Добавил:
Upload Опубликованный материал нарушает ваши авторские права? Сообщите нам.
Вуз: Предмет: Файл:

топология / Tom Dieck T. Algebraic topology (EMS, 2008)

.pdf
Скачиваний:
57
Добавлен:
16.04.2015
Размер:
2.72 Mб
Скачать

548 Bibliography

[148]Peano, G., Sur une courbe, qui remplit tout une aire plane. Math. Ann. 36 (1890), 157–160.

[149]Pommerenke, Ch., Boundary behaviour of conformal maps. Grundlehren Math. Wiss. 299, Berlin, Springer 1992. 251

[150]Poincaré, H., Sur la généralisation d’un théorème d’Euler relatif aux polyèdres. Compt. Rend. Acad. Sci. Paris 117 (1893), 144–145. 310

[151]Poincaré, H., Analysis situs. Journal de l’École Polytechnique 1 (1895), 1–121. 42

[152]Poincaré, H., Complément à l’Analysis Situs. Rend. Circ. Mat. Palermo 13 (1899), 285–343. 310

[153]Poincaré, H., Cinquième complément à l’Analysis Situs. Rend. Circ. Mat.Palermo 18 (1904), 45–110.

[154]Poincaré, H., Œuvres de Henri Poincaré VI. Paris, Gauthier-Villars 1953.

[155]Puppe, D., Homotopiemengen und ihre induzierten Abbildungen. I. Math. Z. 69 (1958), 299–344. 33, 81, 95, 195

[156]Puppe, D., Bemerkungen über die Erweiterung von Homotopien. Arch. Math. 18 (1967), 81–88.

[157]Puppe, D., Some well known weak homotopy equivalences are genuine homotopy equivalences. Symposia Mathematica 5 (1971), 363–374.

[158]Puppe, D., Homotopy cocomplete classes of spaces and the realization of the singular complex. In: Topological Topics (I. M. James ed.), London Math. Soc. Lecture Note Ser. 86, Cambridge University Press, 1983, 55–69.

[159]von Querenburg, B., Mengentheoretische Topologie. Berlin–Heidelberg–New York, Springer 1979.

[160]Quillen, D., Elementary proofs of some results of cobordism using Steenrod operations. Adv. Math. 7 (1971), 29–56. 527, 536

[161]Radó, T., Über den Begriff der Riemannschen Fläche. Acta Sci. Math. (Szeged) 2 (1924), 101–121. 198

[162]Ranicki, A., Algebraic and geometric surgery. Oxford Math. Monographs, Oxford University Press, 2002. 390

[163]Ravenel, D. C., Complex cobordism and stable homotopy groups of spheres. New York, Academic Press 1986. 517

[164]Schubert, H., Topologie. Stuttgart, Teubner, 4. Auflage 1975.

[165]Segal, G., Classifying spaces and spectral sequences. Inst. Hautes Études Sci. Publ. Math. 34 (1968), 105–112.

[166]Seifert, H., Konstruktion dreidimensionaler geschlossener Räume. Ber. Sächs. Akad. Wiss. 83 (1931), 26–66. 45

[167]Seifert, H., Threllfall, W., Lehrbuch der Topologie. Leipzig, Teubner 1934. 56, 311, 438

[168]Selick, P., Introduction to homotopy theory. Fields Institute Monogr. 9, Amer. Math. Soc., Providence, R.I. , 1997.

Bibliography 549

[169]Serre, J.-P., Homologie singulière des espaces fibrés. Ann. of Math. 54 (1951), 425–505.

[170]Serre, J.-P., Groupes d’homotopie et classes de groupes abéliens. Ann. of Math. 58 (1953), 258–294. 504

[171]Serre, J.-P., Trees. Berlin–Heidelberg–New York, Springer 1980. 52

[172]Spanier, E. H., Algebraic Topology. New York, McGraw–Hill 1966.

[173]Sperner, E., Neuer Beweis für die Invarianz der Dimensionszahl und des Gebietes.

Abh. Math. Sem. Univ. Hamburg 6 (1928), 265–272. 139

[174]Steen, L. A., Seebach, J. A. jr., Counterexamples in topology. Holt, Rinehart ans Winston 1970.

[175]Steenrod, N., The topology of fibre bundles. Princeton Math. Ser. 14, Princeton University Press, 1951.

[176]Steenrod, N., A convenient category of topological spaces. Michigan Math. J. 14 (1967), 13–152.

[177]Sternberg, S., Lectures on differential geometry. Englewood Cliffs, Prentice Hall 1964. 365

[178]Stöcker, R., Zieschang, H., Algebraische Topologie. Mathematische Leitfäden, Stuttgart, Teubner 1988.

[179]Strøm, A., Note on cofibrations. Math. Scand. 19 (1966), 11–14.

