Weak Cayley fibrations

Slides for a talk at the Geometry and Analysis seminar in Oxford can be found here.

Instanton Floer homology

Floer homology is in essence the extension of the usual Morse homology of closed finite-dimensional manifolds to certain infinite-dimensional situations, where the naive Morse index of a critical point is not necessarily finite, and compactness becomes a subtle issue. In these notes we focus on the instanton Floer homology $\mathbf{HF}^\bullet(N)$ of a homology three-sphere $N$ , which is the Morse homology of the Chern-Simons functional $\mathbf{CS}$ . Defined on the infinite dimensional space of gauge equivalence classes of $\mathrm{SU}(2)$ -connections on $N$ and being circle-valued, this functional brings in further features not present in classical Morse homology. For one, due to the presence of reducible connections, the gauge equivalence classes of connections only form a manifold away from a collection of singular points. Furthermore, since $\mathbf{CS}$ is circle-valued it can occur that gradient flow lines form loops, making the trajectory spaces more subtle to describe.

In order to address the difficulties arising from the definition of the Morse index, we will introduce the spectral flow of a family of self-adjoint operators between separable Hilbert spaces, a tool common to all Floer-type theories. Using it, we will be able to define a relative index between two critical points of $\mathbf{CS}$ . This index will only be $\Z_8$ -valued however, due to looping gradient flow lines. To circumvent the issues arising from reducible connections, we will restrict ourselves to homology three-spheres, for which achieving transversality is easier than for other classes of three-manifolds. A feature of $\mathbf{CS}$ which makes it particularly suited for Morse homology is that its critical points and gradient trajectories have geometrical interpretations. The critical points of $\mathbf{CS}$ are flat connections on $N$ , and the flow lines are instantons over the Riemannian tube $\R \times N$ which join two connections. As we will see, instantons are a special case of Yang-Mills connections, which allows us to use Uhlenbeck’s compactness theorem to derive the compactness up to broken trajectories of the trajectory spaces. The invariant we will obtain in the end takes the form of a $\Z_8$ -graded vector space over the field $\Z_2$ . For the usual three-sphere $S^3$ and the Poincaré homology sphere $P$ they are given by:

$\begin{equation*} \boxed {\mathbf{HF}^\bullet(S^3) = 0, \quad \mathbf{HF}^\bullet(P) = (0, \Z_2, 0, 0, 0, \Z_2, 0, 0).} \end{equation*}$

Thus the Floer homology groups will be unrelated to the usual homology groups, and provide a stronger invariant. To conclude, we will investigate further properties of the Floer groups, such as their relation to the representation-theoretic Casson invariant, which appears as the Euler characteristic of the Floer groups, and $(3+1)$ -dimensional topological quantum field theories.

We will move according to the following outline. In the first chapter, we introduce electromagnetism as an example of an abelian gauge theory, familiarise ourselves with the concepts we will later encounter in the non-abelian setting and provide a link between gauge theory and topology of three-manifolds via Hodge theory. In the second chapter, we will then recall principal bundles properly, fix the notation, and derive the relation between three- and four-dimensional $\mathrm{SU}(2)$ -gauge theory. Next, we will introduce the Chern-Simons and Yang-Mills functionals $\mathbf{CS}$ and $\mathbf{YM}$ in the third chapter and analyse the relation between $\mathbf{CS}$ -gradient flow lines and instantons, which are the local minimizers of $\mathbf{YM}$ . We will also investigate the local picture around a critical point of these functionals. In the last chapter we will construct the trajectory spaces and the Floer homology groups from a chain complex generated by flat connections, emphasising the Fredholm analysis. We will show independence of auxiliary data and of the perturbation chosen, where it will be important to restrict to the case of homology three-spheres.

We assume that the reader is familiar with differential geometry (bundles and connections, Riemannian geometry, de Rham cohomology, Morse homology), algebraic topology (especially computational tools in (co)homology theories, Poincaré duality, intersection theory, some homotopy theory), as well as functional analysis (Sobolev spaces on open domains, differential operators on $\R^n$ ).

These notes do not contain original work, but merely adapt and combine the literature on the subject, while supplementing explanations where deemed useful. We mostly follow Donaldson’s monograph on the topic, and apply the treatment of Morgan’s lecture notes to give the analytical foundations for the Chern-Simons and Yang-Mills functionals. The first chapter on electromagnetism reshuffles the first part of the excellent lecture notes by Evans on the Yang-Mills equations, and the appendix bundles statements from multiple sources. For references see the PDF.

I would like to thank my supervisor Prof. Will Merry for guiding me through this endeavour with frequent and useful meetings. His (virtual) office door was always open to me, and I appreciated both his technical knowledge in geometry and analysis and his expertise in the wider world of Floer theories, which often provided helpful analogies an additional angles of attack during my study of the instanton flavoured theory.

The entire thesis can be found here.

