nLab
stratifold

Contents

Idea

Stratifolds are a generalization of smooth manifolds – a notion of generalized smooth space – which were introduced by Kreck; see his lecture notes.

Stratifolds comprise one of the many variants of the concept of a stratified space, and they may include some types of singularities.

Definition

Stratifolds form a class of differential modules, which are pairs (S,C)(S,C) of a topological space SS together with an subalgebra CC of the algebra of continuous real-valued functions SRS\to R such that

  1. CC is locally detectable, i.e. for all continuous functions f:SRf: S\to R, ff is in CC iff for every xSx\in S there exist an open neighborhood UxU\ni x and gCg\in C such that g U=f Ug|_U = f|_U,

  2. let f 1,,f nf_1,\ldots,f_n be elements of CC and ϕ:R nR\phi : R^n\to R a smooth function. Then ϕ(f 1,,f n):SR\phi \circ (f_1,\ldots,f_n) : S\to R is in CC.

Local detectability is equivalent to requiring that CC is an algebra of global sections of a given subsheaf of the sheaf of all continuous functions on SS; in particular germs at every point can be defined.

For a manifold C=C (S)C= C^\infty(S). For a differentiable space S=(S,C)S = (S,C) a tangent space T xST_x S can be defined at each xSx\in S. Define S i={xSdimT xS=i}S^i = \{x\in S | \mathrm{dim} T_x S = i\}. By construction, SS decomposes into a disjoint union S= i=0 S iS = \biguplus_{i=0}^\infty S^i. There is an induced stratifold structure on the topological subspace S iSS^i\subset S, which we denote by (S i,C(S i))(S^i, C(S^i)).

Definition

A kk-dimensional stratifold (S,C)(S,C) is a differential space such that

  • SS is locally compact Hausdorff space with countable basis,

  • T xSkT_x S \leq k for all xSx\in S (i.e. S= i=0 kS iS = \bigcup_{i=0}^k S^i),

  • (S i,C(S i))(S^i,C(S^i)) is isomorphic to a smooth manifold,

  • the restriction map C(S)C(S i)C(S)\to C(S^i) induces an isomorphism of stalks of germs C(S) xC(S i) x=C (S i) xC(S)_x\to C(S_i)_x = C^\infty(S^i)_x in all points xS ix\in S^i,

  • for all ySy\in S, and all UyU\ni y open, there is a “bump function” ρC\rho\in C nonvanishing at yy, but whose support is contained in UU.

References

Revised on May 29, 2012 13:16:48 by Urs Schreiber (131.130.239.199)