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Theorem in homotopy theory
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms
Whitehead_theorem
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
Open 3-manifold that is contractible but not homeomorphic to R3
by J. H. C. Whitehead (1935) while trying to prove the Poincaré conjecture, correcting an error in an earlier paper Whitehead (1934, theorem 3) where he
Whitehead_manifold
Branch of mathematics
fixed-point theorem Leray–Hirsch theorem Poincaré duality theorem Seifert–van Kampen theorem Universal coefficient theorem Whitehead theorem Algebraic K-theory
Algebraic_topology
mapping theorem (algebraic topology) Whitehead theorem (homotopy theory) Whitney–Graustein Theorem (algebraic topology) Alexander's theorem (knot theory)
List_of_theorems
cofibration. Hovey (1999), Definition 2.4.3. Hatcher (2002), Theorem 4.32. Is there the Whitehead theorem for cohomology theory? Strøm (1972). Beke (2000), Proposition
Weak equivalence (homotopy theory)
Weak_equivalence_(homotopy_theory)
Topological manifold with a piecewise linear structure on it
canonical PL structure — it is uniquely triangulizable, by Whitehead's theorem on triangulation (Whitehead 1940) — but a PL manifold might not have a smooth structure
Piecewise_linear_manifold
English mathematician and philosopher (1861–1947)
Alfred North Whitehead (15 February 1861 – 30 December 1947) was an English mathematician and philosopher. He created the philosophical school known as
Alfred_North_Whitehead
Subfield of automated reasoning and mathematical logic
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving
Automated_theorem_proving
Type of topological space
always contained in a finite subcomplex. CW complexes satisfy the Whitehead theorem: a map between CW complexes is a homotopy equivalence if and only
CW_complex
Branch of topology
However these two spaces are not homotopy equivalent. So by the Whitehead theorem, the Warsaw circle does not have the homotopy type of a CW complex
Shape_theory_(mathematics)
then X is a contractible space, as follows from the Whitehead theorem and the Hurewicz theorem. Acyclic spaces occur in topology, where they can be used
Acyclic_space
contractible. Indeed, contractibility of a universal cover is the same, by Whitehead's theorem, as asphericality of it. And it is an application of the exact sequence
Aspherical_space
Continuous, position-preserving mapping from a topological space into a subspace
homotopy-theoretic pathologies of arbitrary topological spaces. For example, the Whitehead theorem holds for ANRs: a map of ANRs that induces an isomorphism on homotopy
Retraction_(topology)
Gives a homomorphism from homotopy groups to homology groups
_{1}(A\cap B)} and the generalised Whitehead products. The proof of this theorem uses a higher homotopy van Kampen type theorem for triadic homotopy groups,
Hurewicz_theorem
Two tame knots with homeomorphic complements are the same or mirror images
In mathematics, the Gordon–Luecke theorem on knot complements states that if the complements of two tame knots are homeomorphic, then the knots are equivalent
Gordon–Luecke_theorem
Limitative results in mathematical logic
Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Mathematics glossary
inclusion of the base point is a cofibration. Whitehead 1. J. H. C. Whitehead. 2. Whitehead's theorem says that for CW complexes, the homotopy equivalence
Glossary of algebraic topology
Glossary_of_algebraic_topology
Algebraic topology uses abstract algebra to study topological spaces
groups of spheres Plus construction Whitehead theorem Weak equivalence Hurewicz theorem H-space Künneth theorem De Rham cohomology Obstruction theory
List of algebraic topology topics
List_of_algebraic_topology_topics
Concept in topology
is exactly the Whitehead torsion τ (W, M) of the inclusion M ↪ W {\displaystyle M\hookrightarrow W} . Precisely, the s-cobordism theorem (the s stands
H-cobordism
the s-cobordism theorem states that if the manifolds are not simply-connected, an h-cobordism is a cylinder if and only if the Whitehead torsion of the
Whitehead_torsion
Subject area in mathematics
h-cobordism theorem because the simple connectedness hypotheses imply that the relevant Whitehead group is trivial. In fact the s-cobordism theorem implies
Algebraic_K-theory
