The fact that this is an isomorphism of chain complexes is a proof of Poincaré Duality.

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

This formulation of Poincaré Duality has become quite popular as it provides a means to define Poincaré Duality for any generalized homology theories provided one has a Thom Isomorphism for that homology theory.

: From the point of view of the definition, this is an expression of the Poincaré Duality isomorphism where is the quotient of the field of fractions of by, considered as a-module, and where is the conjugate-module to ie: as an abelian group it is identical to but the covering transformation acts by.

This behaviour under CPT is the same as the statement that the particle and its antiparticle lie in the same irreducible representation of the Poincaré group.

The intersection of the periodic orbit with the Poincaré section is a fixed point of the Poincaré map F. By a translation, the point can be assumed to be at x = 0.

Small changes in the vector field will only produce small changes in the Poincaré map and these small changes will reflect in small changes in the position of the eigenvalues of J in the complex plane, implying that the map is still hyperbolic.

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.

After important contributions of Hendrik Lorentz and Henri Poincaré, in 1905, Albert Einstein solved the problem with the introduction of special relativity, which replaces classical kinematics with a new theory of kinematics that is compatible with classical electromagnetism.

One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand drawn images of a similar function, which is now called the Koch curve ( see Figure 2 ).

In mathematics, more specifically algebraic topology, the fundamental group ( defined by Henri Poincaré in his article Analysis Situs, published in 1895 ) is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other.

( The defining symmetry of special relativity is the Poincaré group which also includes translations and rotations.

As Henri Poincaré famously said, mathematics is not the study of objects, but instead, the relations ( isomorphisms for instance ) between them.

Like all subatomic particles, hadrons are assigned quantum numbers corresponding to the representations of the Poincaré group: J < sup > PC </ sup >( m ), where J is the spin quantum number, P the intrinsic parity ( or P-parity ), and C, the charge conjugation ( or C-parity ), and the particle's mass, m. Note that the mass of a hadron has very little to do with the mass of its valence quarks ; rather, due to mass – energy equivalence, most of the mass comes from the large amount of energy associated with the strong interaction.

* 1913 – Raymond Poincaré is elected President of France.

* The Poincaré group is a 10 dimensional Lie group of affine isometries of the Minkowski space.

The more general set of transformations that also includes translations is known as the Poincaré group.

Henri Poincaré is regarded as the last mathematician to excel in every field of the mathematics of his time.

From a fundamental physics perspective, mass is the number describing under which the representation of the little group of the Poincaré group a particle transforms.

In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.

The Poincaré conjecture asserts that the same is true for 3-dimensional spaces.

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 ).

The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere.

The Poincaré conjecture is the only solved Millennium problem.

Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, but nonetheless, the statement that it does is known as the Poincaré conjecture.

This space is closely related to the Poincaré sphere, which is the spherical surface occupied by completely polarized states in the space of the vector

The stability of a periodic orbit of the original system is closely related to the stability of the fixed point of the corresponding Poincaré map.

A more rigorous approach was adopted by Henri Poincaré in his 1895 set of papers Analysis situs where the related concepts of homology and the fundamental group were also introduced.

More specifically, there is a general Poincaré duality theorem for generalized homology theories which requires a notion of orientation with respect to a homology theory, and is formulated in terms of a generalized Thom Isomorphism Theorem.

The Thom Isomorphism Theorem in this regard can be considered as the germinal idea for Poincaré duality for generalized homology theories.

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.

In a September 1904 lecture in St. Louis named The Principles of Mathematical Physics, Poincaré draw some consequences from Lorentz's theory and defined ( in modification of Galileo's Relativity Principle and Lorentz's Theorem of Corresponding States ) the following principle: " The Principle of Relativity, according to which the laws of physical phenomena must be the same for a stationary observer as for one carried along in a uniform motion of translation, so that we have no means, and can have none, of determining whether or not we are being carried along in such a motion.

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.

Concerning the observables, and states | v ), we get a representation U ( a, L ) of Poincaré group, on integer spin subspaces, and U ( a, A ) of the inhomogeneous SL ( 2, C ) on half-odd-integer subspaces, which acts according to the following interpretation:

It also maps the interior of the unit sphere to itself, with points outside the orthogonal sphere mapping inside, and vice-versa ; this defines the reflections of the Poincaré disc model if we also include with them the reflections through the diameters separating hemispheres of the unit sphere.

We assume there is a pure state called the vacuum such that the Hilbert space associated with it is a unitary representation of the Poincaré group compatible with the Poincaré covariance of the net such that if we look at the Poincaré algebra, the spectrum with respect to energy-momentum ( corresponding to spacetime translations ) lies on and in the positive light cone.

The Poincaré duality theorem gives a natural isomorphism, which we can use to transfer the ring structure from cohomology to homology.

Recall that any vector space V over a field has a basis ; this is a set S such that any element of V is a unique ( finite ) linear combination of elements of S. In the formulation of Poincaré – Birkhoff – Witt theorem we consider bases of which the elements are totally ordered by some relation which we denote ≤.

With the benefit of hindsight, we can see that Couturat was in broad agreement with the logicism of Russell and Whitehead, while Poincaré anticipated Brouwer's intuitionism.

If we assume also that M is compact, Poincaré duality identifies this with

From this Poincaré argues that if we fail to establish the consistency of Peano's axioms for natural numbers without falling into circularity, then the principle of complete induction is not provable by general logic.

He states that his 6 or so ( famous ) examples of paradoxes ( antinomies ) are all examples of impredicative definition, and says that Poincaré ( 1905 – 6, 1908 ) and Russel ( 1906, 1910 ) " enunciated the cause of the paradoxes to lie in these impredicative definitions " ( p. 42 ), however, " parts of mathematics we want to retain, particularly analysis, also contain impredicative definitions.

Poincaré went on to note that Rømer also had to assume that Jupiter's moons obey Newton's laws, including the law of gravitation, whereas it would be possible to reconcile a different speed of light with the same observations if we assumed some different ( probably more complicated ) laws of motion.

According to Poincaré, this illustrates that we adopt for the speed of light a value that makes the laws of mechanics as simple as possible.

In his Palermo paper ( 1906 ), Poincaré called this " the postulate of relativity “, and although he stated that it was possible this principle might be disproved at some point ( and in fact he mentioned at the paper's end that the discovery of magneto-cathode rays by Paul Ulrich Villard ( 1904 ) seems to threaten it ), he believed it was interesting to consider the consequences if we were to assume the postulate of relativity was valid without restriction.

where we have now made used of translational covariance, which is part of the Poincaré covariance.

If let's say gravity is an emergent theory of a fundamentally flat theory over a flat Minkowski spacetime, then by Noether's theorem, we have a conserved stress-energy tensor which is Poincaré covariant.