleton of T with ∂E=v-w we associate a (k+1)-chain with boundary f(v)-f(w) of minimal mass. We proceed this way by induction over the skeleton of T. The sum of all chains corresponding to m-dimensional faces of T will be the desired (m+k)-cycle C. Even though the choices of triangulation and minimal mass fillings were not unique, they all result in an (m+k)-cycle in the same homology class.
[Guth, L. The Width-Volume Inequality. GAFA Geom. funct. anal. 17, 1139–1179 (2007)]
References
Further reading
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* A. Neves, "New applications of Min-max Theory", Proceedings of International Congress of Mathematics, (2014), 939-957
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External links
*{{cite web , url=https://web.math.princeton.edu/~yl15/notes/2019-min-max/Notes.pdf , s2cid=221792677 , title=Yangyang Li's notes on Almgren-Pitts min-max, first1=Yangyang, last1= Li, year=2019
Theorems in algebraic topology