## 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