Branch of mathematics
are abelian. Universal coefficient theorem Dold–Thom theorem See also: Characteristic class, Postnikov tower, Whitehead torsion There are several specific
Homotopy_theory
Question in abstract algebra
incompleteness theorem of 1931, previous examples of undecidable statements (such as the continuum hypothesis) had all been in pure set theory. The Whitehead problem
Whitehead_problem
Limit of spheres in algebraic topology
infinite-dimensional sphere inherits a CW structure from the spheres, Whitehead's theorem claims that it is sufficient to show that it is weakly contractible
Infinite-dimensional_sphere
Topological space consisting of trivial homotopy groups
contractible space is weakly contractible; conversely, it follows from Whitehead's theorem that every weakly contractible CW-complex is contractible. For general
Weakly_contractible_space
Theorem in algebraic geometry
Deligne (1980). Milnor 1963, Theorem 7.3 and Corollary 7.4 Voisin 2003, Theorem 1.23 Lefschetz 1924 Griffiths, Spencer & Whitehead 1992 Andreotti & Frankel
Lefschetz_hyperplane_theorem
Exact homotopy case
resp. Gn(Ck+1). Thus EU(n) (and also Gn(C∞)) is a CW-complex. By Whitehead Theorem and the above Lemma, EU(n) is contractible. Proposition: The cohomology
Classifying_space_for_U(n)
1956 computer program written by Allen Newell, Herbert A. Simon and Cliff Shaw
intelligence program". Logic Theorist proved 38 of the first 52 theorems in chapter two of Whitehead and Bertrand Russell's Principia Mathematica, and found new
Logic_Theorist
In mathematics, a partition of a manifold into submanifolds
yα. Since B is assumed to support a C∞ structure, according to the Whitehead theorem one can fix a Riemannian metric on B and choose the atlas U {\displaystyle
Foliation
Concept in math
from there to the homotopy category. Results of J.H.C. Whitehead, in particular Whitehead's theorem and the existence of CW approximations, give a more explicit
Homotopy_category
For simply connected spaces, this theorem recovers a well-known corollary to the Whitehead and Hurewicz theorems. Nilpotent spaces are of great interest
Nilpotent_space
British mathematician who proved Fermat's Last Theorem
detail in Simon Singh's popular book Fermat's Last Theorem. In 1988, Wiles was awarded the Junior Whitehead Prize of the London Mathematical Society (1988)
Andrew_Wiles
Theorem in geometric topology
conjecture (UK: /ˈpwæ̃kæreɪ/, US: /ˌpwæ̃kɑːˈreɪ/, French: [pwɛ̃kaʁe]) is a theorem about the characterization of the 3-sphere (the hypersphere that bounds
Poincaré_conjecture
{\mathfrak {g}}} over a field of characteristic zero. The theorem is an easy consequence of Whitehead's lemma, which says V → Der ( g , V ) , v ↦ ⋅ v {\displaystyle
Weyl's theorem on complete reducibility
Weyl's_theorem_on_complete_reducibility
which are also attributed to Whitehead. The first Whitehead lemma is an important step toward the proof of Weyl's theorem on complete reducibility. Without
Whitehead's lemma (Lie algebra)
Whitehead's_lemma_(Lie_algebra)
Yes-or-no question that cannot ever be solved by a computer
term. This result was later generalized by Rice's theorem. In 1973, Saharon Shelah showed the Whitehead problem in group theory is undecidable, in the first
Undecidable_problem
Subfield of computer science and logic
addition to proving the theorems, the program found a proof for one of the theorems that was more elegant than the one provided by Whitehead and Russell. After
Automated_reasoning
Problem in computer science
Minsky notes: ...the magnitudes involved should lead one to suspect that theorems and arguments based chiefly on the mere finiteness [of] the state diagram
Halting_problem
3-volume treatise on mathematics, 1910–1913
Language – first computational demonstration of theorems in PM Introduction to Mathematical Philosophy Whitehead, Alfred North; Russell, Bertrand (1963). Principia
Principia_Mathematica
Impossible task in computing
impossible by Alonzo Church and Alan Turing in 1936. By the completeness theorem of first-order logic, a statement is universally valid if and only if it
Entscheidungsproblem
Type of mathematical knot
The class of satellite knots include composite knots, cable knots, and Whitehead doubles. A satellite link is one that orbits a companion knot K in the
Satellite_knot
Propositional logic theorem
