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simple homotopy equivalence is its Whitehead torsion τ ( f ) {\displaystyle \tau (f)} which is an element in the Whitehead group Wh ( π 1 ( Y ) ) {\displaystyle
Whitehead_torsion
Topics referred to by the same term
torsion, de Rham torsion, Ray-Singer torsion), a topological invariant of manifolds Whitehead torsion, in geometric topology Torsion field (pseudoscience)
Torsion
British mathematician (1904–1960)
John Henry Constantine Whitehead (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy
J._H._C._Whitehead
Topological invariant of manifolds that can distinguish homotopy-equivalent manifolds
can be used to classify lens spaces. Reidemeister torsion is closely related to Whitehead torsion; see (Milnor 1966). It has also given some important
Analytic_torsion
Topics referred to by the same term
complex or PL-manifold A, equal to Wh(π1(A)); see Whitehead torsion. All named after J. H. C. Whitehead. This disambiguation page lists mathematics articles
Whitehead_group
Question in abstract algebra
abelian group Whitehead torsion List of statements undecidable in ZFC Statements true in L Shelah, S. (1974). "Infinite Abelian groups, Whitehead problem and
Whitehead_problem
Subject area in mathematics
group ring of π. Later John Milnor used Reidemeister torsion, an invariant related to Whitehead torsion, to disprove the Hauptvermutung. The first adequate
Algebraic_K-theory
American mathematician (born 1930)
Regular homotopy Sard's theorem Sphere eversion Structural stability Whitehead torsion Diffeomorphism Awards Wolf Prize (2007) National Medal of Science
Stephen_Smale
Branch of mathematics
Dold–Thom theorem See also: Characteristic class, Postnikov tower, Whitehead torsion There are several specific theories simple homotopy theory stable
Homotopy_theory
a homotopy equivalence being a simple homotopy equivalence is the Whitehead torsion, τ ( f ) . {\displaystyle \tau (f).} A homotopy theory that studies
Simple-homotopy_equivalence
Concept in topology
connected): An h-cobordism is a cylinder if and only if Whitehead torsion τ (W, M) vanishes. The torsion vanishes if and only if the inclusion M ↪ W {\displaystyle
H-cobordism
Branch of mathematics
algebraic K-theory was the work of J. H. C. Whitehead and others on what later became known as Whitehead torsion. There followed a period in which there were
K-theory
Three groups
Farrell–Jones conjecture. It is even unknown if the Whitehead group of F (see Whitehead torsion) or the projective class group of F (see Wall's finiteness
Thompson_groups
Commutative group (mathematics)
T(A)} is its torsion subgroup, then the factor group A / T ( A ) {\displaystyle A/T(A)} is torsion-free. However, in general the torsion subgroup is not
Abelian_group
structure set depending on the category (DIFF, PL or TOP) and whether Whitehead torsion is taken into account or not. Let X be a closed smooth (or PL- or
Surgery_structure_set
(see for example Wall's finiteness obstruction, surgery obstruction, Whitehead torsion). So suppose a group G {\displaystyle G} satisfies the Farrell–Jones
Farrell–Jones_conjecture
American mathematician (1947–2019)
His thesis was supervised by Stephen M. Gersten, and was titled Whitehead Torsion of Finite Abelian Groups. After temporary positions at Brown University
Roger_C._Alperin
{\widetilde {K}}_{0}(\mathbb {Z} [\pi _{1}(X)])} . Algebraic K-theory Whitehead torsion Milnor, John (1959), "On spaces having the homotopy type of a CW-complex"
Wall's_finiteness_obstruction
Techniques in topology used to produce one finite-dimensional manifold from another
the corresponding manifold. s-cobordism theorem h-cobordism theorem Whitehead torsion Dehn surgery Manifold decomposition Orientation character Plumbing
Surgery_theory
Set of mathematical functions concerning algebraic group isomorphism
Moriah, Yoav (1993), "Generating systems of groups and Reidemeister-Whitehead torsion", Journal of Algebra, 157 (1): 170–198, doi:10.1006/jabr.1993.1096
Nielsen_transformation
Algebraic topology uses abstract algebra to study topological spaces
K-theory Topological K-theory Adams operation Algebraic K-theory Whitehead torsion Twisted K-theory Cobordism Thom space Suspension functor Stable homotopy
List of algebraic topology topics
List_of_algebraic_topology_topics
space. It was originated by J. H. C. Whitehead in his 1950 paper "Simple homotopy types". Whitehead torsion Whitehead 1950 Cohen, M. M. (1973). A Course
