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al case of the suspension isomorphism. Corollary. For any n and q, ˜Eq(Sn) ∼= Eq−n(∗). Of course, for the theory H∗(X; π), this was immediate from our construction in terms of cellular chains. 110 DERIVATIONS OF PROPERTIES FROM THE AXIOMS 4. Axioms for reduced homology In the study of generalized homology theories, it ... |
groups. Then there is a short exact sequence iAi where α(ai) = ai − fi(ai) for ai ∈ Ai and the restriction of β to Ai is the canonical P map given by the definition of a colimit. 0 −→ iAi P β −→ colim Ai −→ 0, α−→ By the additivity axiom, we may as well assume that X and the Xi are path connected. The proof makes use of... |
UNIQUENESS THEOREMS Proof. In view of our comparison of theories on pairs of spaces and theories on pairs of CW complexes and our comparison of theories on pairs with reduced theories, it suffices to obtain a natural isomorphism of reduced theories on based CW complexes X. By the additivity axiom, we may as well assume ... |
n = ∆n × {0} is a subcomplex. We can then lift h simplex by simplex to a simplicial map ˜h : (∆n × I, ∂∆n × I ∪ ∆n × {1}) −→ (ΓX, |x, 1|) such that ˜h restricts to ˜g′ on ∆n × {0} and γ ◦ ˜h = h. 4. Simplicial objects in algebraic topology A simplicial set K∗ is a sequence of sets Kn, n ≥ 0, connected by face and degen... |
dule M we have α : H∗(X) ⊗ M −→ H∗(X ⊗ M ). We omit the proof of the following standard result, but we shall shortly give the quite similar proof of a cohomological analogue. Recall that an R-module M is said to be flat if the functor M ⊗ N is exact (that is, preserves exact sequences in the variable N ). We say that a ... |
ful to remember that, for any Noetherian ring R, the dual Hom(F, R) of a free R-module is a flat R-module. As indicated above, if Y = Hom(X, M ) and Y ′ = Hom(X ′, M ′) for chain complexes X and X ′ and R-modules M and M ′, then we also have the map of cochain complexes ω : Hom(X, M ) ⊗ Hom(X ′, M ′) −→ Hom(X ⊗ X ′, M ⊗... |
(x) is the non-zero element of H 1(RP m; Z2). By the naturality of cup products (f ∗(x))m = f ∗(xm). However, the left side is non-zero in H m(RP m; Z2) and the right side is zero since xm = 0 by our assumption that m > n. The contradiction establishes the conclu- sion. We use this fact together with covering space the... |
−→ ˜E∗(Xi). i • WEAK EQUIVALENCE If f : X −→ Y is a weak equivalence, then Q f ∗ : ˜E∗(Y ) −→ ˜E∗(X) is an isomorphism. The reduced form of the dimension axiom would read ˜H 0(S0) = π and ˜H q(S0) = 0 for q 6= 0. Theorem. A cohomology theory E∗ on pairs of spaces determines and is de- termined by a reduced cohomology t... |
o give an evaluation pairing H p(X; π) ⊗ Hp(X; ρ) −→ π ⊗ ρ. Taking π = ρ to be a commutative ring R and using its product, there results a pairing H p(X; R) ⊗R Hp(X; R) −→ R. It is usually written hα, xi for α ∈ H p(X; R) and x ∈ Hp(X; R). When R is a field and the Hp(X; R) are finite dimensional vector spaces, the adjoi... |
. An R-fundamental class of M at a subspace X is an element z ∈ Hn(M, M − X) such that, for each x ∈ X, the image of z under the map Hn(M, M − X) −→ Hn(M, M − x) induced by the inclusion (M, M − X) −→ (M, M − x) is a generator. If X = M , we refer to z ∈ Hn(M ) as a fundamental class of M . An R-orientation of M is an ... |
mmmmmmmmmmmmm Hn(V, U ∩ V ) / Hn(U ∪ V, U ∩ V ) ∂ vlllllllllllll ∂ ˜Hn−1(U ∩ V ) ∂ Hn(U, U ∩ V ) Hn(M ) 0 / Hn(M, M − y) ∼= Hn(U, U − y) i∗ ˜Hn−1(V ). Let r ∈ ker i∗. Since ˜Hn−1(U ) = 0, the bottom map ∂ is an epimorphism and there exists s ∈ Hn(U, U ∩ V ) such that ∂(s) = r. We claim that s maps to zero in Hn(U, U − ... |
index 2. This implies the first statement, and the second statement is clear. We can use homology with coefficients in a commutative ring R to construct an analogous R-orientation cover. It depends on the units of R. For example, if R = Z2, then the R-orientation cover is the identity map of M since there is a unique unit... |