intuitionistic logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: ∗ 4 ⋅ 13 . ⊢
Double_negation
Establishes the concept of stabilization of homotopy groups
Cambridge University Press, ISBN 0-521-79540-0. Whitehead, G. W. (1953), "On the Freudenthal Theorems", Annals of Mathematics, 57 (2): 209–228, doi:10
Freudenthal suspension theorem
Freudenthal_suspension_theorem
British mathematician (born 1944)
system, an illustration of Gödel's first incompleteness theorem. Paris was awarded the Whitehead Prize in 1983 and elected a Fellow of the British Academy
Jeff_Paris_(mathematician)
Homotopy invariant of maps between n-spheres
contains 2 Z {\displaystyle 2\mathbb {Z} } . Moreover, the image of the Whitehead product of identity maps equals 2, i. e. h ( [ i n , i n ] ) = 2 {\displaystyle
Hopf_invariant
_{n}X} is abelian for n ≥ 2 {\displaystyle n\geq 2} . An analog of Whitehead's theorem holds: a map f {\displaystyle f} between Kan complexes is a homotopy
Simplicial_homotopy
In mathematics, a topological construction
after, Mikhail Postnikov. There is a similar construction called the Whitehead tower (defined below) where instead of having spaces X n {\displaystyle
Postnikov_system
Mathematical logician and philosopher
theorem in 1929 as part of his dissertation to earn a doctorate at the University of Vienna, and the publication of Gödel's incompleteness theorems two
Kurt_Gödel
Commutative group (mathematics)
the Whitehead problem: are all Whitehead groups of infinite order also free abelian groups? In the 1970s, Saharon Shelah proved that the Whitehead problem
Abelian_group
Generalization of Dehn's lemma in the topology of 3-manifolds
mathematics, in the topology of 3-manifolds, the loop theorem is a generalization of Dehn's lemma. The loop theorem was first proven by Christos Papakyriakopoulos
Loop_theorem
a Mahlo cardinal. This theorem of Shelah answers a question of H. Friedman. In 1973, Saharon Shelah showed that the Whitehead problem ("is every abelian
List of statements independent of ZFC
List_of_statements_independent_of_ZFC
truth-functional tautology or theorem of propositional logic. The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica
Absorption_(logic)
Mathematical theorem
are not homeomorphic to the ball (e.g., the Whitehead continuum). The analogue of the Riemann mapping theorem in several complex variables is also not true
Riemann_mapping_theorem
How many linearly independent smooth nowhere-zero vector fields can be on an n-sphere
classical problem of differential topology, beginning with the hairy ball theorem, and early work on the classification of division algebras. Specifically
Vector_fields_on_spheres
Logical principle
2)" (ibid p 421) The principle was stated as a theorem of propositional logic by Russell and Whitehead in Principia Mathematica as: ✸2.1 ~p ∨ p From the
Law_of_excluded_middle
Theorem in topology
using his "tower construction". He also generalized the theorem to the loop theorem and sphere theorem. Papakyriakopoulos proved Dehn's lemma using a tower
Dehn's_lemma
British mathematician (born 1938)
supervision of Christopher Zeeman. In 1991, Lickorish received the Senior Whitehead Prize from the London Mathematical Society. Lickorish and Kenneth Millett
W._B._R._Lickorish
which thus get lost on the way. In certain situations (such as descent theorems for fundamental groups à la Van Kampen) it is much more elegant, even indispensable
Fundamental_groupoid
Whitehead manifold Invariants Fundamental group Heegaard genus tri-genus Analytic torsion Orientable manifold Connected sum Jordan-Schönflies theorem
List of geometric topology topics
List_of_geometric_topology_topics
On when elements of the 2nd homotopy group of a 3-manifold can be embedded spheres
In mathematics, in the topology of 3-manifolds, the sphere theorem of Christos Papakyriakopoulos (1957) gives conditions for elements of the second homotopy
Sphere_theorem_(3-manifolds)
German mathematician (1862–1943)
Hilbert–Burch theorem Hilbert's irreducibility theorem Hilbert's Nullstellensatz Hilbert's theorem (differential geometry) Hilbert's Theorem 90 Hilbert's
David_Hilbert
British mathematician (born 1931)
W. S. Cassels. More influenced by Harold Davenport, he proved Birch's theorem, one of the results to come out of the Hardy–Littlewood circle method.
