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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:

What is the topology on $K_n$?

As morphisms in the above mentioned category he takes chain homotopy equivalences between these complexes and claims that these also form a topological space. So:

What is the topology on Mor($E,F$) for two chain complexes $E$ and $F$ as described above?

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$K_n$ is the subset of $\oplus_{i}{\rm Hom}_{\bf R}(E_i,E_{i+1})= \oplus_i M^{n_{i+1}\times n_{i}}({\bf R})$ (where $M^{n_{i+1}\times n_{i}}({\bf R})$ are the $n_{i+1}\times n_{i}$-matrices with real coefficients) consisting of the elements $(d_i)$ such that $d_{i+1}\circ d_i=0$. So this suggests that the topology should be induced by the natural topology of the real vector space $\oplus_i M^{n_{i+1}\times n_{i}}({\bf R})$. – Damian Rössler Sep 28 '11 at 7:45
up vote 7 down vote accepted

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.

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.

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.

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Thanks! That was much easier than I thought. – Ulrich Pennig Sep 28 '11 at 8:21

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