m and n are congruent to 0 mod 4, the conclusion holds since the signature of the tensor product of two symmetric forms is the product of their signatures. We leave the detailed verifications of these algebraic statements as exercises for the reader. 3. Manifolds with boundary Let Hn = {(x1, . . ., xn)|xn ≥ 0} be the u... |
that q(x) = 0 if x ∈ W . Since q(x1) = 0 and φ(x1, y1) = 1, q(ax1 + y1) = 2a + q(y1) for a ∈ R. Taking a = (1 − q(y1))/2, we find q(ax1 + y1) = 1. If n = 1, this gives r ≥ 1 and completes the proof. If n > 1, define ω : V −→ R2 by ω(x) = (φ(x, x1), φ(x, y1)). Since ω(x1) = (0, 1) and ω(y1) = (1, q(y1)), ω is an epimorph... |
S If A is a subcomplex of X, then the sequence [X/A, Z] −→ [X, Z] −→ [A, Z] is exact. • ADDITIVITY If X is the wedge of a set of based CW complexes Xi, then the inclusions Xi −→ X induce an isomorphism [X, Z] −→ [Xi, Z]. If Z has a multiplication φ : Z × Z −→ Z such that the basepoint ∗ of Z is Q a two-sided unit up to... |
), 1), pn+1 is the fibration induced from the path space fibration over an EilenbergMac Lane space K(πn+1(X), n + 2) by a map kn+2 : Xn −→ K(πn+1(X), n + 2), and αn induces an isomorphism πq(X) → πq(Xn) for q ≤ n. πq(Xn) = 0 for q > n. The system can be displayed diagrammatically as follows: It follows that ... Xn+1 kn+3... |
t h : A × I −→ B be a homotopy f0 ≃ f1. Then the restrictions of h∗E over A × {0} and A × {1} can be identified with f ∗ 0 E and f ∗ 1 E. Thus we change our point of view and consider a general n-plane bundle p : E −→ B × I. It suffices to show that the restrictions E0 and E1 of E over B × {0} and B × {1} are equivalent. ... |
d p∗ (1) w0(ξ) = 1 and wi(ξ) = 0 if i > dim ξ. (2) w1(γ1) 6= 0, where γ1 is the universal line bundle over RP ∞. (3) wi(ξ ⊕ ε) = wi(ξ). (4) wi(ζ ⊕ ξ) = i j=0 wj(ζ) ∪ wi−j (ξ). Every mod 2 characteristic class for n-plane bundles can be written uniquely as a polynomial in the Stiefel-Whitney classes {w1, . . ., wn}. P T... |
construction is functorial with respect to maps of vector bundles. Remark. If we give the bundle ξ a Euclidean metric and let D(E) and S(E) denote its unit disk bundle and unit sphere bundle, then there is an evident homeomorphism between T ξ and the quotient space D(E)/S(E). In turn, D(E)/S(E) is homotopy equivalent t... |
map of Thom spaces carries the orientation of the target bundle to the orientation of the source bundle. The universal bundle ˜γn has a canonical orientation which determines an orientation on f ∗ ˜En for any map f : B −→ BSO(n). Theorem. The natural transformation Φ : [−, BU (n)] −→ E Un(−) obtained by sending the hom... |
n element of O(q), and the isotropy group of x0 is O(q − n). Thus the action of O(q) is transitive, and evaluation on x0 induces a homeomorphism O(q)/O(q − n) −→ Vn(Rq) of O(q)-spaces. The action of O(n) ⊂ O(q) is free, and passage to orbits gives a homeomorphism O(q)/O(n)×O(q−n) −→ Gn(Rq). It is intuitively clear and ... |
ΓE of sections of E is a vector space under fiberwise addition and scalar multiplication. Using a partition of unity argument, one can show that there is a finite dimensional vector subspace V of Γ(E) such that the map g : X × V −→ E specified by g(x, s) = s(x) is an epimorphism of bundles over X. The resulting short exa... |
inclusion of CP ∞ = BU (1) in BU . One can calculate H∗(ΩSU ) and see that it too is a polynomial algebra with an explicitly given generator in each even degree. A direct inspection of the map β shows that it carries generators to generators. In any case, it should now be clear that we have a periodic Ω-prespectrum an... |
ritten as a polynomial in the elementary symmetric polynomials p∗(ck(E)). Application of this polynomial to the ck(E) gives ˆf (E). (For vector bundles E over non-connected spaces X, we add the elements obtained by restricting E to the components of X.) By the universal property of K(X), ˆf extends to a homomorphism ˆf... |
nsional reasons, hence ˜H ∗(X) is free Abelian on generators x = [i] and y = [j]. Define an integer h(f ), the Hopf invariant of f , by x2 = h(f )y. We usually regard h(f ) as defined only up to sign (thus ignoring problems of orientations of cells). Note that h(f ) depends only on the homotopy class of f . If n is odd, ... |
