If $G$ is a graph with edge set $E$, let $W$ be the $\mathbb{Z}/2$-vector space generated by the elements of $E$. If $A = \{a_1, \dots, a_n\} \subset E$, let $\bar{A} = a_1 + \dots + a_n \in V$; then $\bar{A}_1 + \bar{A}_2 = \overline{A_1 \Delta A_2}$, where $\Delta$ indicates symmetric difference.
I'll define the cycle space of $G$ to be the subspace of $W$ generated by simple cycles of $G$. More precisely, the cycle space of $G$ is the subspace of $W$ generated by the set $\{\bar{C} \mid C \text{ is a simple cycle of } G\}$. We could also view the cycle space as the first simplicial homology group of $G$ over $\mathbb{Z}/2$. It is not difficult to show that the dimension of the cycle space of $G$ is the corank of the cycle matroid of $G$.
Given any matroid $M$ with ground set $E$, we could define the circuit space of $M$ in a completely analogous way, just using the word "circuit" instead of "simple cycle." My question is: is it always true that the dimension of the circuit space of $M$ is the corank of $M$? If not, for what types of matroids is this true? Finally, can anyone recommend good resources that deal with this sort of thing?
Thanks!
This is a nice question, and it turns out to have a nice answer. In the following, I will use $r$ and $r^*$ to denote the rank and corank (i.e. the rank of the dual matroid), respectively.
First, let me recall a proof for the graphic case. Consider a spanning forest $F \subseteq E$ of the graph $G = (V,E)$. For each edge $e \in E - F$ there is a unique circuit $C_e$ in $F + e$, the fundamental circuit of $e$ with respect to $F$. The set of vectors $\overline{C_e}, e \in E - F$ is clearly linearly independent, since for each $e \in E - F$, $\overline{C}_e$ is non-zero in the coordinate corresponding to $e$, while $\overline{C}_f$ is zero in the same coordinate for $f \neq e$. This gives us $r^*(G)$ linearly independent vectors in the cycle space.
Thus, it is enough to show that the dimension of the cycle space is at most $r^*(G)$. Each edge $f \in F$ determines a cut $C^*_f \subseteq E - (F - f)$, the fundamental cut of $e$ with respect to $F$. As above, $\{\overline{C^*_f}, f \in F\}$ is linearly independent. Finally, note that a cycle always crosses a cut an even number of times. Translated into linear algebra, this means that for every cycle $C$ and cut $C^*$ of $G$, $\overline{C}$ is orthogonal to $\overline{C^*}$. Thus the cycle space is in the orthogonal complement of the subspace generated by $\{\overline{C^*_f}, f \in F\}$. Since this subspace has dimension $r(G)$, the cycle space has dimension at most $r^*(G)$, as required.
Now let's see whether this argument works for a general matroid $M$. Here, we start with a base $B$ and consider the fundamental circuits $C_e, e \in E - B$. As before, we get that the cycle space has dimension at least $r^*(M)$. What about the second part of the argument? Instead of fundamental cuts, we take fundamental cocircuits: for $f \in B$, let $C^*_f$ be the fundamental circuit of $f$ with respect to the dual base $E - B$ in the dual matroid $M^*$. Now we can repeat the same argument, and it works modulo the following question: is it true that for every circuit $C$ and cocircuit $C^*$, $|C \cap C^*|$ is even?
This turns out to be true if and only if $M$ is a binary matroid (see Theorem 9.1.2 in the second edition of Oxley's book). So in this case, our argument shows that the cycle space has dimension $r^*(M)$.
What about non-binary matroids? It turns out that in this case the dimension of the cycle space is always more than the corank! Indeed, we can find a circuit $C$ and a cocircuit $C^*$ such that $|C \cap C^*|$ is odd, or in other words, $\overline{C}$ and $\overline{C^*}$ are not orthogonal. Let us choose a basis $B$ such that $C^* = C^*_f$ for some $f \in B$. We can do this by choosing $f \in C^*$ arbitrarily, extending $C^* - f$ into a dual basis and taking the complement. The crucial observation is that for every fundamental circuit $C_e$, $|C_e \cap C^*|$ is even. Indeed, the intersection is a subset of $\{e,f\}$ and it is a basic result that in any matroid, the intersection of a circuit and a cocircuit cannot be a singleton (this is Oxley, Proposition 2.1.11). This shows that the subspace generated by $\{\overline{C_e}, e \in E - B\}$ is in the orthogonal complement of $\overline{C^*}$. Since $\overline{C}$ is not in this complement, the dimension of the cycle space is strictly larger than $r^*(M)$.
For a concrete example, you can take the uniform matroid $U_{2,4}$ (the smallest non-binary matroid). Here, the corank is two, but the cycle space has dimension four: every three-element subset of the ground set is a circuit and the corresponding vectors are linearly independent. I do not know how to compute the dimension of the cycle space for an arbitrary matroid. As for a resource, Section 9.2 of Oxley's book deals with the circuit and cocircuit spaces of binary matroids; in particular, Proposition 9.2.2 gives an affirmative answer to your question in the binary case.