Chain complexes of vector bundles - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T12:49:31Z http://mathoverflow.net/feeds/question/76608 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76608/chain-complexes-of-vector-bundles Chain complexes of vector bundles Ulrich Pennig 2011-09-28T07:20:54Z 2011-09-28T07:55:17Z <p>In his paper "Categories and cohomology theories" Graeme Segal considers the category of finite length chain complexes of finite dimensional vector spaces: Let $n = (n_i)_{i \in \mathbb{Z}}$ be a sequence of positive integers almost all zero. Then he claims that the space $K_n$ of chain complexes $E$ with $E^i = \mathbb{R}^{n_i}$ is a real algebraic variety. I am no algebraic geometer, but I suspect this to be similar to flag varieties somehow. So my first question is:</p> <blockquote> <p>What is the topology on $K_n$?</p> </blockquote> <p>As morphisms in the above mentioned category he takes <em>chain homotopy equivalences</em> between these complexes and claims that these also form a topological space. So:</p> <blockquote> <p>What is the topology on Mor($E,F$) for two chain complexes $E$ and $F$ as described above?</p> </blockquote> http://mathoverflow.net/questions/76608/chain-complexes-of-vector-bundles/76610#76610 Answer by Jeffrey Giansiracusa for Chain complexes of vector bundles Jeffrey Giansiracusa 2011-09-28T07:55:17Z 2011-09-28T07:55:17Z <p>The space $K_n$ sits inside the space of sequences of linear maps $$L_n = \Pi_i Hom(E^i,E^{i+1}).$$ This is just a space of sequences of matrices, so it is a real vector space of dimension $\sum_i (n_i \cdot n_{i+1})$. We give it the usual euclidean topology for real vector spaces.</p> <p>The subspace $K_n$ consists of those sequences of linear maps ${f_i}$ which form a chain complex - i.e., $f_{i+1} \circ f_i = 0$. This condition is polynomial in the entries in the matrices, so $K_n$ is a real algebraic affine subvariety inside $L_n$. We give $K_n$ the subspace topology.</p> <p>The topology on the space of morphism can be described similarly by embedding the morphism set $Mor(E,F)$ into the space of sequences of linear maps $\Pi_i Hom(E^i,F^i)$, which is again a real vector space. The condition of being a morphism of chain complexes is real algebraic so the morphism space is again a real algebraic affine variety.</p>