* Poincaré duality theorem

The concept of dual cell structure, which Henri Poincaré used in his proof of his Poincaré duality theorem, contained the germ of the idea of cohomology, but this was not seen until later.

This ( in rather special cases ) provided an interpretation of Poincaré duality and Alexander duality in terms of group characters.

From 1936 to 1938, Hassler Whitney and Eduard Čech developed the cup product ( making cohomology into a graded ring ) and cap product, and realized that Poincaré duality can be stated in terms of the cap product.

By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call, and the cap product can be computed by taking the cup product of and α and capping with the fundamental class of X.

#( Functional equation and Poincaré duality ) The zeta function satisfies < dl >< dd ></ dl > or equivalently < dl >< dd ></ dl > where E is the Euler characteristic of X.

The functional equation for the zeta function follows from Poincaré duality for l-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between l-adic and ordinary cohomology for complex varieties.

Poincaré duality relates H ( E < sup > k </ sup >) to H ( E < sup > k </ sup >), which is in turn the space of covariants of the monodromy group, which is the geometric fundamental group of U acting on the fiber of E < sup > k </ sup > at a point.

: Poincaré duality then implies that

: Poincaré duality then implies that

The Weil conjectures for the cohomology below the middle dimension follow from this by applying the weak Lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from Poincaré duality.

This gives an upper bound on the absolute values of the eigenvalues of Frobenius, and Poincaré duality then shows that this is also a lower bound.

However it does when a suitable form of Poincaré duality holds, for example if f is smooth and proper, or if one works with perverse sheaves rather than sheaves as in.

under conditions ( a closed and oriented manifold ); see Poincaré duality.

The connection with the definition given above is via three basic results, de Rham's theorem and Poincaré duality ( when those apply ), and the universal coefficient theorem of homology theory.

If the variety is defined over the complex numbers, this yields different information from Poincaré duality, which relates H < sup > i </ sup > to H < sup > 2n − i </ sup >, considering V as a real manifold of dimension 2n.

The burden and success of the general theory was certainly both to integrate all this information, and to prove general results such as Poincaré duality and the Lefschetz fixed point theorem in this context.

They satisfy a form of Poincaré duality on non-singular projective varieties, and the ℓ-adic cohomology groups of a " reduction mod p " of a complex variety tend to have the same rank as the singular cohomology groups.

This is the analogue of Poincaré duality for étale cohomology.

* Poincaré duality

In mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds.

* The Poincaré theorem ( proven by Grigori Perelman )

For systems where the volume is preserved by the flow, Poincaré discovered the recurrence theorem: Assume the phase space has a finite Liouville volume and let F be a phase space volume-preserving map and A a subset of the phase space.

The Poincaré recurrence theorem was used by Zermelo to object to Boltzmann's derivation of the increase in entropy in a dynamical system of colliding atoms.

In mathematics, the Poincaré conjecture ( ; ) is a theorem about the characterization of the three-dimensional sphere ( 3-sphere ), which is the hypersphere that bounds the unit ball in four-dimensional space.

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary ( a closed 3-manifold ).

In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture for dimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem.

Likewise, Banach's fixed point theorem, based on earlier methods developed by Charles Émile Picard, was included in his dissertation, and was later extended by his students ( for example in the Banach – Schauder theorem ) and other mathematicians ( in particular Bouwer and Poincaré and Birkhoff ).

The theorem was later utilized, popularized, generalized and further developed by James Clerk Maxwell, Lord Rayleigh, Henri Poincaré, Subrahmanyan Chandrasekhar, Enrico Fermi, Paul Ledoux and Eugene Parker.

He then cemented his reputation with a proof of the Poincaré conjecture for all dimensions greater than or equal to 5, published in 1961 ; in 1962 he generalized the ideas in a 107 page paper that established the h-cobordism theorem.

* Poincaré recurrence theorem, Henri Poincaré's theorem on dynamical systems

These symmetries are grouped into the Poincaré group and internal symmetries and the Coleman – Mandula theorem showed that under certain assumptions, the symmetries of the S-matrix must be a direct product of the Poincaré group with a compact internal symmetry group or if there is no mass gap, the conformal group with a compact internal symmetry group.

Some well-known difficult abstract problems that have been solved relatively recently are the four-colour theorem, Fermat's Last Theorem, and the Poincaré conjecture.

* Poincaré recurrence theorem

The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventually revisit the set.

It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the Differentiable sphere theorem by Brendle and Schoen.

" However, there are a number of paradoxes regarding breakage of the Second Law of Thermodynamics, one of them due to the Poincaré recurrence theorem.

Hence the theorem about Poincaré invariant is a generalization of the Liouville's theorem.

1, 57-78 ( expository article explains the eight geometries and geometrization conjecture briefly, and gives an outline of Perelman's proof of the Poincaré conjecture )

The fundamental Poincaré – Birkhoff – Witt theorem gives a precise description of U ( L ); the most important consequence is that L can be viewed as a linear subspace of U ( L ).

A dual version of this gives the Poincaré lemma.

This result is an application of Poincaré Duality together with the Universal coefficient theorem which gives an identification and.

Rotation of 1 / 10 gives the Poincaré homology sphere, and rotation by 5 / 10 gives 3-dimensional real projective space.

Blanchfield's formulation of Poincaré duality for gives a canonical isomorphism where denotes the 2nd cohomology group of with compact supports and coefficients in.

These oscillations and the associated mixing of particles gives insight into the realization of discrete parts of the Poincaré group, i. e., parity ( P ), charge conjugation ( C ) and time reversal invariance ( T ).

Noether's theorem gives a current which is conserved and Poincaré covariant, but not gauge invariant.