dg.differential geometry – How does the instantaneous Floer homology of the Seifert fibrations relate to that of a trivial fibration?

My question focuses on the relationship of the Chern-Simons theories about a Seifert fibration and the trivial product space. $ Sigma_g times S ^ 1 $, and its implication for the instantaneous homology of Floer. Here, $ Sigma_g $ It is a Riemann gender surface. $ g $.

Leave $ M_ {g, p} $ be the Seifert variety that presents itself as a degree $ -P $, $ U (1) $
package on a Riemann surface of the genus $ g $.
In this Thompson article, it is explained that the Chern-Simons theory for $ M_ {g, p} $ Y $ Sigma_g times S ^ 1 $ They are closely related. This is essentially due to the fact that each 3-manifold has a contact structure, and for $ M_ {g, p} $ the natural contact structure, $ kappa $, is the $ U (1) $ connection in the $ U (1) $ Package used in its definition. As explained earlier in equation 2.2 of Thompson's paper, this corresponds to the obvious structure in $ Sigma_g times S ^ 1 $.

This leads to similar forms for the partition functions of the two theories, as shown on page 2 of Thompson. That is, for $ Sigma_g times S ^ 1 $, one finds the Hirzebruch-Riemann-Roch theorem for fundamental beam power in the space of flat connection modules in $ Sigma_g times S ^ 1 $, While for $ M_ {g, p} $ There is a simple generalization.

This brings me to my question. Since the Instanton Floer homology is defined using the Chern-Simons function, Is there a simple relationship between the Floer instant homology of $ M_ {g, p} $ Y $ Sigma_g times S ^ 1 $?

Algebraic topology – 2 out of 3 property of fibrations?

Leave $ X xrightarrow {f} Y xrightarrow {g} Z $ Be a diagram in the category of topological spaces. Yes $ g circ $ and one of $ f, g $ Are the fibrations, we can conclude that this is the other?

In this question it is stated that if $ f $ it is surjective and $ g circ $ Y $ f $ they are fibrations then $ g $ It is a fibration. However, no test is given there. Is this true? How can I show it? I understand that the main difficulty is that an arbitrary map $ W to Y $ it can not be lifted along $ f $ to a map $ W to X $. I'm not sure if this elevation property is true …

The motivation for this problem is that I want to show that $ mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n} $ Y $ mathbb {CP} ^ {2n + 1} to mathbb {HP} ^ n $ they are the fibrations. I know there are fibrations (in fact fiber bundles) $ mathbb {S} ^ {2n + 1} a mathbb {RP} ^ {2n + 1} $ Y $ mathbb {S} ^ {2n + 1} to mathbb {CP} ^ n $ and the factorization of these two maps gives the map. $ mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n} $. I wonder if we could use the previous property to directly deduce that $ mathbb {RP} ^ {2n + 1} to mathbb {CP} ^ {n} $ It is a fibration.

Etale-analytic comparison without elementary fibrations.

A theorem due to Artin states that for a smooth scheme $ X $ of finite type about $ mathbb {C} $ and a constant locally constructible sheaf $ F $ we have an isomorphism
$$
H ^ * et (X, F) approx H ^ * (X ( mathbb {C}), F)
$$

where the LHS is ethnic cohomology and the RHS is cohomology functor derived from $ F $ In the analytical topology. One of the tests that I know comes from establishing that there is an open cover of $ X $ by $ K ( pi, 1) $ However, the appearance of $ K ( pi, 1) $I feel a little weird. Is there any alternative test that is not based on $ K ( pi, 1) $ neighborhoods