s cobordant to ∅. Thus M = −M and Nn is a vector space over Z2. Cartesian product of manifolds defines a multiplication Nm × Nn −→ Nm+n. This operation is bilinear, associative, and commutative, and the zero dimensional manifold with a single point provides an identity element. We conclude that N∗ is a graded Z2-algebra... |
t square commutes up to homotopy by the definition of φq,r. Sm+q ∧ Sn+r t∧t / T νm ∧ T νN / T O(q) ∧ T O(r) ∼= φq,r Sm+q+n+r t / T (νM×N ) / T O(q + r). This implies the claimed multiplicativity of the maps α. 3. Prespectra and the algebra H∗(T O; Z2) Calculation of the homotopy groups π∗(T O) proceeds by first computing... |
izing the unit of an algebra, a Z2-coalgebra is required to have a counit ε : C −→ Z2. We understand all of these algebraic structures to be graded, and we say that a coalgebra is connected if Ci = 0 for i < 0 and ε : C0 −→ Z2 is an isomorphism. When considering the Hurewicz homomorphism of π∗(T O), we shall need the f... |
X −→ ΩΣX and ΣΩX −→ X are isomorphisms in ¯hS . In particular, up to isomorphism, every object in the category ¯hS is a suspension, hence a double suspension. This implies that each [X, Y ] is an Abelian group and composition is bilinear. Moreover, for any map f : X −→ Y , the canonical map F f −→ ΩCf and its adjoint ... |
hem; the reader can find more of the original references in the sources given. 1. A classic book and historical references The axioms for homology and cohomology theories were set out in the classic: S. Eilenberg and N. Steenrod. Foundations of algebraic topology. Princeton University Press. 1952. I believe the only his... |
5), 211–264. The classic analysis of the structure of the Steenrod algebra as a Hopf algebra: J. Milnor. The Steenrod algebra and its dual. Annals of Math. 67(1958), 150–171. Two classic papers of Adams; the first constructs the Adams spectral sequence relating the Steenrod algebra to stable homotopy groups and the seco... |
cal product of the spaces .Xj ; Oj /. The next proposition shows that X D Xj together with the projections prj is a categorical product of the family .Xj / in the category TOP. Note that for infinite J , open sets in the product are quite large; a Uj , Uj Xj open, is then in general not an open subset of product Xj . Q ... |
s again locally compact. Let X be a Hausdorff space and assume that each point has a compact neighbourhood. Let U be a neighbourhood of x and K a compact neighbourhood. Since K is normal, K \ U contains a closed neighbourhood L of x in K. Then L is compact and a neighbourhood of x in X. Therefore X is locally compact. ... |
ubgroup is often denoted by 1 (in a multiplicative notation) or by 0 (in an additive notation). The neutral element will also be denoted 1 or 0. The symbol H C G is used for a normal subgroup H of G. The notation H K or H G K means that H and K are conjugate subgroups of G. A homomorphism f W G ! H between topological ... |
x and hence gx D p.g/ an interior point of gW 2x V x D p.V /. This shows that p is open. (1.8.7) Corollary. Let the locally compact Hausdorff group G with countable basis act on a locally compact Hausdorff space X. An orbit is locally compact if and only if it is locally closed. An orbit is a homogeneous space with res... |
oup. This symmetry influences almost every other tool of algebraic topology (although we do not always carry out this influence in this text). The chapter contains two sections on point-set topology. We discuss standard spaces like spheres, disks, cells, simplices; they will be used in many different contexts. We present... |
/ to g.x/ is always contained in A. Then H.x; t/ D .1 t/f .x/ C tg.x/ is a homotopy from f to g (linear homotopy). It will turn out that many homotopies are constructed from linear homotopies. A set A Rn is star-shaped with respect to a0 2 A if for each a 2 A the line-segment from a0 to a is contained in A. If A is sta... |
