Euler product

In mathematics, the theory of fiber bundles with a structure group $G$ (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from $F_1$ to $F_2$, which are both topological spaces with a group action of $G$. For a fibre bundle F with structure group G, the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems U_α and U_β are given as a G-valued function g_αβ on U_α∩U_β. One may then construct a fibre bundle F′ as a new fibre bundle having the same transition functions, but possibly a different fibre.

An example

A simple case comes with the Möbius strip, for which $G$ is the cyclic group of order 2, ${\mathbb{\mathbb{Z}}}_2$. We can take as $F$ any of: the real number line $\mathbb{ℝ}$, the interval $[-1,1]$, the real number line less the point 0, or the two-point set $\{-1,1\}$. The action of $G$ on these (the non-identity element acting as $x\to -x$ in each case) is comparable, in an intuitive sense. We could say that more formally in terms of gluing two rectangles $[-1,1]\times I$ and $[-1,1]\times J$ together: what we really need is the data to identify $[-1,1]$ to itself directly at one end, and with the twist over at the other end. This data can be written down as a patching function, with values in G. The associated bundle construction is just the observation that this data does just as well for $\{-1,1\}$ as for $[-1,1]$.

Construction

In general it is enough to explain the transition from a bundle with fiber $F$, on which $G$ acts, to the associated principal bundle (namely the bundle where the fiber is $G$, considered to act by translation on itself). For then we can go from $F_1$ to $F_2$, via the principal bundle. Details in terms of data for an open covering are given as a case of descent.

This section is organized as follows. We first introduce the general procedure for producing an associated bundle, with specified fibre, from a given fibre bundle. This then specializes to the case when the specified fibre is a principal homogeneous space for the left action of the group on itself, yielding the associated principal bundle. If, in addition, a right action is given on the fibre of the principal bundle, we describe how to construct any associated bundle by means of a fibre product construction.

Associated bundles in general

Let π : E → X be a fibre bundle over a topological space X with structure group G and typical fibre F. By definition, there is a left action of G (as a transformation group) on the fibre F. Suppose furthermore that this action is effective. There is a local trivialization of the bundle E consisting of an open cover U_i of X, and a collection of fibre maps

φ_i : π^-1(U_i) → U_i × F

such that the transition maps are given by elements of G. More precisely, there are continuous functions g_ij : (U_i ∩ U_j) → G such that

ψ_ij(u,f) := φ_i o φ_j^-1(u,f) = (u,g_ij(u)f) for each (u,f) ∈ (U_i ∩ U_j) × F.

Now let F′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated to E with fibre F′ is a bundle E′ with a local trivialization subordinate to the cover U_i whose transition functions are given by

ψ′_ij(u,f′) = (u, g_ij(u) f′) for (u,f′) ∈(U_i ∩ U_j) × F′

where the G-valued functions g_ij(u) are the same as those obtained from the local trivialization of the original bundle E.

This definition clearly respects the cocycle condition on the transition functions, since in each case they are given by the same system of G-valued functions. (Using another local trivialization, and passing to a common refinement if necessary, the g_ij transform via the same coboundary.) Hence, by the fiber bundle construction theorem, this produces a fibre bundle E′ with fibre F′ as claimed.

Principal bundle associated to a fibre bundle

As before, suppose that E is a fibre bundle with structure group G. In the special case when G has a free and transitive left action on F′, so that F′ is a principal homogeneous space for the left action of G on itself, then the associated bundle E′ is called the principal G-bundle associated to the fibre bundle E. If, moreover, the new fibre F′ is identified with G (so that F′ inherits a right action of G as well as a left action), then the right action of G on F′ induces a right action of G on E′. With this choice of identification, E′ becomes a principal bundle in the usual sense. Note that, although there is no canonical way to specify a right action on a principal homogeneous space for G, any two such actions will yield principal bundles which have the same underlying fibre bundle with structure group G (since this comes from the left action of G), and isomorphic as G-spaces in the sense that there is a globally defined G-valued function relating the two.

In this way, a principal G-bundle equipped with a right action is often thought of as part of the data specifying a fibre bundle with structure group G, since to a fibre bundle one may construct the principal bundle via the associated bundle construction. One may then, as in the next section, go the other way around and derive any fibre bundle by using a fibre product.

Fiber bundle associated to a principal bundle

Let π : P → X be a principal G-bundle and let ρ : G → Homeo(F) be a continuous left action of G on a space F (in the smooth category, we should have a smooth action on a smooth manifold). Without loss of generality, we can take this action to be effective.

Define a right action of G on P × F via

$(p,f)\cdot g=(p\cdot g,\rho (g^{-1})f).$

We then identify by this action to obtain the space E = P ×_ρ F = (P × F) /G. Denote the equivalence class of (p,f) by [p,f]. Note that

$[p\cdot g,f]=[p,\rho (g)f]\text{ for all }g\in G.$

Define a projection map π_ρ : E → X by π_ρ([p,f]) = π(p). Note that this is well-defined.

Then π_ρ : E → X is a fiber bundle with fiber F and structure group G. The transition functions are given by ρ(t_ij) where t_ij are the transition functions of the principal bundle P.