Bryan_John_Birch
as of September 2022[update]. The conjecture terminology may persist: theorems often enough may still be referred to as conjectures, using the anachronistic
List_of_conjectures
American mathematician (born 1930)
homotopy type as the special orthogonal group of 3 × 3 matrices. Smale's theorem has been reproved and extended a few times, notably to higher dimensions
Stephen_Smale
British mathematician (born 1977)
with Terence Tao, they proved a structure theorem for approximate groups, generalising the Freiman-Ruzsa theorem on sets of integers with small doubling
Ben_Green_(mathematician)
Concept in geometric topology
coefficients in Z [ π 1 ( X ) ] {\displaystyle Z[\pi _{1}(X)]} .) By Whitehead's theorem, the map f {\displaystyle f} is a homotopy equivalence if and only
Normal_invariant
English mathematician
The units of group-rings, was written under the direction of J. H. C. Whitehead. From 1960 to 1984 he was the Waynflete Professor of Pure Mathematics
Graham_Higman
Upper bound on intersecting set families
In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common.
Erdős–Ko–Rado_theorem
Two interlinked loops with five structural crossings
In knot theory, the Whitehead link, named for J. H. C. Whitehead, is one of the most basic links. It can be drawn as an alternating link with five crossings
Whitehead_link
American mathematician (1911–1996)
representation theorem Birkhoff's HSP theorem Birkhoff's theorem (equational logic) Birkhoff–von Neumann theorem Birkhoff–Kakutani theorem Pierce–Birkhoff
Garrett_Birkhoff
British mathematician (1923–2010)
DPhil in 1949 from Oxford University under the supervision of John Henry Whitehead. His dissertation was "Calculation of the homotopy groups of A n 2 {\displaystyle
Peter_Hilton
Basic framework of mathematics
Russell and Alfred North Whitehead in 1913. Mathematical logic led to unexpected results, such as Gödel's incompleteness theorems, which, roughly speaking
Foundations_of_mathematics
Every knot or link can be represented as a closed braid
In mathematics Alexander's theorem states that every knot or link can be represented as a closed braid; that is, a braid in which the corresponding ends
Alexander's_theorem
Theorem relating Milnor K-theory and Galois cohomology
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively
Norm residue isomorphism theorem
Norm_residue_isomorphism_theorem
Describes a periodicity in the homotopy groups of classical groups
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by Raoul Bott (1957, 1959)
Bott_periodicity_theorem
English mathematician (1831–1907)
physics, he also contributed original research such as the Routh–Hurwitz theorem. Central tenets of modern control systems theory relied upon the Routh
Edward_Routh
Mathematical property
under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3. However, it is a theorem of John R. Stallings that for n ≥ 5 {\displaystyle
Simply_connected_at_infinity
Concept from mathematics
But by the long exact sequence of the pair (Mf, p−1(b)) and by Whitehead's theorem, this is equivalent to f being a homotopy equivalence. For topological
Quasi-fibration
Knot invariant
_{K_{1}}(t)\Delta _{K_{2}}(t)} . If K {\displaystyle K} is an untwisted Whitehead double, then Δ K ( t ) = ± 1 {\displaystyle \Delta _{K}(t)=\pm 1} . The
Alexander_polynomial
Greek philosopher (c. 570 – c. 495 BC)
with mathematical and scientific discoveries, such as the Pythagorean theorem, Pythagorean tuning, the five regular solids, the theory of proportions
Pythagoras
American mathematician (1880–1960)
relativity. He proved the Jordan curve theorem in 1905; while this was long considered the first rigorous proof of the theorem, many now also consider Camille
