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| In [[differential topology]], a branch of [[mathematics]], a '''stratifold''' is a generalization of a [[differentiable manifold]] where certain kinds of [[singularity theory|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]].
| | Looking for some simple solutions that will help you have your pole in the water tomorrow? Rest easy, as this activity is fun and simple. Because fishing is a diverse hobby, it accommodates those of varied levels of knowledge and skill. Here are some tips to help make fishing a fun and successful activity, whatever your skill level.<br><br>People that wish to fish in a stream need to cast upstream and allow the current to sweep the lure into the fishing hole. This makes your bait more natural, which increases chances of catching a fish. Try to always keep your line taught, and reduce the slack in your line so you can feel the fish bite.<br><br>Stream fisherman would be wise to start their fishing trips downstream, and make their way upstream as the day progresses. This is simply because fish tend to face against the direction of the current. By moving upstream you are lowering your chances of being seen or heard by your prey.<br><br>Do not ever forget to wear a life jacket when you are fishing. Make sure that it is securely fastened. You also want to make sure that the one you have is the right size. This is very important,because if it is too large it can slip over your head if you fall into the water.<br><br>When fishing alone, be careful not to go into water that is too deep. This is especially true when fishing near large rivers as a sudden undercurrent could pull you under and drown you in even fairly shallow water. It is typically better to fish with at least one other friend.<br><br>If you are going to be using live bait to go fishing, be sure your bait is not kept in the sun for too long. Most fish prefer bait that is cooler, rather than warmer. Have the bait kept in an insulated container until you are ready to use it.<br><br>A great tip for all fisherman is to stay relatively quite while you are on the water. Everyone has heard that loud voices can scare away fish, and while this is slightly exaggerated, making a lot of noise on the surface, particularly disturbing the surface, can indeed spook fish.<br><br>The introduction stressed the fact that not knowing how to fish isn't a bad thing. Picking up fishing is not difficult at all. Even though there are many levels to it, there are many entry points for the hobby, and if you use the tips from this article, you can start fishing now.<br><br>In case you beloved this short article as well as you wish to receive more info relating to [http://www.latestsms.in/latest-funny-sms-2013.htm Friendship SMS] i implore you to visit the web site. |
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| ==Definitions==
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| 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 <math>X\to\mathbb{R}</math> such that a function is in ''C'' if it is locally in ''C'' and <math>g\circ(f_1,\dots, f_n): X\to \mathbb{R}</math> is in C for <math>g:\mathbb{R}^n\to \mathbb{R}</math> smooth and <math>f_i\in C</math>. A simple example takes for ''X'' a smooth manifold and for ''C'' just the smooth functions.
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| 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]] <math>T_x X</math> as the [[vector space]] of all [[Derivation (abstract algebra)|derivations]] of function [[Germ (mathematics)|germs]] at ''x''. Define strata <math>X_i = \{x\in X\colon T_x X</math> has dimension i<math>\}</math>. For an ''n''-dimensional manifold ''M'' we have that <math>M_n = M</math> 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:
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| 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:
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| [[Image:suspension.svg|thumb|upright|right|The suspension]]
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| # The <math>(S_i, C|_{S_i})</math> are ''i''-dimensional smooth manifolds.
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| # For all ''x'' in ''S'', restriction defines an [[isomorphism]] [[Stalk (sheaf)|stalks]] <math>C_x \to C^{\infty}(S_i)_x</math>.
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| # All tangent spaces have dimension ≤ ''k''.
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| # For each ''x'' in ''S'' and every neighbourhood ''U'' of ''x'', there exists a function <math>\rho\colon U \to \R</math> with <math>\rho(x) \neq 0</math> and <math>\text{supp}(\rho) \subset U</math> (a bump function).
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| 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 <math>(T,\partial T)</math> of topological spaces such that <math>T-\partial T</math> is an ''n''-dimensional stratifold and <math>\partial T</math> is an (''n'' − 1)-dimensional stratifold, together with an equivalence class of [[collar (topology)|collars]].
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| 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.
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| ==Examples==
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| There are plenty of examples of stratifolds. The first example to consider is the open [[cone (topology)|cone]] over a manifold ''M''. We define a continuous function from ''S'' to the reals to be in ''C'' [[iff]] it is smooth on ''M'' × (0, 1) and it is locally constant around the cone point. The last condition is automatic by point 2 in the definition of a stratifold. We can substitute ''M'' by a stratifold ''S'' in this construction. The cone is oriented if and only if ''S'' is oriented and not zero-dimensional. If we consider the (closed) cone with bottom, we get a stratifold with boundary ''S''.
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| Other examples for stratifolds are [[one-point compactification]]s and [[suspension (topology)|suspensions]] of manifolds, (real) algebraic varieties with only isolated singularities and (finite) simplicial complexes.