Reduction of the structure group

DTZ gives a comprehensive integrated property and services administration resolution for buyers, corporate house for sale In singapore owners, management firms and occupiers of property whatever their needs with the only goal of optimising and enhancing the investment worth of their property. We at the moment make use of a staff of more than 70 skilled staffs who are well-trained and dedicated to collectively achieving our purchasers' objectives.

Actual estate agency specialising in non-public condos and landed properties island vast. 10 Winstedt Highway, District 10, #01-thirteen, Singapore 227977. Property providers for enterprise relocation. Situated at 371 Beach Street, #19-10 KeyPoint, Singapore 199597. Property agents for homes, town houses, landed property, residences and condominium for sales and rentals of properties. Administration letting services for property homeowners. is there a single authority in singapore who regulates real property agents that i can file a complaint with for unethical behaviour? or is CASE is simply route? The 188 pages of Secrets and techniques of Singapore Property Gurus are full of professional knowledge and life altering wisdom. Asian industrial property market outlook Property Listing Supervisor Property Advertising Services

Should sellers go along with an agent who claims to specialize in your space? His experience might turn out to be useful, but he is probably additionally advertising a number of models within the neighbourhood – and so they're all your rivals. Within the worst-case state of affairs, your house may be used as a "showflat" as house owner YS Liang found. "Weekend after weekend, our agent would convey a stream of individuals to speed-go to our apartment, leaving within minutes. She did not even try to promote our condominium. It felt like we were just one of the many tour stops for her clients," he complains.

Step one in direction of conducting enterprise as an actual property company in Singapore is to include an organization, or if you happen to're going the partnership or sole-proprietorship route, register your Limited Legal responsibility Partnership or sole-proprietorship with the ACRA (Accounting and Company Regulatory Authority of Singapore) Whether or not you might be considering to promote, let, hire or buy a new industrial property, we're right here to assist. Search and browse our commercial property section. Possess not less than 3 years of working expertise below a Singapore licensed real-property agency; Sale, letting and property administration and taxation companies. three Shenton Means, #10-08 Shenton Home, Singapore 068805. Real property agents for purchasing, promoting, leasing, and renting property. Caveat Search

Firstly, the events might take into account to rescind the sale and buy agreement altogether. This avenue places the contracting events to a position as if the contract didn't happen. It's as if the contract was terminated from the start and events are put back into place that they were before the contract. Any items or monies handed are returned to the respective original house owners. As the worldwide real property market turns into extra refined and worldwide real property investments will increase, the ERA real estate network is well equipped to offer professional recommendation and guidance to our shoppers in making critical actual estate decisions. Relocationg, leasing and sales of properties for housing, food and beverage, retail and workplace wants.

Pasir Panjang, Singapore - $5,000-6,000 per 30 days By likelihood one among our buddies here in Singapore is an agent and we made contact for her to help us locate an residence, which she did. days from the date of execution if the doc is signed in Singapore; Be a Singapore Citizen or PR (Permanent Resident); The regulations also prohibit property agents from referring their shoppers to moneylenders, to discourage irresponsible shopping for. Brokers are additionally prohibited from holding or dealing with money on behalf of any party in relation to the sale or purchase of any property situated in Singapore, and the lease of HDB property. - Negotiate To Close A Sale together with sale and lease of HDB and private properties) Preparing your house for sale FEATURED COMMERCIAL AGENTS Property Guides

i) registered as a patent agent or its equal in any nation or territory, or by a patent workplace, specified within the Fourth Schedule; The business-specific tips for the true property agency and telecommunication sectors have been crafted to address considerations about scenarios that particularly apply to the two sectors, the PDPC stated. Mr Steven Tan, Managing Director of OrangeTee real property company, nonetheless, felt that it was a matter of "practising until it becomes part of our knowledge". "After a while, the agents ought to know the spirit behind the (Act)," he stated. Rising office sector leads real property market efficiency, while prime retail and enterprise park segments moderate and residential sector continues in decline Please choose an attendee for donation.

The companion concept to associated bundles is the reduction of the structure group of a $G$-bundle $B$. We ask whether there is an $H$-bundle $C$, such that the associated $G$-bundle is $B$, up to isomorphism. More concretely, this asks whether the transition data for $B$ can consistently be written with values in $H$. In other words, we ask to identify the image of the associated bundle mapping (which is actually a functor).

Examples of reduction

Examples for vector bundles include: the introduction of a metric resulting in reduction of the structure group from a general linear group GL(n) to an orthogonal group O(n); and the existence of complex structure on a real bundle resulting in reduction of the structure group from real general linear group GL(2n,R) to complex general linear group GL(n,C).

Another important case is finding a decomposition of a vector bundle V of rank n as a Whitney sum (direct sum) of sub-bundles of rank k and n-k, resulting in reduction of the structure group from GL(n,R) to GL(k,R) × GL(n-k,R).

One can also express the condition for a foliation to be defined as a reduction of the tangent bundle to a block matrix subgroup - but here the reduction is only a necessary condition, there being an integrability condition so that the Frobenius theorem applies.

See also

• Spinor bundle

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

Books

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534