Simple_homotopy_theory
Tool to classify manifolds within a homotopy type in dim > 4
with: smooth (DIFF), PL, or topological manifolds and whether we take Whitehead torsion into account or not (decorations s {\displaystyle s} or h {\displaystyle
Surgery_exact_sequence
Mathematics glossary
homotopy equivalence is a simple-homotopy equivalence if and only if its Whitehead torsion vanishes. simplicial approximation See simplicial approximation theorem
Glossary of algebraic topology
Glossary_of_algebraic_topology
Homotopy invariant of maps between n-spheres
trivial (since π 2 n − 1 ( S n ) {\displaystyle \pi _{2n-1}(S^{n})} is torsion). If n {\displaystyle n} is even, the image of h {\displaystyle h} contains
Hopf_invariant
American mathematics professor
of seminars and public lectures across Australia. In 2020, she won a Whitehead Prize of the London Mathematical Society "for her deep contributions to
Holly_Krieger
Classical theory of gravitation
removing the zero torsion constraint from the Palatini action, which results in the additional set of equations for spin and torsion, as well as the addition
Einstein–Cartan_theory
Left-right asymmetry of the spine's curvature
Jeanne; Mitchell, Laura E; Pasquariello, Patrick S; Sutton, Leslie N; Whitehead, Alexander S (2004). "Spina bifida". The Lancet. 364 (9448): 1885–1895
Scoliosis
How spheres of various dimensions can wrap around each other
the first place that p-torsion occurs in the homotopy groups of n dimensional spheres, by showing that πn+k(Sn) has no p-torsion if k < 2p − 3, and has
Homotopy_groups_of_spheres
Graph manifolds Knot complements Whitehead manifold Invariants Fundamental group Heegaard genus tri-genus Analytic torsion Orientable manifold Connected
List of geometric topology topics
List_of_geometric_topology_topics
Topological space with only one nontrivial homotopy group
classifying space of the group G {\displaystyle G} . Note that if G has a torsion element, then every CW-complex of type K(G,1) has to be infinite-dimensional
Eilenberg–MacLane_space
In mathematics, a module that has a basis
the set-theoretic sense. A famous example is the Whitehead problem, which asks whether a Whitehead group is free or not. As it turns out, the problem
Free_module
Ferrari's first Formula One car
double wishbone suspension with a transverse leaf spring in front and a torsion bar in the rear which was upgraded to a de Dion tube for 1950. Worm and
Ferrari_125_F1
American mathematician
Ph.D. in 1952 from Brown University under the supervision of George W. Whitehead. Moore began his career at Princeton University as an instructor, and
John_Coleman_Moore
Proposed theories of gravity
derivative gravity is a covariant theory of gravity, quadratic in curvature, torsion free and parity invariant, L = − g [ M p 2 R + R f 1 ( ◻ M s 2 ) R + R
Alternatives to general relativity
Alternatives_to_general_relativity
Suspension system for a vehicle
Today, coil springs are used in most cars. In 1920, Leyland Motors used torsion bars in a suspension system. In 1922, independent front suspension was
Car_suspension
Hoyle First theorised existence of black holes, binary stars; invented torsion balance – John Michell Spiral galaxies – William Parsons, 3rd Earl of Rosse
List of British innovations and discoveries
List_of_British_innovations_and_discoveries
Tensor field in Riemannian geometry
(called, in this context, flat space). Since the Levi-Civita connection is torsion-free, its curvature can also be expressed in terms of the second covariant
Riemann_curvature_tensor
Merel, Loïc (1996). ""Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields]"
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
British mathematician
chaired Professor appointment in Mathematics. Bowditch was awarded a Whitehead Prize by the London Mathematical Society in 1997 for his work in geometric
Brian_Bowditch
Double cover Lie group of the special orthogonal group
The essential dimension of spin groups is OEIS:A280191. Grothendieck's "torsion index" is OEIS:A096336. Karoubi, Max (2008). K-Theory. Springer. pp. 210–214
Spin_group
Representation of mathematical space
Reidemeister torsion. Suppose further that the modification into L 1 ′ , L 2 ′ {\displaystyle L'_{1},L'_{2}} does not affect Reidemeister torsion but such
Triangulation_(topology)
Theorem relating Milnor K-theory and Galois cohomology
