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{{DISPLAYTITLE:Semi-''s''-cobordism}}
In [[mathematics]], a [[cobordism]] (''W'', ''M'', ''M''<sup>&minus;</sup>) of an (''n''&nbsp;+&nbsp;1)-dimensionsal [[manifold]] (with boundary) ''W'' between its boundary components, two ''n''-manifolds ''M'' and ''M''<sup>&minus;</sup> (n.b.: the original creator of this topic, Jean-Claude Hausmann, used the notation ''M''<sub>&minus;</sub> for the right-hand boundary of the cobordism), is called a '''semi-''s''-cobordism''' if (and only if) the inclusion <math>M \hookrightarrow W</math> is a [[Homotopy equivalence|simple homotopy equivalence]] (as in an [[s-cobordism|''s''-cobordism]]) but the inclusion <math>M^- \hookrightarrow W</math> is not a homotopy equivalence at all.
 
A consequence of (''W'', ''M'', ''M''<sup>&minus;</sup>) being a semi-''s''-cobordism is that the [[Kernel (algebra)|kernel]] of the derived [[homomorphism]] on [[fundamental group]]s <math>K = \ker(\pi_1(M^{-}) \twoheadrightarrow \pi_1(W))</math> is [[Perfect group|perfect]]. A corollary of this is that <math>\pi_1(M^{-})</math> solves the [[Group extension|group extension problem]] <math>1 \rightarrow K \rightarrow \pi_1(M^{-}) \rightarrow \pi_1(M) \rightarrow 1</math>. The solutions to the group extension problem for proscribed [[quotient group]] <math>\pi_1(M)</math> and kernel group K are classified up to congruence (see ''Homology'' by MacLane, e.g.), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-''s''-cobordism with proscribed left-hand boundary M and superperfect kernel group K.
 
Note that if (''W'', ''M'', ''M''<sup>&minus;</sup>) is a semi-''s''-cobordism, then (''W'',&nbsp;''M''<sup>&minus;</sup>,&nbsp;''M'') is a [[Plus construction|Plus cobordism]]. (This justifies the use of ''M''<sup>&minus;</sup> for the right-hand boundary of a semi-''s''-cobordism, a play on the traditional use of ''M''<sup>+</sup> for the right-hand boundary of a Plus cobordism.) Thus, a semi-''s''-cobordism may be thought of as an inverse to Quillen's Plus construction in the manifold category. Note that (''M''<sup>&minus;</sup>)<sup>+</sup> must be [[diffeomorphism|diffeomorphic]] (respectively, [[Piecewise linear homeomorphism|piecewise-linearly (PL) homeomorphic]]) to ''M'' but there may be a variety of choices for (''M''<sup>+</sup>)<sup>&minus;</sup> for a given closed [[Differentiable manifold|smooth]] (respectively, [[Piecewise linear manifold|PL]]) manifold ''M''.
 
==References==
*{{citation | last=MacLane | year=1963 | title=Homology | pages=124–129 | isbn=0-387-58662-8}}
*{{citation| first=Jean-Claude | last=Hausmann | title=Homological Surgery | journal=[[The Annals of Mathematics]], 2nd Ser. | volume=104 | issue=3 | year=1976 | pages=573–584|doi=10.2307/1970967 | jstor=1970967}}.
*{{citation|url=http://www.springerlink.com/content/d0740ht7537775h5/ | first=Jean-Claude | last=Hausmann | first2=Pierre | last2=Vogel | title=The Plus Construction and Lifting Maps from Manifolds | journal=[[Proceedings of Symposia in Pure Mathematics]] | volume=32 | year=1978 | pages=67–76}}.
*{{citation|url=http://www.ams.org/bookstore-getitem/item=PSPUM-32 | first=Jean-Claude | last=Hausmann | title=Manifolds with a Given Homology and Fundamental Group | journal=[[Commentarii Mathematici Helvetici]] | volume=53 | issue=1 | year=1978 | pages=113–134|doi=10.1007/BF02566068}}.
 
[[Category:Manifolds]]
[[Category:Geometric topology]]
[[Category:Algebraic topology]]
[[Category:Homotopy theory]]
 
 
{{topology-stub}}

Latest revision as of 19:58, 22 August 2013

In mathematics, a cobordism (W, M, M) of an (n + 1)-dimensionsal manifold (with boundary) W between its boundary components, two n-manifolds M and M (n.b.: the original creator of this topic, Jean-Claude Hausmann, used the notation M for the right-hand boundary of the cobordism), is called a semi-s-cobordism if (and only if) the inclusion MW is a simple homotopy equivalence (as in an s-cobordism) but the inclusion MW is not a homotopy equivalence at all.

A consequence of (W, M, M) being a semi-s-cobordism is that the kernel of the derived homomorphism on fundamental groups K=ker(π1(M)π1(W)) is perfect. A corollary of this is that π1(M) solves the group extension problem 1Kπ1(M)π1(M)1. The solutions to the group extension problem for proscribed quotient group π1(M) and kernel group K are classified up to congruence (see Homology by MacLane, e.g.), so there are restrictions on which n-manifolds can be the right-hand boundary of a semi-s-cobordism with proscribed left-hand boundary M and superperfect kernel group K.

Note that if (W, M, M) is a semi-s-cobordism, then (WMM) is a Plus cobordism. (This justifies the use of M for the right-hand boundary of a semi-s-cobordism, a play on the traditional use of M+ for the right-hand boundary of a Plus cobordism.) Thus, a semi-s-cobordism may be thought of as an inverse to Quillen's Plus construction in the manifold category. Note that (M)+ must be diffeomorphic (respectively, piecewise-linearly (PL) homeomorphic) to M but there may be a variety of choices for (M+) for a given closed smooth (respectively, PL) manifold M.

References

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

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    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

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  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    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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Template:Topology-stub