[180]Strøm, A., Note on cofibrations II. Math. Scand. 22 (1968), 130–142. 103

[181]Stong, R. E., Notes on cobordism theory. Princeton University Press, 1968. 536

[182]Sweedler, M. E., Hopf algebras. Math. Lecture Note Ser., New York, Benjamin 1980. 485

[183]Switzer, R. M., Algebraic Topology – Homotopy and Homology. Grundlehren Math. Wiss. 212, Berlin–Heidelberg–New York, Springer 1975.

[184]Thom, R., Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28 (1954), 17–86. 521, 527, 528

[185]Toda, H., Composition methods in homotopy groups of spheres. Ann. of Math. Stud. 49, Princeton University Press, 1962. 517

[186]Vogt, R., Convenient categories of topological spaces for homotopy theory. Arch. Math. 22 (1971), 545–555.

[187]Waldhausen F., Algebraic K-theory of spaces. In: Proc. Conf., Rutgers New Brunswick 1983, Lecture Notes in Math. 1126, Springer, 1985, 318–419.

[188]Wall, C. T. C., Finiteness conditions for CW-complexes. Ann. of Math. 81 (1965), 56–69; Finiteness conditions for CW-complexes II, Proc. London Math. Soc. 1966, 129–139.

[189]Wall, C. T. C., On the exactness of interlocking sequences. Enseign. Math. 12 (1966), 95–100. 245

[190]Wall, C. T. C., A geometric introduction to topology. Reading, Mass., Addison-Wesley 1972.

550 Bibliography

[191]Wall, C. T. C., Ranicki, A., Surgery on compact manifolds. Math, Surveys Monogr, 69, Providence, R.I., Amer. Math. Soc. 1999. 390

[192]Whitehead, G. W., Elements of homotopy theory. Grad. Texts in Math. 61, New York, Springer 1978. 214

[193]Whitehead, J. H. C., On C 1-complexes. Ann. of Math. 41 (1940), 809–824. 198

[194]Whitehead, J. H. C., Combinatorial homotopy. Bull. Amer. Math. Soc. 55 (1949), 213–245.

[195]Whitehead, J. H. C., Mathematical Works. Vol I– IV. Oxford, Pergamon Press 1962.

[196]Whitney, H., On regular closed curves in the plane. Compositio Math. 4 (1937), 276–286.

[197]Zieschang, H., Vogt, E., Coldewey, H.-D., Surfaces and planar discontinuous groups. Lecture Notes in Math. 835, Berlin–Heidelberg–New York, Springer 1980. 198

Symbols

Numbers

 

N

natural numbers f1; 2; 3; : : : g

N0

N [ f0g

Z

integers f0; ˙1; ˙2; : : : g

Q

rational numbers

R, RC, R

real numbers, non-negative, non-positive

C

complex numbers

C

C X 0, non-zero numbers

H

quaternions

Z=m D Z=mZ

integers modulo m

Fq

field with q elements

Categories

 

TOP

category of topological spaces and continuous maps

TOP0

pointed spaces and pointed maps

TOPK

spaces under the space K

TOPB

spaces over the space B

TOP.2/

pairs .X; A/ of a space X and a subspace A

G- TOP

spaces with an action of the topological group G

COVB

covering spaces of the space B

h-TOP

homotopy category of TOP

h-C

homotopy category associated to a category C

 

with homotopy notion

R- MOD

left modules over the ring R

ABEL

abelian groups

SET

sets

G- SET

sets with left G-action

….X/

fundamental groupoid of X

Or.G/

orbit category of the group G

552

Symbols

ŒC; D

 

category of functors C ! D and natural transformations

TRAB

 

transport category Œ….B/; SET

Ker, Ke

kernel

Coker, Ko

cokernel

Im

 

image

id; 1

 

identity

pr

 

projection onto a factor of a product

.fj /

 

map into a product with components fj

h fj i

 

map from a sum with components fj

Spaces

 

 

Rn

 

Euclidean n-space

Sn

 

unit sphere in RnC1

Dn

 

unit disk in Rn

En

 

Dn X Sn 1 unit cell

I D Œ0; 1

unit interval

I n

 

n-fold Cartesian product I I I

@I n

 

combinatorial boundary of I n

S.n/

 

I n=@I n

S.n/

 

Rn [ f1g

n D Œn

n-dimensional standard simplex

@ n

 

combinatorial boundary of n

RP n

 

n-dimensional real projective space

CP n

 

n-dimensional complex projective space

Vk .F n/

Stiefel manifold of orthonormal k-frames in F n

Gk .F n/

Grassmann manifold of k-dimensional subspaces of F n

EG

 

universal free G-space

BG

 

classifying space of the topological group G

E=G; GnE

orbit space of a G-action on E

E G F

balanced product

XH

 

H -fixed point set

K. ; n/

Eilenberg–Mac Lane space of type . ; n/

 

Symbols

553

X=A

X with A X identified to a point

 

XC

X with additional base point

 

Groups

 

