And in 1902, Henri Poincaré published the philosophical and popular-science book " Science and Hypothesis ", which included: philosophical assessments on the relativity of space, time, and simultaneity ; the opinion that a violation of the Relativity Principle can never be detected ; the possible non-existence of the aether but also some arguments supporting the aether ; many remarks on non-Euclidean geometry.

However, Poincaré himself never abandoned the ether hypothesis and stated in 1900: " Does our ether actually exist?

Henri Poincaré declared in 1905 that the impossibility of demonstrating absolute motion ( principle of relativity ) is apparently a law of nature.

In fact, whether one can smooth certain higher dimensional spheres was, until recently, an open problem — known as the smooth Poincaré conjecture.

In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivial fundamental group had to be a 3-sphere.

For n = 3, the h-cobordism theorem for smooth manifolds has not been proved and, due to the Poincaré conjecture, is equivalent to the hard open question of whether the 4-sphere has non-standard smooth structures.

It is determined by whether the particle transforms in a right or left-handed representation of the Poincaré group.

The Poincaré conjecture for PL manifolds has been proved for all dimensions other than 4, but it is not known whether it is true in 4 dimensions ( it is equivalent to the smooth Poincaré conjecture in 4 dimensions ).

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.

The two basic questions of surgery theory are whether a topological space with n-dimensional Poincaré duality is homotopy equivalent to an n-dimensional manifold, and also whether a homotopy equivalence of n-dimensional manifolds is homotopic to a diffeomorphism.

It is not known ( as of 2009 ) whether or not there are any exotic 4-spheres ; such an exotic 4-sphere would be a counterexample to the smooth generalized Poincaré conjecture in dimension 4.

In 1908 he believed that he had found a proof of the Poincaré conjecture, but Tietze found an error.

Like Poincaré, Alfred Bucherer ( 1903 ) believed in the validity of the relativity principle within the domain of electrodynamics, but contrary to Poincaré, Bucherer even assumed that this implies the nonexistence of the aether.

Only a few theoretical physicists like Lorentz, Poincaré, Abraham or Langevin, still believed in the existence of an aether in any form.

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.

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.

Poincaré first approached it assuming all functions to be analytic and in the process discovered the non-resonant condition.

Eventually, in 1953 Einstein described the advances of his theory ( although Poincaré already stated in 1905 that Lorentz invariance is a general condition for any physical theory ):

The gauge condition of the Multipolar gauge ( also known as the Line gauge, point gauge or Poincaré gauge ) is:

The gauge condition of the Fock – Schwinger gauge ( sometimes called the relativistic Poincaré gauge ) is:

Similar to Poincaré, Einstein concluded in 1906 that the inertia of ( electromagnetic ) energy is a necessary condition for the center of mass theorem to hold in systems, in which electromagnetic fields and matter are acting on each other.

At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology — what would later be called combinatorial topology and then algebraic topology.

Henri Poincaré gave a keynote address on mathematical physics, including an outline for what would eventually became known as special relativity.

Translations, however, cannot be the usual Poincaré generators as it would be in contradiction with postulate 2 ).

Thus, proving the elliptization conjecture would prove the Poincaré conjecture as a corollary.

This is because, among other reasons, according to the principle of mass-energy equivalence, if the Earth was absorbing the energy of the ultramundane flux at the rate necessary to produce the observed force of gravity ( i. e. by using the values calculated by Poincaré ), its mass would be doubling in each fraction of a second.

In the spring of 1914 an exceptionally radical chamber was elected, and for a while it seemed that they would be unable to agree upon any one for Premier, but finally, he was appointed Prime Minister on 13 July 1914, by President Poincaré.

For example, with a value of 4, there is only one point on the Poincaré map, because the function yields a periodic orbit of period one, or if the value is set to 12. 8, there would be six points corresponding to a period six orbit.

Poincaré often argued to the British that letting the Germans defy Versailles in regards to the reparations would create a precedent that would lead to the Germans dismantling the rest of the Versailles treaty.

Finally, Poincaré argued that once the chains that had bound Germany in Versailles were destroyed, it was inevitable that Germany would plunge the world into another world war.

Emperor Karl I, using his brother-in-law Prince Sixte of Bourbon-Parma as his intermediary, had secretly assured French President Poincaré by a letter dated 24 March 1917 that he would support France's " just demand " for the return of Alsace-Lorraine.

On 18 September, on his return to London, Curzon pointed out that this would enrage the pro-Turkish Prime Minister of France, Raymond Poincaré and left for Paris to attempt to smooth things over.

However, Poincaré later said the translation of physics into the language of four-dimensional metry would entail too much effort for limited profit, and therefore he refused to work out the consequences of this notion.

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.

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.

Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3-sphere.

To establish that the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new topological invariant, the fundamental group, and showed that the Poincaré sphere had a fundamental group of order 120, while the 3-sphere had a trivial fundamental group.

In 1958 Bing proved a weak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a 3-ball, then the manifold is homeomorphic to the 3-sphere .< ref > Bing also described some of the pitfalls in trying to prove the Poincaré conjecture.

The Poincaré conjecture, proved in 2003 by Grigori Perelman, provides that the 3-sphere is the only three-dimensional manifold ( up to homeomorphism ) with these properties.

A positively curved universe is described by spherical geometry, and can be thought of as a three-dimensional hypersphere, or some other spherical 3-manifold ( such as the Poincaré dodecahedral space ), all of which are quotients of the 3-sphere.

* Poincaré dodecahedral space, a positively curved space, colloquially described as " soccerball-shaped ", as it is the quotient of the 3-sphere by the binary icosahedral group, which is very close to icosahedral symmetry, the symmetry of a soccer ball.

Examples include the 3-sphere, the Poincaré homology sphere, Lens spaces.