ted contractible with respect to .0; 0/. Let Y D X be another comb space. Then X [ Y Š X _ Y is not contractible. Since .X [ Y /=Y is homeomorphic to X, we see that it does not suffice in (2.1.7) to assume that A is contractible. These counterexamples indicate the need for base points with additional (local) proper- j X... |
.0/ is homotopy equivalent to the product of k factors S 1. j 2.4 Mapping Spaces and Homotopy It is customary to endow sets of continuous maps with a topology. In this section we review from point-set topology the compact-open topology. It enables us to consider a homotopy H W X I ! Y as a family of paths in Y , param... |
he constant path. (5) ku.0/ u ' u ' u ku.1/. 2 implies u1 u2 ' u0 1 and u2 ' u0 1 u0 2. Proof. (1) H W .s; t/ 7! u.s.1 t/ C t˛.s// is a homotopy from u to u˛. ˛.t/ D 2t for t 1 (2) The relation .u1 .u2 u3//˛ D .u1 u2/ u3 holds for ˛ defined as 4 t 1 4 for 1 2 for 1 2 t 1. i then G W u1 u2 ' u0 2 for G defined as G.s; t /... |
to H on the upper boundary path from .0; 0/ to .1; 1/. These paths differ from H0 and H1 by composition with a constant path. Finally, from the construction we conclude that is a functor. (2.6.2) Theorem (Seifert–van Kampen). Let X0 and X1 be subspaces of X such that the interiors cover X. Let i W X01 D X0 \ X1 ! X and... |
espect to w.0/ and w.1/ are equal. The complement C n f .S 1/ decomposes into open path components, and the winding number with respect to x is constant as long as x stays within a component. Let u W I ! C X fxg be a loop. Then there exists a unique continuous map f W S 1 ! C X fxg such that f ı p0 D u. The winding num... |
punctured half-spaces Þ f.x; y/ j x < 1=3; x 6D 1g and f.x; y/ j 1=3 < x; x 6D 1g . In the example in 2.8.6 one cannot apply (2.6.2) directly to the covering of S 1_S 1 by the two summands, since the interiors do not cover the space. The general method in cases like this is to first “thicken” the subspaces up to h-equi... |
X @I . Proof. ˆ.x; s; t/ D K.x; .1 t/s C t˛.s// is a suitable homotopy. (2.9.2) Proposition. The data for ….X; Y / satisfy the axioms of a category. The category is a groupoid. Proof. The associativity of the composition follows, because .K L/ M D K .L M / ı .id ˛/; 4 , ˛.t/ D t C 1 4 for 1 4 t 1 2 , ˛.t/ D t 2 C 1 2 w... |
pen set U such that q is trivial over U Œti ; tiC1. For the classification of covering spaces we need spaces with suitable local properties. A space X is called locally connected (locally path connected) if for each x 2 X and each neighbourhood U of x there exists a connected (path connected) neighbourhood V of x which ... |
with initial point y. Then V .1/ 2 Fb, and V shows 0.i/.ŒV .1/ D Œy. There is more algebraic structure in the sequence. (3.2.7) Proposition. The pre-images of elements under @x are the left cosets of 1.B; b/ with respect to p1.E; x/. The pre-images of 0.i/ are the orbits of the 1.B; b/-action on 0.Fb; x/. Proof. Let @x... |
tors ….C / ! SET yields a functor TRA.f / W TRAC ! TRAB . These functors are compatible TB ı COV.f / D TRA.f / ı TC W COVC ! TRAB . 3.4 Connected Groupoids In this section the space B is assumed to be path connected. A functor ….B/ ! SET is an algebraic object. The category of these functors has an equivalent descripti... |
p/1.E E/ p1.E/. This inclusion holds, since (using (2.7.3)) m.pp/Œ.w1; w2/ D Œpw1pw2 D Œpw1pw2 D Œp.w1w2/ D pŒw1w2: From the uniqueness of liftings one shows that M is associative. In a similar manner we see that (passage to) the inverse in X has a lifting to E, and uniqueness of liftings shows that the result is an i... |
erefore the inclusion S 1 ! M.S 1/ is an h-equivalence. The space H.S 1/ of homeomorphisms of S 1 is h-equivalent to O.2/. Chapter 4 Elementary Homotopy Theory Further analysis and applications of the homotopy notion require a certain amount of formal consideration. We deal with several related topics. (1) The construc... |
omotopies ˛ˇ ' id and ˇ˛ ' id are induced by a linear homotopy in the I -coordinate. The reader should verify that ˇ and the homotopies are continuous. The covering X˙ of X is numerable if the projection pN has a section. A section is determined by its second component s W X ! Œ0; 1, and a function of this type defines ... |
