Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister (Reidemeister 1935) for 3-manifolds and generalized to higher dimensions by Wolfgang Franz (1935) and Georges de Rham (1936).
Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger (1977, 1979) and Werner Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds.
Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field. It can be used to classify lens spaces.
Reidemeister torsion is closely related to Whitehead torsion; see (Milnor 1966). It has also given some important motivation to arithmetic topology; see (Mazur). For more recent work on torsion see the books (Turaev 2002) and (Nicolaescu 2002, 2003).
dimensions by Wolfgang Franz (1935) and Georges de Rham (1936). Analytictorsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Daniel...
a ring Torsion-free module, in algebra See also Torsion-free (disambiguation) Analytictorsion (Reidemeister torsion, R-torsion, Franz torsion, de Rham...
derived analytically that the torsion constant of a beam is identical to the second moment of area normal to the section Jzz, which has an exact analytic equation...
computer science. It was finally proved in 2013. Singer also developed analytictorsion with D.B. Ray and with Henry McKean introduced heat equation formulae...
36. 1980. 91–146. L² cohomology Cheeger, Jeff (1977), "AnalyticTorsion and Reidemeister Torsion", Proceedings of the National Academy of Sciences, 74...
can define the determinant using zeta function regularization. See "analytictorsion." Hawking (1977) suggested using this idea to evaluate path integrals...
dimensions, the torsion of a curve measures how sharply it is twisting out of the osculating plane. Taken together, the curvature and the torsion of a space...
Whitehead manifold Invariants Fundamental group Heegaard genus tri-genus Analytictorsion Orientable manifold Connected sum Jordan-Schönflies theorem Signature...
the determinant line bundle, a Hermitian metric defined using the analytictorsion of a family of differential operators. Quillen metric Quillen, Daniel...
D. B. Ray's Ph.D. dissertation Ray, D. B.; Singer, I. M. (1973), "Analytictorsion for complex manifolds.", Ann. of Math., 2, 98 (1), Annals of Mathematics:...
by the differential-geometric invariants called the curvature and the torsion of a curve. The fundamental theorem of curves asserts that the knowledge...
characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to (Q/Z)2g. Hence, its n-torsion part is isomorphic to (Z/nZ)2g...
zeta function and infinite determinant is intimately related to the analytictorsion of the Laplacian D t ∗ D t {\displaystyle D_{t}^{*}D_{t}} . In the...
numerically, if not analytically) yields the curve. If a pair of curves are in different positions but have the same curvature and torsion, then they are congruent...
equations. The latter objects can be studied using algebra and the calculus of analytic functions, and this allows one to indirectly understand the broad shape...
usability. The torsion balance is one of the most mechanically accurate of analog balances. Pharmacy schools still teach how to use torsion balances in the...
with torsion a study of the relationship between asymptotic symmetries in asymptotically flat spacetimes, soft theorems and memory effects an analytic calculation...
Since h ^ ( P ) = 0 {\displaystyle {\hat {h}}(P)=0} if and only if P is a torsion point, the Bogomolov conjecture generalises the Manin-Mumford conjecture...
encompass global analysis and local Atiyah–Singer–index theory (analytic Ray–Singer torsion, Eta forms, elliptic genera), Bergman kernels and geometric quantization...
champion". As for the groups constituting the torsion subgroup of E(Q), the following is known: the torsion subgroup of E(Q) is one of the 15 following...
engineering able to apply a lot of his theories and solutions, including torsion bar suspension for tracked vehicles such as tanks. Most of his research...
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