I'm currently studying this article by Farber about Configuration Spaces and Motion Planning Algorithms for a seminar. I'm having some trouble with some arguments because I'm also learning singular (co)homology for this article and I feel that I didn't absorb the intuition of these theories enough to understand some ideas.
Definitions and notations
For some context, we consider graphs to be CW-complexes and a tree is said to have an essential vertex if there is a vertex with valency of three or more. For each essential vertex on a tree $\Gamma$ we can choose topological embedding of the $Y$-letter graph on $\Gamma$ that send the center vertex on $Y$ to the essential vertex. We denote by $m(\Gamma)$ the number of essential vertices, and also denote by $F(X, n)$ the n-pointed configuration space of $X$.
We are dealing with (co)homology classes over a field $K$ and $F(Y, 2) \simeq_H \mathbb{S}^1$.
Associated to a tree $\Gamma$ (with a fixed root $u$) with $m(\Gamma)>0$, we construct the following CW-complex $Q_\Gamma$. For every $\Gamma$, $Q_\Gamma$ has two vertices and there is an edge for each ordered triple $(v, e_1, e_2)$ where $v$ is an essential vertex of $\Gamma$ and $e_1, e_2$ are distinct edges incident to $v$ that are both not in the path connecting $v$ to $u$.
First, the author takes a tree $\Gamma$ with $m$ essential vertices and fix $m$ topological embedding of $\{Y_i\}_{i = 1}^m$ on $\Gamma$ associated with the vertices. These embeddings are nice enough that their image don't intersect on $\Gamma$. It also said that further on, $F(Y_i, 2)$ we be considered a subspace of $F(\Gamma, 2)$ and I will denote by $\iota_i$ the inclusion.
What is said in the text
In some point of the text (pg. 27) the author evoke $m$ cohomology classes $\alpha_j \in H^1(F(\Gamma, 2))$ associated with the $m$ essential vertices. He states that those classes exists due to the following Theorem.
Theorem 11.1 For a tree $\Gamma$ having an essential vertex the configuration space $F(\Gamma, 2)$ is $\mathbb{Z}_2$-equivariantly homotopy equivalent to the complex $Q_\Gamma$.
Better yet he states that due to the same theorem, these clases can be required to statisfy $$\alpha_i|_{F(Y_i, 2)} \neq 0 \in H^1(F(Y_i, 2)) = \mathbb{Z}$$ $$\alpha_j|_{F(Y_i, 2)} \quad \text{if} i \neq j$$
After proving the stated theorem, the author also gives the following to corollary that I believe that may play a role in the preceding proposition
What I didn't understood
- By $\alpha_j|_{F(Y_i, 2)}$ he meant $\iota_i^*(\alpha_j)$, where $\iota_i^*$ is the homomorphism $H^1(F(\Gamma, 2)) \to H^1(F(Y_i, 2))$ induced by $\iota_i$?
- I cannot see in the proof of the corollary 11.3 the fact that $Q_{\Gamma'}$ be a subcomplex of $Q_{\Gamma}$ imply that the induced map is a monomorphism. For example, the inclusion of a circle in a disk does not induce a monomorphism, right?
- In the corollary 11.4, what are these so-called homology classes corresponding to all such embeddings? I thought of something like, because $F(\Gamma_i, 2) \simeq_H \mathbb{S}^1$, a map $F(\Gamma_i, 2) \to F(\Gamma, 2)$ can be though of a singular simplex of degree one, then we take its homology class or something like that.
- How does the theorem (and possibly the corollaries) guarantee the existence of such cohomology classes? I believe the idea is to use the homology classes of the corollary 11.4 and construct the cohomology classes with them somehow
I'm sorry about the HUGE question, but I'm really stuck.
The answer to your first question is yes.
If $X$ is a CW-complex of dimension $n$, then $H_n(X)$ is the kernel of $d\colon C_n(X)\to C_{n-1}(X)$ (cellular chains), hence a free abelian group. Since $\mathrm{Ext}(F,A)=0$ for any free abelian group it follows from the universal coefficient theorem that $H^n(X)\cong \mathrm{Hom}(H_n(X),\mathbb{Z})$.
If $Y$ is also a CW-complex of dimension $n$ that contains $X$ as a subcomplex, then it follows from the above considerations that the map $H_n(X)\to H_n(Y)$ is injective, hence $\mathrm{Hom}(H_n(Y),\mathbb{Z})\to \mathrm{Hom}(H_n(X),\mathbb{Z})$ is surjective (use freeness of the homology groups), in particular it is nonzero. Since the exact sequence of the universal coefficient theorem is natural, we see that the map $H^n(Y)\to H^n(X)$ is surjective.
In the case at hand $Q_\Gamma$ is a graph and the topological embeddings $\Gamma'\to \Gamma$ Farber considers give rise to inclusions of graphs $Q_{\Gamma'}\to Q_\Gamma$, hence an injection on $H_1$ and surjection on $H^1$. Note that one can determine a basis of $H_1(Q_\Gamma)$ by choosing a spanning/maximal tree $T$. Every edge not in $T$ gives rise to a cycle, hence a homology class and one can show that these form a basis. Using this it shouldn't be too hard to show that the different embeddings yield pairwise linearly independent homology classes in $H_1(Q_\Gamma)$ so that you can construct the desired cohomology classes using the isomorphism $H^1(Q_\Gamma)\cong \mathrm{Hom}(H_1(Q_\Gamma),\mathbb{Z})$.