map. A coassociative comonoid with coinverse in h-TOP0 is called a cogroup in h-TOP0. Let .C; / be a coassociative comonoid and Y a space. A left coaction of C on Y (up to homotopy) is a map W Y ! C _ Y such that .id _/ ' . _ id/ and prY ' id. The suspension †X is such a cogroup. We define the comultiplication W †X ! †... |
V 0 g ŒB; W 0 is exact. If we apply this to id.U /, we see that gf is null homotopic. We need the dual form of the cone. Let F .Y / D fw 2 Y I j w.0/ D g be the space of paths which start in the base point of Y , with the constant path k.t/ D as base point, and evaluation e1 W F Y ! Y , w 7! w.1/. Via adjunction we hav... |
teresting fact that one need not assume A to be closed. Strøm [180, Theorem 2] proved that an inclusion A X is a cofibration if and only if the subspace X 0 [ A I is a retract of X I . If we multiply a retraction by id.Y / we obtain again a retraction. Hence AY ! X Y is a (closed) cofibration for each Y , if i W A ! X is... |
id//# ı 0 D 1, if we set iŒf D Œ i f . Proof. Use that t f is an extension of t g and apply the definition. 108 Chapter 5. Cofibrations and Fibrations (5.2.4) Proposition. Let f W X ! Y be an ordinary homotopy equivalence and i W K ! A a cofibration. Then f W Œ.A; i/; .X; g/K ! Œ.A; i/; .Y; fg/K is bijective. Proof. Let g... |
ofibre of f . 3. h i0; i1 i W X _ X ! XI is an embedding. 4. Let f W X ! Y and g W Y ! Z be pointed maps. We have canonical maps ˛ W C.f / ! C.gf / and ˇ W C.gf / ! C.g/; ˛ is the identity on the cone and maps Y by g, and ˇ is the identity on Z and maps the cone by f id. Show that ˇ is the pointed homotopy cofibre of ˛. ... |
ll subcategory of h-TOPC with objects the fibrations over C . (5.5.9) Proposition. Let p W X ! B be a fibration. We assign to f W C ! B the induced fibration pf W Yf ! C and to the morphism Œ' W f ! g in ….C; B/ the morphism Œ'. This yields a functor ….C; B/ ! h-FIBC . Since ….C; B/ is a groupoid, Œ' is always an isomorph... |
classes of maps of triples f W .I nC1; @I nC1; J n/ ! .X; A; /. (Recall that this means f .@I nC1/ A; f .J n/ fg, and for homotopies H we require Ht for each t 2 I to be a map of triples.) Thus, with notation introduced earlier, nC1.X; A; / D Œ.I nC1; @I nC1; J n/; .X; A; /: A group structure Ci , 1 i n is defined agai... |
v# is always bijective, homotopy groups associated to base points in the same path component are isomorphic. We list some naturality properties of the transport functors. As a special case of the functor property we obtain right actions of the fundamental groups: n.X; x/ 1.X; x/ ! n.X; x/; n.X; A; a/ 1.A; a/ ! n.X; A;... |
6) Example. Recall the Hopf fibration p W S 2nC1 ! CP n (14.1.9). The exact sequence (6.3.2) and i .S 1/ D 0 for i > 1 yield the isomorphisms p W i .S 2nC1/ Š i .CP n/; for i 3I and in particular i .S 3/ Š i .S 2/ for i 3, since CP 1 is homeomorphic to S 2 Þ (the Riemann sphere). (6.3.7) Example. From linear algebra one... |
determinant. Thus it suffices to show that for some A with det.A/ D 1 we have d.LA/ D 1. By the preceding discussion and (6.1.4) we see that .x1; : : : ; xn/ 7! .x1; x2; : : : ; xn/ has degree 1. The stereographic projection (6.1.4) now shows that the map S n ! S n which changes the sign of the first coordinate has degr... |
ause J is finite. Each Bj is contained in some Ak, hence the Ak cover n. Moreover, by construction, Ak \ @kn D ;. We can therefore apply part (9) of (6.6.1) and find an x in the intersection of the Ak. Hence for each 6.7. Higher Connectivity 141 k there exists ik such that x 2 Bik . Since each Bj is contained in exactly ... |
f . Let now W be an homotopic relative @I n to F jK n-dimensional cube, say with W K0. Then @W K 0 and h.@W / D h0.@W / X0. Since .Y0; X0/ is n-connected, we can deform the map relative to @W to a map into X0. 01 and yield a map 6.7.10) Corollary. Let f W X ! Y be an n-connected map between well-pointed spaces. Then †... |
nt, that ˆ can be deformed as an admissible map into a map with image in Y1. We apply the preparation theorem to ˆ and obtain a certain map ‰. The deformation in (6.9.2) stays inside admissible maps. Consider the projection W I n I ! I n. We claim that the images of ‰1.Y X Y1/ and ˆ1.Y X Y2/ under are disjoint. Let y 2... |