Oswald_Veblen
Group that is a topological space with continuous group operations
Hewitt & Ross 1970, Theorem 27.40. Mackey 1976, section 2.4. Banaszczyk 1983. Hatcher 2001, Theorem 4.66. Hatcher 2001, Theorem 3C.4. Edwards 1995, p
Topological_group
Theory of algebraic structures in general
coordinatewise. The isomorphism theorems, which encompass the isomorphism theorems of groups, rings, modules, etc. Birkhoff's HSP Theorem, which states that a class
Universal_algebra
British-American mathematician (born 1962)
Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves. He received the Whitehead Prize in 1990, the Fermat Prize and
Richard Taylor (mathematician)
Richard_Taylor_(mathematician)
English mathematician (born 1957)
result that would establish his fame and later become known as Donaldson's theorem. He published the result in a paper "Self-dual connections and the topology
Simon_Donaldson
Academic journal
Doob, Abraham Taub, George W. Whitehead, and Oscar Zariski. The journal published the proof of the four color theorem by Kenneth Appel and Wolfgang Haken
Illinois Journal of Mathematics
Illinois_Journal_of_Mathematics
Mathematical term; concerning axioms used to derive theorems
known as lemmas or theorems. A mathematical theory is an expression used to refer to an axiomatic system and all its derived theorems. A proof within an
Axiomatic_system
Category of mathematical proof
In mathematics, an impossibility theorem is a theorem that demonstrates a problem or general set of problems cannot be solved. These are also known as
Proof_of_impossibility
Australian mathematician
Erdős. He was awarded the London Mathematical Society's Whitehead Prize in 1979 and Senior Whitehead Prize in 2017, and is joint winner of the 2003 Euler
Peter_Cameron_(mathematician)
Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
chart (i.e. a knot and its mirror image are considered equivalent). A theorem due to Horst Schubert (1919–2001) states that every knot can be uniquely
Prime_knot
products of the spaces according to the Hilton–Milnor theorem. The map is defined by generalised Whitehead products (Baues & Quintero 2001). If Y is a group-like
Generalised_Whitehead_product
Subfield of mathematics
mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary
Mathematical_logic
Branch of mathematics
of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, a problem that was stated in terms of elementary arithmetic, and remained
Geometry
Type of mathematical link
[clarification needed] As a consequence of Thurston's hyperbolic Dehn surgery theorem, performing Dehn surgeries on a hyperbolic link enables one to obtain many
Hyperbolic_link
Family of mathematical knots
loop and then linking the ends together. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite family of knots
Twist_knot
Algebraic manipulation of "true" and "false"
the theorem proved by the proof. Every nonempty initial segment of a proof is itself a proof, whence every proposition in a proof is itself a theorem. An
Boolean_algebra
Branch of metaphysics
Marquis (2013) and Whitehead's early work is discussed in Kneebone (1963: ch. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929 Process and
Mereotopology
Link that consists of finitely many unlinked unknots
components such that any proper sublink is an unlink (a Brunnian link). The Whitehead link and Borromean rings are such examples for n = 2, 3. Linking number
Unlink
Ancient Greek philosopher
Christian, Jewish and Islamic philosophy. In modern times, Alfred North Whitehead said: "the safest general characterization of the European philosophical
Plato
WHITEHEAD THEOREM
WHITEHEAD THEOREM
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).