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| ==Bordism theories==
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| [[Image:Pair of pants cobordism (pantslike).svg|thumb|right| An example of a bordism relation]]
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| In this section, we will assume all stratifolds to be regular. We call two maps <math>S,S' \to X</math> from two oriented compact ''k''-dimensional stratifolds into a space ''X'' ''[[cobordism|bordant]]'' if there exists an oriented (''k'' + 1)-dimensional compact stratifold ''T'' with boundary ''S'' + (−''S''<nowiki>'</nowiki>) such that the map to ''X'' extends to ''T''. The set of equivalence classes of such maps <math>S\to X</math> is denoted by <math>SH_k X</math>. The sets have actually the structure of abelian groups with disjoint union as addition. One can develop enough [[differential topology]] of stratifolds to show that these define a [[homology theory]]. Clearly, <math>SH_k(\text{point}) = 0</math> for ''k'' > 0 since every oriented stratifold ''S'' is the boundary of its cone, which is oriented if dim(''S'') > 0. One can show that <math>SH_0(\text{point})\cong\mathbb{Z}</math>. Hence, by the [[Eilenberg–Steenrod axioms|Eilenberg–Steenrod]] uniqueness theorem, <math>SH_k(X) \cong H_k(X)</math> for every space ''X'' homotopy-equivalent to a [[CW-complex]], where ''H'' denotes [[singular homology]]. It should be noted, however, that for other spaces these two homology theories need not be isomorphic (an example is the one-point compactification of the surface of infinite genus).
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| There is also a simple way to define [[equivariant homology theory|equivariant homology]] with the help of stratifolds. Let ''G'' be a compact [[Lie group]]. We can then define a bordism theory of stratifolds mapping into a space ''X'' with a ''G''-action just as above, only that we require all stratifolds to be equipped with an orientation-preserving free ''G''-action and all maps to be G-equivariant. Denote by <math>SH_k^G(X)</math> the bordism classes. One can prove <math>SH_k^G(X)\cong H_{k-\dim(G)}^G(X)</math> for every X homotopy equivalent to a CW-complex.
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| ==Connection to the theory of genera==
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| A [[Genus of a multiplicative sequence|genus]] is a ring homomorphism from a bordism ring into another ring. For example the [[Euler characteristic]] defines a ring homomorphism <math>\Omega^O(\text{point})\to \mathbb{Z}/2[t]</math> from the [[Bordism#Unoriented_cobordism|unoriented bordism ring]] and the [[Signature (topology)|signature]] defines a ring homomorphism <math>\Omega^{SO}(\text{point})\to \mathbb{Z}[t]</math> from the [[Bordism#Oriented_cobordism|oriented bordism ring]]. Here ''t'' has in the first case degree ''1'' and in the second case degree ''4'', since only manifolds in dimensions divisible by ''4'' can have non-zero signature. The left hand sides of these homomorphisms are homology theories evaluated at a point. With the help of stratifolds it is possible to construct homology theories such that the right hand sides are these homology theories evaluated at a point, the Euler homology and the Hirzebruch homology respectively.
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| ==Umkehr maps==
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| Suppose, one has a closed embedding <math>i: N\hookrightarrow M</math> of manifolds with oriented normal bundle. Then one can define an [[umkehr map]] <math>H_k(M)\to H_{k+\dim(N)-\dim(M)}(N)</math>. One possibility is to use stratifolds: represent a class <math>x\in H_k(M)</math> by a stratifold <math>f:S\to M</math>. Then make ''ƒ'' transversal to ''N''. The intersection of ''S'' and ''N'' defines a new stratifold ''S''<nowiki>'</nowiki> with a map to ''N'', which represents a class in <math>H_{k+\dim(N)-\dim(M)}(N)</math>. It is possible to repeat this construction in the context of an embedding of [[Hilbert manifold]]s of finite codimension, which can be used in [[string topology]].
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| ==References==
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| *M. Kreck, ''Differential Algebraic Topology: From Stratifolds to Exotic Spheres'', AMS (2010), ISBN 0-8218-4898-4
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| *[http://www.hausdorff-research-institute.uni-bonn.de/kreck-stratifolds The stratifold page]
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| *[http://arxiv.org/abs/math/0606558 Euler homology]
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| [[Category:Generalized manifolds]]
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| [[Category:Homology theory]]
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Looking for some simple solutions that will help you have your pole in the water tomorrow? Rest easy, as this activity is fun and simple. Because fishing is a diverse hobby, it accommodates those of varied levels of knowledge and skill. Here are some tips to help make fishing a fun and successful activity, whatever your skill level.
People that wish to fish in a stream need to cast upstream and allow the current to sweep the lure into the fishing hole. This makes your bait more natural, which increases chances of catching a fish. Try to always keep your line taught, and reduce the slack in your line so you can feel the fish bite.
Stream fisherman would be wise to start their fishing trips downstream, and make their way upstream as the day progresses. This is simply because fish tend to face against the direction of the current. By moving upstream you are lowering your chances of being seen or heard by your prey.
Do not ever forget to wear a life jacket when you are fishing. Make sure that it is securely fastened. You also want to make sure that the one you have is the right size. This is very important,because if it is too large it can slip over your head if you fall into the water.
When fishing alone, be careful not to go into water that is too deep. This is especially true when fishing near large rivers as a sudden undercurrent could pull you under and drown you in even fairly shallow water. It is typically better to fish with at least one other friend.
If you are going to be using live bait to go fishing, be sure your bait is not kept in the sun for too long. Most fish prefer bait that is cooler, rather than warmer. Have the bait kept in an insulated container until you are ready to use it.
A great tip for all fisherman is to stay relatively quite while you are on the water. Everyone has heard that loud voices can scare away fish, and while this is slightly exaggerated, making a lot of noise on the surface, particularly disturbing the surface, can indeed spook fish.
The introduction stressed the fact that not knowing how to fish isn't a bad thing. Picking up fishing is not difficult at all. Even though there are many levels to it, there are many entry points for the hobby, and if you use the tips from this article, you can start fishing now.
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