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In differential topology, a branch of mathematics, a stratifold is a generalization of a differentiable manifold where certain kinds of singularities are allowed. More specifically a stratifold is stratified into differentiable manifolds of (possibly) different dimensions. Stratifolds can be used to construct new homology theories. For example, they provide a new geometric model for ordinary homology. The concept of stratifolds was invented by Matthias Kreck. The basic idea is similar to that of a topologically stratified space, but adapted to differential topology. ==Definitions== Before we come to stratifolds, we define a preliminary notion, which captures the minimal notion for a smooth structure on a space: A ''differential space'' (in the sense of Sikorski) is a pair (''X'', ''C''), where ''X'' is a topological space and ''C'' is a subalgebra of the continuous functions such that a function is in ''C'' if it is locally in ''C'' and is in C for smooth and . A simple example takes for ''X'' a smooth manifold and for ''C'' just the smooth functions. For a general differential space (''X'', ''C'') and a point ''x'' in ''X'' we can define as in the case of manifolds a tangent space as the vector space of all derivations of function germs at ''x''. Define strata . For an ''n''-dimensional manifold ''M'' we have that and all other strata are empty. We are now ready for the definition of a stratifold, where more than one stratum may be non-empty: A ''k''-dimensional ''stratifold'' is a differential space (''S'', ''C''), where ''S'' is a locally compact Hausdorff space with countable base of topology. All skeleta should be closed. In addition we assume: # The are ''i''-dimensional smooth manifolds. # For all ''x'' in ''S'', restriction defines an isomorphism stalks . # All tangent spaces have dimension ≤ ''k''. # For each ''x'' in ''S'' and every neighbourhood ''U'' of ''x'', there exists a function with and (a bump function). A ''n''-dimensional stratifold is called ''oriented'' if its (''n'' − 1)-stratum is empty and its top stratum is oriented. One can also define stratifolds with boundary, the so-called ''c-stratifolds''. One defines them as a pair of topological spaces such that is an ''n''-dimensional stratifold and is an (''n'' − 1)-dimensional stratifold, together with an equivalence class of collars. An important subclass of stratifolds are the ''regular'' stratifolds, which can be roughly characterized as looking locally around a point in the ''i''-stratum like the ''i''-stratum times a (''n'' − ''i'')-dimensional stratifold. This is a condition which is fulfilled in most stratifold one usually encounters. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「stratifold」の詳細全文を読む スポンサード リンク
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