First of all, please excuse my English. I'm not native Englsih speaker, so you will see so many grammer mistakes, I just only hope that my mistkae wouldn't effect what I want to mean.

Hi, recently I'm taking graduate course Algebraic Topology, with text Bott and Tu, "Differential Forms in Algebriac Topology." I think this text is really great, but I'm having hard time with some minor (I think) problems and details. By background in this field is just first two chapter of Hatcher's Algebraic Topology book. So I'm studying cohomology first time with this book.

  1. On page 63p. Can anyone give me the construction of homotopy operator $$K : \Omega^\ast_{cv}(M \times \mathbb{R}^n) \rightarrow \Omega^{\ast-1}_{cv}(M\times \mathbb{R}^n)$$ For type (II) case I found that $$\pi^\ast \phi f(x,t) dt_1\cdots dt_n \mapsto \pi^\ast \sum_i (-1)^{i-1} \left(\int_{-\infty}^{t_i} f(x,t) dt_i\right) dt_{1\cdots \hat{i},\cdots n} - \pi^\ast\phi\sum_i (-1)^{i-1} \left(\int_{-\infty}^{t_i}e(t)dt_i\right) \left(\int_{\mathbb{R}^n}f(x,t)dt_{1\cdots n} \right)dt_{1\cdots \hat{i}\cdots n} $$ But I don't know how to define this kind thing to Type (I).

  2. On page 75 Example 6.44.1, Finding transition function was not hard, But I don't know how to directly use equation (6.38). Isn't this equation only valid only when we reduced the structure group into $SO(2)$?. I cannot see how the function $g_{01} = z$ means rotating angle. I also want to know what dose integration of euler class over base space means. I think this is only possible if the rank of bundle is equal to the dimension of base space. Well, actually I found something in wikipedia and lecture notes that it is related with self intersection number, but I cannot find any published text book.

  3. Last question is about the motivation of compact vertical cohomology and its Poincare duality. When I first saw definition of deRham cohomology was not weird, which means, it as natural enough for me. But deRham cohomology with compact support was not to natural at first time. After I see Poincare duality on this book, the assumption 'compact support' seem to assumed to make interal well defined. I think this conclusion is fair enough because, at least until now, everytime when I faced with function with compact support, there was integral with it. When I first saw a definition of compact vertical cohomology, I thought we are trying to integrate a form in fiber direction, and that was exactly the flow of this book. But still, the definition was not clear to me. If we want to integrate a form in a direcion of fiber, why do we need a assumption that the form should have compact support on the preimage of compact set? So with this kind of view, the definition of compact vertical cohomology looked like we want to do integrate a form over preimage of compact set i.e. for a rank $n$ vector bundle over $m$-manifold $\pi : E \rightarrow M$ with compact submanifold $K^k \subset M$, let me denote $i: \pi^{-1}(K) = E|_K \rightarrow E $. Then with compactly supported in vertical direction form $\omega \in \Omega^{k+n}_cv (E)$ the integral is well defined, $$\int_{E|_K} i^\ast \omega.$$ While, the Poincare duality tells us there is some kind of duality between submanifold and cohomology. In this book, I felt that deRham cohomology let me know what kind of submanifold does the manifold have, in this moment let me assume the manifold is finite type and let me use the word effective submanifold. (I mean, circle in the sphere is not interesting one) Similarly, compact cohomology tells us what kind of effective compact submanifold the manifold have. Hence with natural generalization I thougth I can do this kind things with compact vertical cohomology. Above integral gives us an element in $(H^{n+k}_{cv}(E))^\ast$, so I guess there should be something about this kind of Poincare duality.

But the problem is, I cannot see this kind of Poincare duality in this book. I've looked for other texts with bundle theory, but other texts don't even seem they care much about compact vertical cohomology.

So please help me with these questions.

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