The Serre Spectral Sequence
In this chapter we study the Serre spectral sequence \(\{ E_{*,*}^r (p),d^r \} \) over \(\mathbb{K}\, = \mathbb{Z}\) , and ℤ2, of the path space fibration $$ p\,:\,P(M) \to M, $$ with \(M = \,\mathbb{F}_k (\mathbb{R}^{n + 1} )\) or \(\mathbb{F}_{k + 1} (S
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The Serre Spectral Sequence
In this chapter we study the Serre spectral sequence {E;,*(p), dr } over II{ = Z, and Z2, of the path space fibration p: P(M) -+ M,
with M = IF k (IRn+l) or IF k+ 1 (sn+l ). Here the paths are based at an appropriate basepoint. First, in §1 we take up the case of M = IRn+l. We shall see that the spectral sequence stabilizes at the nth term, in the sense that
E:,!l(p) ~ E~*(p) ~ II{. Consequently, regarding H*({}(M); II{) as a chain algebra, with the trivial differential and II{ as a trivial chain module over it, we interpret the E~ • term of the spectral sequence as an acyclic, free resolution of II{ over H*({}(M); II{). This result, together with the fact that
H.(A(M); II{) ~ Tor H .(n(M);IK) (H*({}(M); II{), II{) (see §6 of Chapter IX), is the primary tool used in Chapter XII to study the module H.(A(M); Z2)' In §§2 and 3 we take up the case when M = sn+l. The situation here is somewhat different, and the results of §1 are adapted suitably.
1 The Case of IFk-r,r, n
>1
In keeping with the notation of Chapter II, put IFr,k-r = IFr(lRn+l - Qk-r),
IRk!~
= IR n+1 - Qk-r.
and consider the path fibration Pk-r,r : P(IFk-r,r) -+ IFk-r,r
that sends a based Moore path (0:, r) to its endpoint o:(r). Throughout this section, we assume that the homology and cohomology groups are with integral coefficients.
E. R. Fadell et al., Geometry and Topology of Configuration Spaces © Springer-Verlag Berlin Heidelberg 2001
226
XI.
The Serre Spectral Sequence
Theorem 1.1 The Serre spectral sequence {E~,*(Pk-r,r), d t } of the path fibration Pk-r,r : P(IF k-r,r) ---+ IF k-r,r has the following properties:
(i) E~,*(Pk-r,r) ~ H.(IFk-r,r) ® H*(Q(IFk-r,r)), (ii) E~,*(Pk-r,r) ~ E~,.(Pk-r,r), for r S n, and (iii) E~:t1(Pk_r,r) ~ E~.(Pk-r,r) = Z, where homology is with integral .coefficients. To start, consider the following segment of the fundamental fiber sequence Fk of IFk(lRn+1) (see Chapter II, §1):
+- IFk-t,t ..I-
+- IFk-t-1,t+1
Pk-t,t lR tn+1
+- ...
Pk-t-1,t+2 lRn+1 t+1
..I-
..I-
..I-
Recall that the vertical maps are the projections on the first nonconstant factor. Observe that
t. S t+1).) !:>! nn+1 ,..., ~t - (v }=1 - (sn)vt for r S t < k, where the notation is that of Chapter II (see Proposition 1.1 of Chapter II). The proof of Theorem 1.1 will be by induction, but first we need some preparatory work. To simplify the notation, put
E = IFk-t,t> B = lRf+1, F = IFk- t- 1,t+b P = Pk-t,t, and consider the fibration
with fiber F. The map i : F ---+ E that imbeds F as the fiber at the basepoint induces the commutative diagram
Q(F) ---+ P(F) ..I-
Vk-t,t:
Q(i)
..I-
---+F
..1-=
Q(E) ---+ Q(E, F) ---+ F
..1-=
.!.
Q(E) ---+ P(E)
..I-
---+ E
of path fibrations, where Q(E,F) = ((x, a) I a(r) = x E F} = i*P(E). The vertical maps are the natural maps induced by i or by the identity. Recall that there is a section p : lRf+1 ---+ IFk-t,t, with restrictions to the spheres {St+1j 11 S j S t} that realize the elements {at+ 1s11 S s < t + I}
1 The Case of Fk-r,r, n
>1
227
(see the pr
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