residue map. Because étale cohomology with mod-ℓ coefficients is an ℓ-torsion group, this map additionally factors through K n M ( k ) / ℓ {\displaystyle
Norm residue isomorphism theorem
Norm_residue_isomorphism_theorem
Theory of gravity
Weitzenböck connection has vanishing curvature, but – in general – non-vanishing torsion. Given the frame field {Xi}, one can also define a metric by conceiving
Teleparallelism
Idea in quantum gravity
look like curvature and torsion in an Einstein–Cartan spacetime. This allows one to create a theory of gravity with torsion from a world crystal model
Induced_gravity
Knot invariant
{\displaystyle \Delta _{M}(1)} is, up to sign, equal to the order of the torsion subgroup of H 1 M {\displaystyle H_{1}M} . The value of the Alexander polynomial
Alexander_polynomial
1956 sports racing car
from a DBR3, while DBR1/5 was a spare chassis sold to privateer Graham Whitehead. This latter car was the only DBR1 to be sold to a privateer during the
Aston_Martin_DBR1
Motor vehicle
versions, survived until 1992. All XK120s had independent Heynes designed torsion bar front suspension, semi-elliptic leaf springs at the rear, recirculating
Jaguar_XK120
Algebra associated to any vector space
the calculus of differential forms. A short while later, Alfred North Whitehead, borrowing from the ideas of Peano and Grassmann, introduced his universal
Exterior_algebra
Classical field theory describing gravitation
theory. However, in the ECKS theory, torsion is algebraic and vanishes in the vacuum. To include a dynamical torsion, Poincaré gauge theory is developed
Poincaré_gauge_theory
p-core Schreier refinement theorem Subgroup Transversal (combinatorics) Torsion subgroup Zassenhaus lemma Automorphism Automorphism group Factor group
List_of_group_theory_topics
Atiyah–Singer index theorem, Atiyah–Hitchin–Singer theorem, Ray–Singer torsion and Kadison–Singer problem David Slepian (B.Sc.), mathematician; Shannon
List of University of Michigan alumni
List_of_University_of_Michigan_alumni
Theory of gravitation as curved spacetime
dilation and frequency shift. Choosing a different connection with non-zero torsion leads to a modified theory known as Einstein–Cartan theory Ehlers 1973
General_relativity
Modern theory of gravitation that combines supersymmetry and general relativity
inequivalent in the sense that their actions and constraints upon the torsion tensor are different, but ultimately equivalent in that we can always perform
Supergravity
Small family car (1938–2003)
a central structural tunnel. The front and rear suspension incorporate torsion bars and a front stabiliser bar, providing independent suspension at all
Volkswagen_Beetle
Shock barrier to contain animals or people
brackets attached to the posts by what may be called a "pivot bind" or "torsion-lock". The weight of the bracket, the attached insulator and the electric
Electric_fence
Group whose operation is a composition of braids
{\displaystyle B_{n}} have infinite order; i.e., B n {\displaystyle B_{n}} is torsion-free. There is a left-invariant linear order on B n {\displaystyle B_{n}}
Braid_group
theorem (number theory) Max Noether's theorem (algebraic geometry) Mazur's torsion theorem (algebraic geometry) Mumford vanishing theorem (algebraic geometry)
List_of_theorems
valuable horse after he diagnoses that it is too far gone with an agonising torsion. Characters: Mrs Hall, Bert Sharpe, Diana Brompton, Mr Biggins, Mr White
List of All Creatures Great and Small (1978 TV series) episodes
List_of_All_Creatures_Great_and_Small_(1978_TV_series)_episodes
Array of numbers
two authors of historical importance. Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) use the word "matrix" in the
Matrix_(mathematics)
weight, made the car out of aluminium; the car had a torsion bar suspension, with eight torsion bars; Flaminio Bertoni, an Italian, was head of exterior
List_of_Equinox_episodes
Algebraic construct classifying topological spaces
know π 4 ( S O ( 4 ) ) {\displaystyle \pi _{4}(\mathrm {SO} (4))} has two-torsion. Milnor used the fact π 3 ( S O ( 4 ) ) = Z ⊕ Z {\displaystyle \pi _{3}(\mathrm
Homotopy_group
Invariant of a knot diagram
to describe the amount a piece of DNA is deformed as a result of this torsional stress. In general, this phenomenon of forming coils due to writhe is
Writhe
Automobile manufacturer
leaf springs and friction dampers with a trailing arm with transverse torsion bars and hydraulic dampers. Three type B ERAs were modified to type C standard
