3 typo

Given a bundle $F \to M \to S^1$, the Mayer-Vietoris sequence corresponding to the decomposition of $M$ coming from writing $S^1$ as the union of two intervals tells you there's a short exact sequence:

$$0 \to coker( f_n - I ) \to H_n(M) \to ker( f_{n+1f_{n-1} - I ) \to 0$$

Here $f_n : H_n F \to H_n F$ is the induced map from the monodromy of the bundle, ie: you think of the bundle as $R \times_f F, f: F --> F$ a homeomorphism / diffeomorphism / whatever. And $I$ is the identity map on $H_n(M)$ and $H_{n+1} H_{n-1} M$ respectively.

There's a similar decomposition for cohomology, and this is what the Serre spectral sequence gives you, too. The short-exact sequence basically encodes the extension problem from the spectral sequence.

2 TeX it up

Given a bundle $F --> \to M --> S^1\to S^1$, the Mayer-Vietoris sequence corresponding to the decomposition of M $M$ coming from writing S^1 $S^1$ as the union of two intervals tells you there's a short exact sequence:

0 -->

$$0 \to coker( f_n - I ) --> \to H_n(M) --> \to ker( f_{n+1} - I ) --> 0\to 0$$

Here $f_n : H_n F --> \to H_n F F$ is the induced map from the monodromy of the bundle, ie: you think of the bundle as Rx_f $R \times_f F, f: F --> F F$ a homeomorphism / diffeomorphism / whatever. And I $I$ is the identity map on H_n(M) $H_n(M)$ and H_{n+1} M $H_{n+1} M$ respectively.

There's a similar decomposition for cohomology, and this is what the Serre spectral sequence gives you, too. The short-exact sequence basically encodes the extension problem from the spectral sequence.

1

Given a bundle F --> M --> S^1, the Mayer-Vietoris sequence corresponding to the decomposition of M coming from writing S^1 as the union of two intervals tells you there's a short exact sequence:

0 --> coker( f_n - I ) --> H_n(M) --> ker( f_{n+1} - I ) --> 0

Here f_n : H_n F --> H_n F is the induced map from the monodromy of the bundle, ie: you think of the bundle as Rx_f F, f: F --> F a homeomorphism / diffeomorphism / whatever. And I is the identity map on H_n(M) and H_{n+1} M respectively.

There's a similar decomposition for cohomology, and this is what the Serre spectral sequence gives you, too. The short-exact sequence basically encodes the extension problem from the spectral sequence.