Jordan's inequality

In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by Template:Harvs. Any foliation on a manifold induces a Haefliger structure, which uniquely determines the foliation.

Definition

A Haefliger structure on a space X is determined by a Haefliger cocycle. A codimension-q Haefliger cocycle consists of a covering of X by open sets U_α, together with continuous maps Ψ_αβ from U_α ∩ U_β to the sheaf of germs of local diffeomorphisms of R^q, satisfying the 1-cocycle condition

${\mathrm{\Psi }}_{\gamma \alpha }(u)={\mathrm{\Psi }}_{\gamma \beta }(u){\mathrm{\Psi }}_{\beta \alpha }(u)$ for $u\in U_{\alpha }\cap U_{\beta }\cap U_{\gamma }.$

More generally, C^r, PL, analytic, and continuous Haefliger structures are defined by replacing sheaves of germs of smooth diffeomorphisms by the appropriate sheaves.

Haefliger structure and foliations

A codimension-q foliation can be specified by a covering of X by open sets U_α, together with a submersion φ_α from each open set U_α to R^q, such that for each α, β there is a map Φ_αβ from U_α ∩ U_β to local diffeomorphisms with

${\varphi }_{\alpha }(v)={\mathrm{\Phi }}_{\alpha ,\beta }(u)({\varphi }_{\beta }(v))$

whenever v is close enough to u. The Haefliger cocycle is defined by

${\mathrm{\Psi }}_{\alpha ,\beta }(u)=$ germ of ${\mathrm{\Phi }}_{\alpha ,\beta }(u)$ at u.

An advantage of Haefliger structures over foliations is that they are closed under pullbacks. If f is a continuous map from X to Y then one can take pullbacks of foliations on Y provided that f is transverse to the foliation, but if f is not transverse the pullback can be a Haefliger structure that is not a foliation.

Classifying space

Two Haefliger structures on X are called concordant if they are the restrictions of Haefliger structures on X×[0,1] to X×0 and X×1.

If f is a continuous map from X to Y, then there is a pullback under f of Haefliger structures on Y to Haefliger structures on X.

There is a classifying space BΓ_q for codimension-q Haefliger structures which has a universal Haefliger structure on it in the following sense. For any topological space X and continuous map from X to BΓ_q the pullback of the universal Haefliger structure is a Haefliger structure on X. For well-behaved topological spaces X this induces a 1:1 correspondence between homotopy classes of maps from X to BΓ_q and concordance classes of Haefliger structures.

References

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  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
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