3
votes
0answers
73 views

How should I imagine the cup product on a topological space/manifold/variety

Let $X$ be a "space" with a vector bundle $E$. Let $n$ be the dimension of $X$. Then there is a map $$H^1(X,E)\otimes \ldots\otimes H^1(X,E)\to H^n(X,\Lambda^n E).$$ This map is given by taken the ...
6
votes
3answers
131 views

introductory reference for Hopf Fibrations

I am looking for a good introductory treatment of Hopf Fibrations and I am wondering whether there is a popular, well regarded, accessible book. ( I should probably say that I am just starting to ...
4
votes
1answer
106 views

Specific homotopy between complex conjugation and the identity.

Consider the set $\mathcal{C} = C^{\infty}(\mathbb{C}^*, \mathbb{C}^*)$, where $\mathbb{C}^* = \mathbb{C}\backslash\{0\}$. Both $f(z) = z$ and $g(z) = \bar{z}$ can be seen as elements in ...
4
votes
0answers
135 views

Can the disk bundle associated to a vector bundle over a finite CW-complex be obtained by attaching cells?

Let $V$ be a vector bundle (with a chosen metric) over a finite CW-complex $X$ and $B(V), S(V)$ the associated (unit) disk resp. sphere bundles. In the paper Clifford-Modules the authors remark that ...