ath connected. The natural maps k.S n/ ! k.Y S n; S n/ ! k.Y S n= S n/ ! k.S n/ are isomorphisms for 1 k n. (6.10.9) Proposition. Suppose i .X/ D 0 for i < p . 0/ and i .Y / D 0 for i < q . 0/. Then i .X ? Y / D 0 for i < p C q C 1. Proof. In the case that p D 0 there is no condition on X. From the definition of the joi... |
somorphic in ST. The image ST.f / of f W X ! Y in ST.X; Y / is called the stable homotopy class of f . Maps f; g W X ! Y are stably homotopic if and only if they represent the same element in ST.X; Y /. The groups ST.S k; S 0/ D Þ colimn nCk.S n/ are the stable homotopy groups of the spheres. 162 Chapter 7. Stable Homo... |
. Proof. In the category of compactly generated spaces C.X; A/ ^ C.Y; B/ is a quotient of under the following relations: .a; 0; y/ .a; y/, .x; b; 0/ .x; b/, .a; 0; b; t/ .a; b; t/, .a; s; b; 0/ .a; s; b/, and A 1 B I [ A I B 1 is identified to a base point . In a first step we show that the smash product is homeomorphic ... |
f and 'gf a scaling function for gf , then 1 W Y Z ! 0; 1Π; .y; z/ 7! 'f .y/; 2 W Y Z ! 0; 1Π; .y; z/ 7! 'gf .z/ are scaling functions for h. We have a factorization D2.h/ D D2 D1 we use Qh D . Qf ; gf / with gf D Qg Qf . The diagram 2.h/.x; y; z/ D .x; y Qf .x/; z/ and D2 2.h/ where 2.h/.x; y; z/ D .x; y; z gf .x// ... |
mutative C n ^ C m .1^/ ^1 C.X; ;/ ^ C.RnjX/ ^ C m C.f /^1^1 C.Y; ;/ ^ C.RnjD/ ^ C.RmjY / 1^D#f C.Y; ;/ ^ C.RnjX/ ^ C m: Proof. We unravel the definitions and deform suitable maps between pairs. The composition .1 ^ D#f /. ^ 1/.1 ^ / is induced by maps Rnj0 Rmj0 RnjD RmjD RnjD RmjY Y jY RnjX Rmj0 ˛ RnjD W jY with ˛.x; y... |
ive g W S nCkC2 ! S 2 ^ Z ^ E.n/. Then f W S nCkC2 g ! S 2 ^ Z ^ E.n/ ! Z ^ E.n/ ^ S 2 e2 ! Z ^ E.n C 2/ represents an element y 2 Ek.Z/. Here e2 is the composition of the spectral structure maps E.n/ ^ S 2 D .E.n/ ^ S 1/ ^ S 1 ! E.n C 1/ ^ S 1 ! E.n C 2/: We show 2 surjective and injective and hence the same holds for... |
. Then k1.f 1.U // is open, and this shows what we want. (2) ) (1). We show that the identity X ! kX is continuous. This holds by (2) and because X and kX have the same test maps. (7.9.5) Proposition. Let f W X ! Y be continuous. Then the same set map kf W kX ! kY is continuous. The assignments X 7! kX, f 7! kf yield a... |
ctly Generated Spaces 193 Proof. It suffices to treat the case g D id, since a composition of quotient maps is a quotient map. Using (7.9.18), the proof is now analogous to (2.4.6). (7.9.20) Proposition. Let f W X ! Y be a quotient map and X a whk-space. Then Y is a whk-space if and only if R D f.x1; x/ j f .x1/ D f .x2... |
d Þ finite subsets of P . (8.1.3) Example. Let K D .E; S/ be a simplicial complex. Define a partial order on S by s t , s t. The simplicial complex K0 associated to this ordered set Þ is called the barycentric subdivision of K. Let K D .E; S/ be a simplicial complex. We denote by jKj the set of functions ˛ W E ! I such t... |
ell e in Y . We have to show that L is closed in Y . We show that L is closed in X. Let f be a cell of X. By (W3), xf is contained in a finite union e1 [ [ ek of cells. Let e1; : : : ; ej be those which are contained in Y . Then xf \ Y e1 [ [ ej xe1 [ [ xej Y since Y is a subcomplex. Hence xf \ Y D . xf \ xe1/ [ [ . xf ... |
ition (8.3.6). We can also form the colimit S 1 of S n S nC1 , a CW-complex with two cells in each Þ dimension. The general topology of adjunction spaces and colimit topologies gives us the next results. (8.3.8) Proposition. Let .X; A/ be a relative CW-complex. If A is a T1-space, then X is a T1-space and a compact sub... |
roof. By the suspension theorem (6.10.4), the map is .2n C 1/-connected. Now use the pointed version of (8.4.2). (8.4.9) Theorem. Let X be a finite pointed CW-complex. Then † W Œ†kX; †kY 0 ! Œ†kC1X; †kC1Y 0 is bijective for dim.X/ k 1. Proof. We have dim †kX D k C dim X. The space †Y is path connected. By the theorem of... |
