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Thom-Gysin sequence

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Algebraic topology

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

The Thom-Gysin sequence is a type of long exact sequence in cohomology induced by a spherical fibration and expressing the cohomology groups of the total space in terms of those of the base plus correction. The sequence may be obtained as a corollary of the Serre spectral sequence for the given fibration. It induces, and is induced by, the Thom isomorphism.

Statement

Proposition

Let RR be a commutative ring and let

S n E π B \array{ S^n &\longrightarrow& E \\ && \downarrow^{\mathrlap{\pi}} \\ && B }

be a Serre fibration over a simply connected CW-complex with typical fiber (exmpl.) the n-sphere.

Then there exists an element cH n+1(E;R)c \in H^{n+1}(E; R) (in the ordinary cohomology of the total space with coefficients in RR, called the Euler class of π\pi) such that the cup product operation c()c \cup (-) sits in a long exact sequence of cohomology groups of the form

H k(B;R)π *H k(E;R)H kn(B;R)c()H k+1(B;R). \cdots \to H^k(B; R) \stackrel{\pi^\ast}{\longrightarrow} H^k(E; R) \stackrel{}{\longrightarrow} H^{k-n}(B;R) \stackrel{c \cup (-)}{\longrightarrow} H^{k+1}(B; R) \to \cdots \,.

(e.g. Switzer 75, section 15.30, Kochman 96, corollary 2.2.6)

Proof

Under the given assumptions there is the corresponding Serre spectral sequence

E 2 s,t=H s(B;H t(S n;R))H s+t(E;R). E_2^{s,t} \;=\; H^s(B; H^t(S^n;R)) \;\Rightarrow\; H^{s+t}(E; R) \,.

Since the ordinary cohomology of the n-sphere fiber is concentrated in just two degees

H t(S n;R)={R fort=0andt=n 0 otherwise H^t(S^n; R) = \left\{ \array{ R & for \; t= 0 \; and \; t = n \\ 0 & otherwise } \right.

the only possibly non-vanishing terms on the E 2E_2 page of this spectral sequence, and hence on all the further pages, are in bidegrees (,0)(\bullet,0) and (,n)(\bullet,n):

E 2 ,0H (B;R),andE 2 ,nH (B;R). E^{\bullet,0}_2 \simeq H^\bullet(B; R) \,, \;\;\;\; and \;\;\; E^{\bullet,n}_2 \simeq H^\bullet(B; R) \,.

As a consequence, since the differentials d rd_r on the rrth page of the Serre spectral sequence have bidegree (r+1,r)(r+1,-r), the only possibly non-vanishing differentials are those on the (n+1)(n+1)-page of the form

E n+1 ,n H (B;R) d n+1 E n+1 +n+1,0 H +n+1(B;R). \array{ E_{n+1}^{\bullet,n} & \simeq & H^\bullet(B;R) \\ {}^{\mathllap{d_{n+1}}}\downarrow \\ E_{n+1}^{\bullet+n+1,0} & \simeq & H^{\bullet+n+1}(B;R) } \,.

Now since the coefficients RR is a ring, the Serre spectral sequence is multiplicative under cup product and the differential is a derivation (of total degree 1) with respect to this product. (See at multiplicative spectral sequence – Examples – AHSS for multiplicative cohomology.)

To make use of this, write

ι1H 0(B;R)E n+1 0,n \iota \coloneqq 1 \in H^0(B;R) \stackrel{\simeq}{\longrightarrow} E_{n+1}^{0,n}

for the unit in the cohomology ring H (B;R)H^\bullet(B;R), but regarded as an element in bidegree (0,n)(0,n) on the (n+1)(n+1)-page of the spectral sequence. (In particular ι\iota does not denote the unit in bidegree (0,0)(0,0), and hence d n+1(ι)d_{n+1}(\iota) need not vanish; while by the derivation property, it does vanish on the actual unit 1H 0(B;R)E n+1 0,01 \in H^0(B;R) \simeq E_{n+1}^{0,0}.)

Write

cd n+1(ι)E n+1 n+1,0H n+1(B;R) c \coloneqq d_{n+1}(\iota) \;\; \in E_{n+1}^{n+1,0} \stackrel{\simeq}{\longrightarrow} H^{n+1}(B; R)

for the image of this element under the differential. We will show that this is the Euler class in question.

To that end, notice that every element in E n+1 ,nE_{n+1}^{\bullet,n} is of the form ιb\iota \cdot b for bE n+1 ,0H (B;R)b\in E_{n+1}^{\bullet,0} \simeq H^\bullet(B;R).

(Because the multiplicative structure gives a group homomorphism ι():H (B;R)E n+1 0,0E n+1 0,nH (B;R)\iota \cdot(-) \colon H^\bullet(B;R) \simeq E_{n+1}^{0,0} \to E^{0,n}_{n+1} \simeq H^\bullet(B;R), which is an isomorphism because the product in the spectral sequence does come from the cup product in the cohomology ring, see for instance (Kochman 96, first equation in the proof of prop. 4.2.9), and since hence ι\iota does act like the unit that it is in H (B;R)H^\bullet(B;R)).

Now since d n+1d_{n+1} is a graded derivation and vanishes on E n+1 ,0E_{n+1}^{\bullet,0} (by the above degree reasoning), it follows that its action on any element is uniquely fixed to be given by the product with cc:

d n+1(ιb) =d n+1(ι)b+(1) nιd n+1(b)=0 =cb. \begin{aligned} d_{n+1}(\iota \cdot b) & = d_{n+1}(\iota) \cdot b + (-1)^{n}\, \iota \cdot \underset{= 0}{\underbrace{d_{n+1}(b)}} \\ & = c \cdot b \end{aligned} \,.