 

GLn.F /

group of .n; n/-matrices with entries in F D R; C; H

 

O.n/

group of orthogonal matrices in GLn.R/

 

SO.n/

matrices of determinant 1 in O.n/

 

U.n/

group of unitary matrices in GLn.C/

 

SU.n/

matrices of determinant 1 in U.n/

 

S1

complex numbers of modulus 1

 

Spin.n/

Spinor group, double covering of SO.n/

 

Sp.n/

symplectic group

 

Sn

symmetric group

 

Relations

end of proof

Þend of numbered item

Xdifference set

Ycup product

Zcap product

equivalent (equivalence relation)

'homotopy-equivalent, homotopic

Šisomorphic, homeomorphic, diffeomorphic

˚direct sum

˝tensor product

_pointed sum (bouquet)

?join

^smash product

induced morphism, general index, base point, free product of groups

C, q

topological sum

kxk

Euclidean norm of x

554 Symbols

jzj jXj

Modules

lim colim lim1

Tor

Ext; Extnƒ.A; B/

Bundles

E. /; E0. /

TM

K.X/ D KC.X/ D KU.X/ KO.X/ D KR.X/

"; n" M. /

M O.n/; M SO.n/

ci . /; wi . /; pi . /

Homotopy

CX †X Z.f / Z.f; g/ C.f /

C.X; A/

H

KL

ŒX; Y

ŒX; Y 0

ŒX; Y K ; ŒX; Y B

absolute value of complex number z cardinality of X

limit (inverse limit) colimit (direct limit) derived functor of lim

torsion product, left derived of Hom module of extensions, right derived of Hom

total space of vector bundle, without zero section tangent bundle of the manifold M

Grothendieck ring of complex vector bundles Grothendieck ring of real vector bundles trivial bundle, of dimension n

Thom space of the bundle

Thom space of the universal n-dimensional (oriented) bundle

i-th Chern, Stiefel–Whitney, Pontrjagin class

(pointed) cone on X suspension of X

(pointed) mapping cylinder of f double mapping cylinder mapping cone of f

mapping cone of A X inverse of the homotopy H

product (concatenation) of homotopies K; L homotopy classes of maps X ! Y

pointed homotopy classes homotopy classes under K, over B

….X/

….X; Y /

n.X; A; /

XY D F .X; Y /

X

hocolim

N.U/

B.U/

Co-Homology

Sq .X/

S .X; A/

Hq .X; AI G/

H q .X; AI G/

Œv0; : : : ; vq h ; h

Q Q h ; h

@

ı

.X/ N .X/.X/

M Ok .X/; M SOk .X/

Symbols 555

fundamental groupoid of X homotopy groupoid

n-homotopy group of the pointed pair .X; A; / space of maps X ! Y with compact-open topology loop space of X

homotopy colimit nerve of the covering U

geometric realization of the nerve

singular q-chains

singular chain complex of .X; A/ ordinary homology with coefficients in G

ordinary cohomology with coefficients in G affine simplex with vertices vj

general homology, cohomology; or its coefficient groups

reduced homology, cohomology boundary operator

coboundary operator Euler characteristic of X unoriented bordism oriented bordism

unoriented, oriented bordism via Thom spectra

Index

accumulation value, 4 action

diagonal, 17 effective, 17 free, 17 left, 17 proper, 329

properly discontinuous, 64 right, 17

transitive, 17 trivial, 17

weakly proper, 329 acyclic, 287, 498 additive invariant, 309 additivity axiom, 245 adjoint map, 38 adjunction space, 7 affinely independent, 198 Alexander duality, 446

Alexander–Whitney map, 240 Alexander–Whitney morphism, 241 algebra, 482

dual, 483 antipodal involution, 21 antipode, 484

approximation of the diagonal, 240 Alexander–Whitney, 240

atlas, 358

orienting, 336, 372 attaching a space, 7 attaching map, 203

barycenter, 231

barycentric coordinates, 198 barycentric subdivision, 197 base change, 117

base point, 31 nondegenerate, 102

base space, 32

basis of a topology, 2 Bernoulli numbers, 493 Betti number, 310 bi-degree, 260 bialgebra, 484 Bockstein operator, 419 bordant

orientable, 403 bordism, 521, 524

oriented, 526 Borsuk–Ulam theorem, 436 boundary, 2, 283

boundary operator, 123, 224, 245, 266, 283, 284

boundary relation, 224 bouquet, 31

Brouwer Fixed Point Theorem, 137 bundle, 62

associated, 334 framing, 532 induced, 331, 337 numerable, 342 principal, 328 projective, 471 universal, 344

bundle atlas, 336 bundle chart, 62, 336

bundle isomorphism, 331 bundle map, 331, 336 bundle morphism, 336

canonical complex line bundle, 340 cap product, 438

category topological, 334

2-category, 60

1-morphism, 61

Соседние файлы в папке топология