n the n-connective covering of X. The induced map i .j X n / W i .Xhni/ ! i .X/ is an isomorphism for i > n and i .Xhni/ D 0 for i n. The universal covering has such properties in the case that n D 1. So we have a generalization, in the realm of fibrations. Objects of this type occur in the theory of Postnikov decomposi... |
t homomorphism. Hence this map is null homotopic. Commutativity is verified in a similar manner by applying n. See also Problem 1. For the construction of the product pairing we need a general result about products for homotopy groups. We take the smash product of representatives f W I m=@I m ! X, g W I n=@I n ! Y and o... |
that a homomorphism from Sn.X/ is determined by its values on the basis elements W n ! X, and these values can be prescribed arbitrarily. The boundary operator @q is defined for q 1 by P @q W Sq.X/ ! Sq1.X/; 7! P q iD0.1/i d q i ; and for q 0 as the zero map. Basic for everything that follows is the 9.1.2 Boundary rela... |
SnC1.X I / such .Kn/ @sX n C sX n1@ D 1 n.X/ 0 n.X/ (chain homotopy), and such that for continuous X ! Y the relations .Nn/ .f id/# ı sX n D sY n ı f# hold (naturality). We construct the sn inductively. n D 0. In this case, s0 sends the 0-simplex W 0 ! fxg X to the 1-simplex s0 W 1 ! X I , .t0; t1/ 7! .x; t1/. Then the... |
\ Sn.Y2/. The inclusion S.Y2/ ! S.Y / induces therefore, by an isomorphism theorem of elementary algebra, Sn.Y2/ Sn.Y1 \ Y2/ D Sn.Y2/ Sn.Y1/ \ Sn.Y2/ Š Sn.Y1/ C Sn.Y2/ Sn.Y1/ D S U n .Y / Sn.Y1/ : By (9.4.5) and (11.2.7) we see, firstly, that S U .Y /=S.Y1/ ! S.Y /=S.Y1/ and, altogether, that S.Y2/=S.Y1 \ Y2/ ! S.Y /=S... |
lly homotopic (similarly for Q; Q0). (2) An Eilenberg–Zilber morphism P is associative and commutative up to natural homotopy, i.e., the natural transformations PXY;Z ı .PX;Y ˝ 1/ and 9.7. The Theorem of Eilenberg and Zilber 239 PX;Y Z ı .1 ˝ PY;Z/ from S.X/ ˝ S.Y / ˝ S.Z/ to S.X Y X/ are naturally homotopic and the tr... |
ferent form. Suppose Y1; Y2 are subspaces of Y such that Y D Y ı 2 . Then the inclusion induces an isomorphism hn.Y1; Y1 \ Y2/ Š hn.Y; Y2/. 1 [Y ı The module hn.X; A/ is the n-th homology group (module) of .X; A/ in the homology theory (we also say in degree or in dimension n). We set hn.X; ;/ D hn.X/. The groups hn.X/... |
an h-equivalence. We can interchange the roles of 0 and 1; let denote this suspension isomorphism. By applying the Hexagon Lemma (11.1.3) with center group hn.I X; I A [ @I X/ we obtain D (draw the appropriate diagram). For some purposes of homotopy theory one has to use a similar suspension isomorphism defined with X I... |
monoid in h-TOP, i.e., W C ! C _ C is a pointed map such that the composition with the inclusions of the summands is pointed homotopic to the identity. Then we have the -sum in each homotopy set ŒC; Y 0, defined by Œf C Œg D Œı.f _ g/ with the folding map ı D hid; id i W Y _ Y ! Y . Let us write h D Qhn. (10.4.4) Propos... |
2 K such that Ux \ K D fxg. In that case L x2K Hn.Ux; Ux X x/ Š Hn.U; U X K/; and Hn.Ux; Ux X x/ Š Z, by excision and Hn.Dn; S n1/ Š Z. We call the integer d.f; x/ defined by fzUx ;x D d.f; x/zS n;p the local degree of f at p. With this notation we therefore have Þ d.f / D P x2K d.f; x/. Let U Rn be an open neighbourhoo... |
1. The computation D.f /.uC uC/ D D.f /vC D fvC D fvC D f.uC u/ D .a b/.uC u/ finally yields the second assertion D.f / D a b. (10.6.5) Example. The map f W S 1 ! S 1, z 7! z2kC1 satisfies the hypothesis of (10.6.4). We know already that D.f / D 2k C 1 and d.f / D 1, hence a D k C 1 and b D k. Let dC denote the singular... |
struct a space, a .2n C 1/-manifold, X by identifying in DnC1 S n C DnC1 S n the point .x; y/ 2 S n S n in the first summand with ˛.x; y/ in the second summand via a homeomorphism ˛ D .˛1; ˛2/ of S n S n as above. The two summands are embedded as X1 and X2 into X. We use the MV-sequence of .XI X1; X2/ to determine the i... |
1B [ 0A/. Let j be the isomorphism j W hn1.1B [ 0A; 0A/ Š hn1.1B/ Š hn1.B/: By diagram chasing one verifies .B/j˛ D Q@ and jˇ .A;B/ D @. (10.9.3) Lemma. Let .A; B; C / be a triple. Then we have an isomorphism h1; 0; i W hn.A; B/˚hn.A; B/˚hn.IB; @IB [IC / ! hn.@IA[IB; @IB [IC /: Here is induced by the inclusion, and by a... |
