1 Transgression Sequence of a Fibration

F → E → B is a fibration of finite spaces and suppose that F and B are m-

and n-connective, respectively. If n > 0, Then there is a Serre spectral sequence

H

p

(B; H

q

(F ; Z)) ⇒ H

p+q

(E)

The r-th page, for r < m, of the spectral sequence features only one non-trivial

differential:

d

for −n < r < m, where d

= 0 when r < 0.

r

is called the r-th transgression map (of the

fibration). We call the exact sequence of Proposition (1.0.1) the transgression

sequence of the underlying fibration.

1.1 Transgression for Loop Spaces

For an n-connective space X, with n > 1, we have transgression maps

is an isomorphism, for r < n.

1.2 Relation With The Boundary Map

π

n+r

(B) π

n+r

∂

d

Proof. d

([f]), where f : S

→ S

corresponds to chasing [f ] up the

d

([f])

But passing through the identification of C(E) with C(B) ⊗ C(F ), which is

implicit in the Serre Spectral sequence, the full diagram chase is equivalent to

a lift,

n+r

r

= ∂[f], which by construction lies

in C(B)

. On the other hand, we could have lifted f to an n + r-cell of E,

1. If X is n-connective, with π

(X) Abelian, then ϵ : H

n+r

(ΣΩX)

n+r

→

H

n+r

(X) is an isomorphism for r < n.

2. If r < n then the following diagram commutes

H

n+r

(ΣΩX) H

H

(X)

(1)

Proof.