Rabi cycle

In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926.

The counting is subject to an imputed multiplicity at a fixed point called the fixed point index. A weak version of the theorem is enough to show that a mapping without any fixed point must have rather special topological properties (like a rotation of a circle).

Formal statement

For a formal statement of the theorem, let

$f:X\to X$

be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number Λ_f of f by

${\mathrm{\Lambda }}_f:=\sum _{k\ge 0}(-1{)}^k\mathrm{T}\mathrm{r}(f_{*}|H_k(X,\mathbb{\mathbb{Q}})),$

the alternating (finite) sum of the matrix traces of the linear maps induced by f on the H_k(X,Q), the singular homology of X with rational coefficients.

A simple version of the Lefschetz fixed-point theorem states: if

${\mathrm{\Lambda }}_f\ne 0$

then f has at least one fixed point, i.e. there exists at least one x in X such that f(x) = x. In fact, since the Lefschetz number has been defined at the homology level, the conclusion can be extended to say that any map homotopic to f has a fixed point as well.

Note however that the converse is not true in general: Λ_f may be zero even if f has fixed points.

Sketch of a proof

First, by applying the simplicial approximation theorem, one shows that if f has no fixed points, then (possibly after subdividing X) f is homotopic to a fixed-point-free simplicial map (i.e., it sends each simplex to a different simplex). This means that the diagonal values of the matrices of the linear maps induced on the simplicial chain complex of X must be all be zero. Then one notes that, in general, the Lefschetz number can also be computed using the alternating sum of the matrix traces of the aforementioned linear maps (this is true for almost exactly the same reason that the Euler characteristic has a definition in terms of homology groups; see below for the relation to the Euler characteristic). In the particular case of a fixed-point-free simplicial map, all of the diagonal values are zero, and thus the traces are all zero.

Lefschetz-Hopf theorem

A stronger form of the theorem, also known as the Lefschetz-Hopf theorem, states that, if f has only finitely many fixed points, then

$\sum _{x\in \mathrm{F}\mathrm{i}\mathrm{x}(f)}i(f,x)={\mathrm{\Lambda }}_f,$

where Fix(f) is the set of fixed points of f, and i(f,x) denotes the index of the fixed point x.

Relation to the Euler characteristic

The Lefschetz number of the identity map on a finite CW complex can be easily computed by realizing that each $f_{\ast }$ can be thought of as an identity matrix, and so each trace term is simply the dimension of the appropriate homology group. Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic χ(X). Thus we have

${\mathrm{\Lambda }}_{\mathrm{i}\mathrm{d}}=\chi (X).$

Relation to the Brouwer fixed point theorem

The Lefschetz fixed point theorem generalizes the Brouwer fixed point theorem, which states that every continuous map from the n-dimensional closed unit disk Dⁿ to Dⁿ must have at least one fixed point.

This can be seen as follows: Dⁿ is compact and triangulable, all its homology groups except H₀ are 0, and every continuous map f : Dⁿ → Dⁿ induces a non-zero homomorphism f_* : H₀(Dⁿ, Q) → H₀(Dⁿ, Q); all this together implies that Λ_f is non-zero for any continuous map f : Dⁿ → Dⁿ.

Historical context

Lefschetz presented his fixed point theorem in [Lefschetz 1926]. Lefschetz's focus was not on fixed points of mappings, but rather on what are now called coincidence points of mappings.

Given two maps f and g from an orientable manifold X to an orientable manifold Y of the same dimension, the Lefschetz coincidence number of f and g is defined as

${\mathrm{\Lambda }}_{f,g}=\sum (-1{)}^k\mathrm{T}\mathrm{r}(D_X\circ g^{*}\circ D_Y^{-1}\circ f_{*}),$

where f_∗ is as above, g^∗ is the mapping induced by g on the cohomology groups with rational coefficients, and D_X and D_Y are the Poincaré duality isomorphisms for X and Y, respectively.

Lefschetz proves that if the coincidence number is nonzero, then f and g have a coincidence point. He notes in his paper that letting X = Y and letting g be the identity map gives a simpler result, which we now know as the fixed point theorem.

Frobenius

Let $X$ be a variety defined over the finite field $k$ with $q$ elements and let $\overline{X}$ be the lift of $X$ to the algebraic closure of $k$. The Frobenius endomorphism (often just the Frobenius), notation $F_q$, of $\overline{X}$ maps a point with coordinates $x_1,\dots ,x_n$ to the point with coordinates $x_1^q,\dots ,x_n^q$ (i.e. $F_q$ is the geometric Frobenius). Thus the fixed points of $F_q$ are exactly the points of $X$ with coordinates in $k$, notation for the set of these points: $X(k)$. The Lefschetz trace formula holds in this context and reads:

$\#X(k)=\sum _i(-1{)}^i\mathrm{t}\mathrm{r}{F}_q|H_c^i(\overline{X},{\mathbb{\mathbb{Q}}}_{\ell }).$

This formula involves the trace of the Frobenius on the étale cohomology, with compact supports, of $\overline{X}$ with values in the field of $\ell$-adic numbers, where $\ell$ is a prime coprime to $q$.

If $X$ is smooth and equidimensional, this formula can be rewritten in terms of the arithmetic Frobenius ${\mathrm{\Phi }}_q$, which acts as the inverse of $F_q$ on cohomology:

$\#X(k)=q^{\dim X}\sum _i(-1{)}^i\mathrm{t}\mathrm{r}{\mathrm{\Phi }}_q^{-1}|H^i(\overline{X},{\mathbb{\mathbb{Q}}}_{\ell }).$

This formula involves usual cohomology, rather than cohomology with compact supports.

The Lefschetz trace formula can also be generalized to algebraic stacks over finite fields.

See also

• Fixed point theorems
• Lefschetz zeta function
• Holomorphic Lefschetz fixed-point formula

References

• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

• Other Sports Official Kull from Drumheller, has hobbies such as telescopes, property developers in singapore and crocheting. Identified some interesting places having spent 4 months at Saloum Delta.
  
  my web-site http://himerka.com/