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Whitehead G.W. Elements of Homotopy Theory (Springer, 1978)(T)(764s)_MDat_.djvu |
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Size 5.9Mb Date May 18, 2002 |
Hopf algebras are determined...
It follows from (ii), above, that
the diagram
G*+1
H^SP^C)) * H(SP(C)),
whose horizontal arrows denote injections, is commutative...
The map
Pn ° 9n '■ (P"» P") -> (S", en) is a relative homeomorphism and the injection
Hn(V, P"~!) -> H"(V) is an isomorphism...
We shall be concerned with the homology and cohomology of the
spaces involved with coefficients in an ^-module M...
The above considerations have allowed us to reduce the proof of
Theorem E.3) to the case where B is a finite simplicial complex...
The sequences E.13), E.14) are called the Gysin sequences of the fibration
P-
The homomorphism /?* has certain multiplicative properties analogous
to those of the homomorphism a* of §3...
Suppose, however, that B and F are homo-
logically (m — l)-connected and (n — l)-connected, respectively, i.e.,
Hp(B) = 0 for all p < m and Hq(F) = 0 for all q < n...
Next, suppose that B is a ^-cellular extension of Bo, with cells {El}, q>m-
By the General Direct Sum Theorem (Theorem B.7), (Chapter II), the
injections
tf,(B0uE*,B0)-/Jr(B,Bo)
represent the groups Hr(B, Bo), Hr(X, Xo) as direct sums, and our result
now follows immediately from the case of only one cell...
Hence
G.6) The homomorphism l^ : Hq(B, C) -> Hq(P, Po) is an isomorphism for
q < m + n — 2 and an epimorphism for q = m + n — 2...
Suppose that B is «-connected (n > 1); then QBis (n — l)-connected, and
we saw in Corollary F.5) of Chapter VII that a^ is an isomorphism for
q < 2n - 1 and an epimorphism for q = In...
But more is true; the operation Sq" is
non-zero in the mapping cone of the k-fold suspension Eka, and therefore the
latter element, too, is non-zero (in other words, the element Eka is detectable
by Sq")...
Let us now introduce the Main Diagram (Figure 1), which will exhibit the
relationships among the homology groups of many of the pairs of subspaces
Hq(F) -^-> Hq+1(B)
U I t ± tlJ^>/^jJVil
Hq-i{F x F^J Ex x e, e x F kj E1 x e) *— /f?_1(F x F)
Hi(El x F,F x Fu £t x e)
!,xF,FxF)
fc4
/^
/f?(F x £0 u £j x F, F x £0 u £x x e) ^— Hq(F x Eo u £x x F,F x £0)
Hq+l(E x £, £0 x £ u £ x £j) < Hq+1(E1 x £0, F x £0 u £x x F)
Hq+1(E,X)
Figure 8.1 (Main diagram)
376 VIII The Homology Suspension
of E and E x E...
C.7) Theorem // B is n-connected and W is m-connected, then
<jL : Hq(F a W) —> Hq+l(B a W) is an isomorphism if q < m + 2n and an epi-
morphism if q = m + In + 1...
□
Just as elements of H" + r(H, n; G) correspond to cohomology operations,
so do elements of Ar(H; G) correspond to stable operations...
For if fi is the product in X, application of Sq' to both sides of the formula
u*x = xxl + lxx
yields
Sql(i*x = Sq'x x 1 + 1 x Sq'x,
the remaining terms vanishing because S^'l = 0 for j > 0...
Another application of the Steenrod process is
G.17) Theorem Let d = 2,4, or 8, and let a e n2d- i(Sd) be the homotopy class
of the Hopfmapf: S2d~1 -♦ Sd...
Let Kn = K{Z2, n); then, by Theorem G.8) of Chapter V,
Hn(Kn; Z) = Z2, Hn+1(Kn; Z) = 0
and therefore
H"(Kn; Z2) * Hom(Hn(Kn; Z), Z2) * Z2,
H" + 1(Kn; Z2) % Ext(Hn(Kn; Z), Z2) % Z2...
This time we apply both sides of (8.4) to the class b" + 2 e Hn+2(K1); of
course,
On the other hand
b+2)B-2+
= (w + 2)Sg2" + 1(b"!+3) = (n
"+ ')
-»()
Since the mod 2 binomial coefficient I I has period 4 in n, we need merely
check that the relation
2 ) + »( 2 )( 2
holds for four consecutive values of n...
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