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.
WHITEHEAD THEOREM
WHITEHEAD THEOREM
Girl/Female
Australian, Danish, Finnish, French, Latin
Rival; Laborious; Eager
Biblical
a moist table
Male
Thai/Siamese
Thai name THUANTHONG means "golden spear."
Girl/Female
Hindu
God has blessed
Boy/Male
African, American, Armenian, British, Christian, Czechoslovakian, Danish, Dutch, English, Finnish, French, German, Hawaiian, Hebrew, Hindu, Indian, Irish, Jamaican, Polish, Portuguese, Swedish, Swiss, Tamil
Asked of God; Told by God; Name of King in Bible; Follower of Jesus; Heard by God
Surname or Lastname
English
English : habitational name from either of two places, in Norfolk and Suffolk, named Brettenham, from Old English Bretta ‘of the Britons’ (genitive of Brettas) + tūn ‘farmstead’.
Boy/Male
Hindu
The Moon
Boy/Male
Australian, Christian, Egyptian, Finnish, French, German, Swedish
Teacher; Builder; The Hidden One; Kindred; Tribal
Boy/Male
Irish
From the Latin patricius “â€nobly born.â€â€ The patron saint of Ireland, it is hard to differentiate between fact and myth. What is probably true is that he was born in Britain around 373 AD and was brought to Ireland as a slave at the age of seven, possibly by Niall of the Nine Hostages (read the legend). Forced to guard sheep on the Slemish Mountains in Country Antrim for six years he had a vision urging him to convert his captors. He escaped to France where he trained as a priest before returning to Ireland where he banished the snakes (i.e. paganism) and converted the population to Christianity. Both Patrick and Padraig are very popular names in Ireland.
Boy/Male
German, Greek
Rich; Song
WHITEHEAD THEOREM
WHITEHEAD THEOREM
WHITEHEAD THEOREM
WHITEHEAD THEOREM
WHITEHEAD THEOREM
n.
The whiteweed (Chrysanthemum Leucanthemum), the plant commonly called daisy in North America; -- called also oxeye daisy. See Whiteweed.
n.
A fatty, solid substance, produced by bees, and employed by them in the construction of their comb; -- usually called beeswax. It is first excreted, from a row of pouches along their sides, in the form of scales, which, being masticated and mixed with saliva, become whitened and tenacious. Its natural color is pale or dull yellow.
v. i.
To become white or whiter; to be whitened or blanched by excluding the light of the sun, as, plants.
v. t.
To formulate into a theorem.
n.
That which is considered and established as a principle; hence, sometimes, a rule.
a.
Alt. of Theorematical
n.
An old man; a graybeard.
a.
Theorematic.
n.
A numerical coefficient in any particular case of the binomial theorem.
a.
Whitened; make white.
n.
The common beam tree of England (Pyrus Aria); -- so called from the white, woolly under surface of the leaves.
n.
A statement of a principle to be demonstrated.
n.
The blue-winged snow goose.
n.
One who constructs theorems.
n.
The enunciation of a self-evident problem, in distinction from an axiom, which is the enunciation of a self-evident theorem.
n.
A perennial composite herb (Chrysanthemum Leucanthemum) with conspicuous white rays and a yellow disk, a common weed in grass lands and pastures; -- called also oxeye daisy.
a.
Of or pertaining to a theorem or theorems; comprised in a theorem; consisting of theorems.
imp. & p. p.
of Whiten
n.
The surf scoter.
superl.
White mixed with black, as the color of pepper and salt, or of ashes, or of hair whitened by age; sometimes, a dark mixed color; as, the soft gray eye of a dove.