This shows that d n+1d_{n+1} is identified with the cup product operation in question:

E n+1 s,n H s(B;R) d n+1 c() E n+1 s+n+1,0 H s+n+1(B;R). \array{ E_{n+1}^{s,n} & \simeq & H^s(B;R) \\ {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}} \\ E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R) } \,.

In summary, the non-vanishing entries of the E E_\infty-page of the spectral sequence sit in exact sequences like so

0 E s,n ker(d n+1) E n+1 s,n H s(B;R) d n+1 c() E n+1 s+n+1,0 H s+n+1(B;R) coker(d n+1) E s+n+1,0 0. \array{ 0 \\ \downarrow \\ E_\infty^{s,n} \\ {}^{\mathllap{ker(d_{n+1})}}\downarrow \\ E_{n+1}^{s,n} & \simeq & H^s(B;R) \\ {}^{\mathllap{d_{n+1}}}\downarrow && \downarrow^{\mathrlap{c \cup (-)}} \\ E_{n+1}^{s+n+1, 0} & \simeq & H^{s+n+1}(B;R) \\ {}^{\mathllap{coker(d_{n+1})}}\downarrow \\ E_\infty^{s+n+1,0} \\ \downarrow \\ 0 } \,.

Finally observe (lemma ) that due to the sparseness of the E E_\infty-page, there are also short exact sequences of the form

0E s,0H s(E;R)E sn,n0. 0 \to E_\infty^{s,0} \longrightarrow H^s(E; R) \longrightarrow E_\infty^{s-n,n} \to 0 \,.

Concatenating these with the above exact sequences yields the desired long exact sequence.

Lemma

Consider a cohomology spectral sequence converging to some filtered graded abelian group F C F^\bullet C^\bullet such that

  1. F 0C =C F^0 C^\bullet = C^\bullet;

  2. F sC <s=0F^{s} C^{\lt s} = 0;

  3. E s,t=0E_\infty^{s,t} = 0 unless t=0t = 0 or t=nt = n,

for some nn \in \mathbb{N}, n1n \geq 1. Then there are short exact sequences of the form

0E s,0C sE sn,n0. 0 \to E_\infty^{s,0} \overset{}{\longrightarrow} C^s \longrightarrow E_\infty^{s-n,n} \to 0 \,.

(e.g. Switzer 75, p. 356)

Proof

By definition of convergence of a spectral sequence, the E s,tE_{\infty}^{s,t} sit in short exact sequences of the form

0F s+1C s+tiF sC s+tE s,t0. 0 \to F^{s+1}C^{s+t} \overset{i}{\longrightarrow} F^s C^{s+t} \longrightarrow E_\infty^{s,t} \to 0 \,.

So when E s,t=0E_\infty^{s,t} = 0 then the morphism ii above is an isomorphism.

We may use this to either shift away the filtering degree

  • if tnt \geq n then F sC s+t=F (s1)+1C (s1)+(t+1)i s1F 0C (s1)+(t+1)=F 0C s+tC s+tF^s C^{s+t} = F^{(s-1)+1}C^{(s-1)+(t+1)} \underoverset{\simeq}{i^{s-1}}{\longrightarrow} F^0 C^{(s-1)+(t+1)} = F^0 C^{s+t} \simeq C^{s+t};

or to shift away the offset of the filtering to the total degree:

  • if 0t1n10 \leq t-1 \leq n-1 then F s+1C s+t=F s+1C (s+1)+(t1)i (t1)F s+tC (s+1)+(t1)=F s+tC s+tF^{s+1}C^{s+t} = F^{s+1}C^{(s+1)+(t-1)} \underoverset{\simeq}{i^{-(t-1)}}{\longrightarrow} F^{s+t}C^{(s+1)+(t-1)} = F^{s+t}C^{s+t}

Moreover, by the assumption that if t<0t \lt 0 then F sC s+t=0F^{s}C^{s+t} = 0, we also get

F sC sE s,0. F^{s}C^{s} \simeq E_\infty^{s,0} \,.

In summary this yields the vertical isomorphisms

0 F s+1C s+n F sC s+n E s,n 0 i (n1) i s1 = 0 F s+nC s+nE s+n,0 C s+n E s,n 0 \array{ 0 &\to& F^{s+1}C^{s+n} &\longrightarrow& F^{s}C^{s+n} &\longrightarrow& E_\infty^{s,n} &\to& 0 \\ && {}^{\mathllap{i^{-(n-1)}}}\downarrow^{\mathrlap{\simeq}} && {}^{\mathllap{i^{s-1}}}\downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{=}} \\ 0 &\to& F^{s+n}C^{s+n} \simeq E_\infty^{s+n,0} &\longrightarrow& C^{s+n} &\longrightarrow& E_\infty^{s,n} &\to& 0 }

and hence with the top sequence here being exact, so is the bottom sequence.

References

Formalization in homotopy type theory is discussed in

Applications:

  • Martin SaralegiA Gysin Sequence for Semifree Actions of S 3S^3, Proceedings of the American Mathematical Society Vol. 118, No. 4 (Aug., 1993), pp. 1335-1345 (jstor)

Last revised on February 4, 2017 at 13:32:00. See the history of this page for a list of all contributions to it.