English_Racing_Automobiles
Algebraic structure used in topology
(finite) dimension. Likewise, the product on integral cohomology modulo torsion with values in H n ( X , Z ) ≅ Z {\displaystyle H^{n}(X,\mathbb {Z} )\cong
Cohomology
Algebraic structure
the bracket of any element with itself is 2-torsion, if it does not actually vanish. See also: Whitehead product. Trell, Erik (1998), "English Translation
Generalization of a Lie algebra
Generalization_of_a_Lie_algebra
historiography, and archaeology." History in Africa 45, no. 1 (2018): 145–160. Whitehead, Clive. "The historiography of British imperial education policy, Part
African_historiography
Concept in physics
the nonsymmetric part of the metric tensor was shown to be equal to the torsion potential, a result following the metricity condition, that the length
Nonsymmetric gravitational theory
Nonsymmetric_gravitational_theory
Mathematical theory of topological spaces
simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by Dennis Sullivan (1977)
Rational_homotopy_theory
Mathematical concept
which the group acts geometrically so it is of type F∞. When the group is torsion-free the action is free, showing that the group has finite cohomological
Hyperbolic_group
Racing car model
racer, but there was one last swan song for the DB3S in the hands of the Whitehead brothers at Le Mans in 1958 with a second place after all the works DBR1s
Aston_Martin_DB3S
Sayão, J. M. (2025). "Wing bone laminarity in pterosaurs: insights into torsional adaptations for flight evolution". Anais da Academia Brasileira de Ciências
2025 in archosaur paleontology
2025_in_archosaur_paleontology
Attempt to describe spacetime and associated phenomena in terms of geometry
Gravity". arXiv:gr-qc/9903072. Pasic, Vedad; Barakovic, Elvis (2015). "Torsion Wave Solutions in Yang-Mielke Theory of Gravity". Advances in High Energy
Geometrodynamics
Élisabeth Lutz (1914–2008), French student of Weil, showed how to compute torsion subgroups of elliptic curves Julie Lutz (born 1944), American astronomer
List_of_women_in_mathematics
Quantum mechanical statistic
ISBN 81-902190-9-X, page 82, eq.(37) ff. Rapoport, Diego L. (2007). "Torsion fields, Cartan-Weyl space-time, and state-space quantum geometries, Brownian
Quantum_potential
double stars might be attracted to each other (1767), and he invented the torsion balance (before 1783). 1843: Existence and position of Neptune predicted
List of English inventions and discoveries
List_of_English_inventions_and_discoveries
Fourth AIACR European Championship season
weight limit. Four inches longer, with a limited-slip differential and torsion-bar suspension now installed, the handling was greatly improved. Alfa Romeo
1936_Grand_Prix_season
Historiography in Bulgaria. East European Monographs. ISBN 9780880335065. Whitehead, Cameron Ean Alfred (2014). "The Bulgarian Horrors : culture and the international
Bulgarian_historiography
Theory of gravity
dimensions, around constant curvature backgrounds, parity invariant and torsion free, can be expressed by: S = ∫ d 4 x − g ( M P 2 R + R F 1 ( ◻ ) R +
Infinite_derivative_gravity
Clive Whitehead, "The historiography of British imperial education policy, Part I: India". History of Education 34#3 (2005): 315–329. Clive Whitehead, "The
Historiography of the British Empire
Historiography_of_the_British_Empire
History and analysis of the history of science
of invention and scientific development Gunpowder and gun transmission Torsion mangonel myth Organizations, publications Heritage registers Historical
Historiography_of_science
WHITEHEAD TORSION
WHITEHEAD TORSION
Surname or Lastname
English
English : metonymic occupational name for a baker or seller of white bread, from Old English hwīt ‘white’ or hwǣte ‘wheat’ + brēad ‘bread’. White bread, considered the best bread, was made from wheat flour.In some cases, perhaps a translation of the German cognate Weisbrot.
Surname or Lastname
English and Scottish
English and Scottish : nickname for someone with fair or prematurely white hair, from Middle English whit ‘white’ + heved ‘head’.Irish (Connacht) : erroneous translation of Ó Ceanndubháin ‘descendant of the little black-headed one’ (see Canavan), as if from Gaelic ceann ‘head’ + bán ‘white’.Translated form of German Weisshaupt (see Weishaupt) or Weisskopf (see Weiskopf).
WHITEHEAD TORSION
WHITEHEAD TORSION
WHITEHEAD TORSION
WHITEHEAD TORSION
WHITEHEAD TORSION
WHITEHEAD TORSION
WHITEHEAD TORSION