diagrams (Ke D kernel, Ko D cokernel, Im D Image). Ke.˛/ a Ke.ˇ/ b Ke./ .1/ D .2/ Ke.a0˛/ a Ke.ˇ/ b Ke./ \ Im.b/ a0 Ko.ˇ/ b0 Ko./ Ko.˛/ .3/ A0 Im.˛/ C Ke.a0/ a0 D B 0 Im.ˇ/ b0 .4/ C 0 Im.b/ 280 Chapter 11. Homological Algebra The morphisms named a; b; a0; b0 are induced by the original morphisms with the same name by ... |
ule. We apply the functor HomR.; R/ to C and obtain a chain complex C D .C n; ın/ of R-modules with C n D HomR.Cn; R/ and the R-linear map ın W C n D HomR.Cn; R/ ! HomR.CnC1; R/ D C nC1 defined by ın.'/ D .1/nC1' ı @nC1 for ' 2 Hom.Cn; R/. For the choice of this sign see 11.7.4. The reader will find different choices of ... |
e). This composition is associative, as it should be. When we use the degree as upper index (e.g., in cohomology), then the agreement Ak D Ak is sometimes Þ suitable. 11.7.2 Graded algebras. A Z-graded R-algebra A is a Z-graded R-module .An j n 2 Z/ together with a family of R-linear maps Ai ˝R Aj ! AiCj ; x ˝ y 7! x y... |
ain complex Hom.P; A/; its i-th cohomology group (i 1) is denoted Exti R.C; A/. Since projective resolutions are unique up to chain equivalence, the Exti R-groups are unique up to isomorphism. For principal ideal domains only Ext1 occurs, since we have resolution of length 1. The notation Ext has its origin in the noti... |
phic to the homology groups of the homology theory (if it satisfies the dimension axiom). From this fact one obtains immediately qualitative results and explicit computations of homology groups. Thus if X has k.n/ n-cells, then Hn.XI Z/ is a subquotient of the free abelian group of rank k.n/. A finite cell complex has fin... |
n/ @ Hn.X n/ Hn.X nC1/ +’’’’’’’’’’’’ Š 0 0 Hn.X/ shows us that we have an induced isomorphism .a/ (Five Lemma). (12.2.2) Corollary. Suppose X has a finite number of n-cells; then Hn.XI Z/ is a finitely generated abelian group. Let X be n-dimensional; then Hk.XI G/ D 0 for k > n. (12.2.3) Example (Real projective space).... |
and C a chain complex of finite length in i0.1/i .Hi .C// the Euler i0.1/i .Ci / D P P M, then we call .C/ D characteristic of C with respect to . (12.4.3) Proposition. Let 0 H 0 0 H0 H 00 0 H 0 1 H1 H 00 1 H 0 2 H2 be an exact sequence of modules in M which consists eventually of zero-modules. i0.1/i .Hi / and similarl... |
ch arise as orbit maps from an action are of a more special type. If x 2 F is a ramification point, then so is each point in p1px, and these points have the same ramification index, since points in the same orbit have conjugate isotropy groups. Let C1; : : : ; Cr be the orbits with non-trivial isotropy group and let nj d... |
K/ of positive real numbers such that, for t1; t2 2 Œ0; 1, t1 < t2 and jt1 t2j < .k/, there exist a j 2 J with Vk Œt1; t2 Uj . Proof. Let .tj j j 2 J / be a numeration of .Uj /. For each r-tuple k D .j1; : : : ; jr / 2 J r , define a continuous map vk W B ! I; x 7! min tji .x; s/ j s 2 rY iD1 i1 rC1 ; iC1 rC1 : S 1 rD1... |
uivalences. Þ (13.3.3) Theorem. Let p W X ! B and q W Y ! B be spaces over B and f W X ! Y a map over B. Let .Uj j j 2 J / be a numerable covering of B. Let fj W p1.Uj / ! q1.Uj / be the map induced by f over Uj . Suppose each fj is a fibrewise homotopy equivalence. Then f is a fibrewise homotopy equivalence. Proof. The ... |
ions rg W E ! E, x 7! xg. This is due to the fact that E G is homeomorphic to the topological sum qg2GE fgg, ifG is discrete. Let E G ! E be a free action and set C.E/ D f.x; xg/ j x 2 E; g 2 Gg. We call t D tE W C.E/ ! G, .x; xg/ 7! g the translation map of the action. (14.1.1) Lemma. Let p W E ! E=G be locally trivia... |
phisms of the fibre which arise from an action of an element of the group G. Let p W Y ! B be a right G-principal bundle. It may happen that there exists a right H -principal bundle q W X ! Y for a subgroup H G and a G-homeomorphism W X H G ! Y over B. In that case .q; / is called a reduction of the structure of p. One ... |
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