The Model of Solovay

The objective of measure theory is to make rigorous the intuitive notions of length, surface area, volume, and measurement of quantity. A mathematical model of all of these can be found in the concept of measure on a set $S$ , which in its simplest form is a function $\mu : 2^S \rightarrow [0,1]$ such that the following hold:

Monotonicity: $X \subset Y \Rightarrow \mu(X) \le \mu(Y)$
$\sigma$ -additivity: $\mu(\sqcup_{n\in\omega} X_n) = \Sigma_{n \in\omega} \mu(X_n)$
Normalization: $\mu(\emptyset) = 0$ and $\mu(S) = 1$
Atomlessness: $\forall a \in S: \mu(\{a\}) = 0$

On the one hand, this notion replicates a number of facts about volume in the physical world. For instance, the first condition can be rephrased as “Bigger objects have bigger volume” and the second one translates to “The combined volume of a collection of disjoint objects is the sum of their respective volumes”. On the other hand, we can interpret a measure as an order-preserving map between partially ordered sets. Here the set $2^S$ is ordered by inclusion, and $\R$ is ordered by $\le$ . From this perspective, (i) just means that $\mu$ is order-preserving, (ii) restricts the value of $\mu$ on certain joins, and (iii) and (iv) are normalization conditions. In analogy of the representation theory of groups, the structure of $2^S$ as a poset is compared trough $\mu$ to the poset structure of $[0,1]$ . This interrelates the two structures and allows to analyze the structure of either one.With this definition in mind, there are a few questions that arise naturally. From the practical point of view, me might ask:

Does euclidean space $\R^n$ admit a measure that is translation invariant and hence reflects our experience of volume?

From the set theoretic point of view, one might ask:

Which sets $S$ admit a measure? In particular, since $S$ admits a measure exactly when $|S|$ does, which cardinals admit measures? This question is known as the Measure Problem.

As we will show in this semester project, the answer to the first question is no under $\ac$ . This has peculiar consequences such as the possibility to decompose a solid ball into a small number of pieces, rearranging them, and ending up with two exact copies of the initial ball. This mathematical possibility of copying a ball without adding any mass is known as the Banach-Tarski paradox, and would be impossible if a measure existed on all of $\R^3$ , for then mass would have to be conserved.

From here, there are essentially two possible directions to proceed. Either one can weaken the requirement that the measure be translation invariant, in which case there may exist a solution, or one can weaken $\ac$ , which is the route we are taking. Robert M. Solovay showed in 1970 that there is a model of set theory in which every set of reals is Lebesgue measurable and where a such a weakening of $\ac$ , the axiom of dependent choice $\dc$ , still holds.

The main goal of this thesis is to construct this model and to discuss how the structure of the real line relates to strong assumptions, such as further axioms of set theory or large cardinal hypotheses.In the first chapter, we introduce essential concepts and tools from descriptive set theory, such as a representation of the reals used in set theory, the Borel sets and alongside Lebesgue measurability two other regularity properties of the reals, the property of Baire and the perfect set property. The second chapter constructs the Model of Solovay and proves that any subset of the real line in it is Lebesgue measurable, has the Property of Baire and the perfect set property. The proof requires the existence of an inaccessible cardinal to define the Lévy collapse, which is a forcing notion that allows to overwrite any bad behaviour the reals might display. With this a possible solution to the measure problem is presented if $\ac$ is not assumed.In the last chapter we present the work of Shelah and Specker, illuminating when the assumption of the inaccessible cardinal is really needed, and in which cases it can be avoided. Finally, we will compare different alternatives to $\ac$ and present their impact on the regularity of the reals.We generally follow the books by Jech and Kanamori.

The rest of this semester project can be found here.

Morse Homology

Morse theory studies the topology of smooth manifolds via the smooth functions that live on it. More precisely, by looking at the critical points of special height functions, so called Morse functions. Consider for instance the example of a sphere:

Example of Morse functions on $S^2$

Let us assign each critical point $p$ a number according to its type, called its $\textbf{index}$ :

$\mu(p) = \left \{\begin{array}{ll} 2 & \text{, if } p \text{ is a local maximum} \\1 & \text{, if } p \text{ is a saddle point } \\0 & \text{, if } p \text{ is a local minimum} \end{array}\right.$

Then we can consider the sum:

$\begin{equation*} \eta(S^2) = \sum_{p \text{ critical point}} (-1)^{\mu(p)} \end{equation*}$

If we calculate $\eta(S^2)$ , we will find that in both cases its equal to $2$ . This turns out not to be a coincidence, since no matter which Morse functions we choose on $S^2$ , this number remains $2$ . If we consider the $2$ -torus $\mathbb{T}^2$ , i.e. a donut, and a Morse function on it, such as the one on the cover, we see that

$\begin{equation*} \eta(\mathbb{T}^2) = 0 \end{equation*}$

Again, changing the Morse function will not change $\eta(\mathbb{T}^2)$ . Now one can play this game also for genus $g$ surfaces $\mathbb{T}g$ and sees that:

$\eta(\mathbb{T}_g) = 2-2g$

Hence, by the classification of $2$ -manifolds (which interestingly can also be proven using Morse theory), we see that for any Riemann surface $M$ , we have:

(1) $\begin{equation*} \eta(M) = \chi(M) \end{equation*}$

where $\chi(M)$ is the Euler characteristic of $M$ . This is again no coincidence! Let us rewrite the formula for $\eta$ as follows:

(2) $\begin{equation*} \eta(M) = \sum_{k \ge 0} (-1)^kC_k, \quad \quad C_k = |\{p : p \text{ critical point of index } k\}|\end{equation*}$

This compares nicely to a formula from aglebraic topology. Let $M_0 \subset M_1 \subset M_2 = M$ be a cell complex structure on $M$ . Then we have:

$\chi(M) = \sum_{k \ge 0} (-1)^kN_k, \quad \quad N_k = \text{Number of } k \text{-cells}$

It turns out that under certain conditions a Morse function will give rise to a cell complex structure with exactly $C_k$ cells of dimension $k$ . If we for assume this result for a moment, we immediately get:

$\chi(M) = \sum_{k \ge 0} (-1)^kN_k = \sum_{k \ge 0} (-1)^kC_k = \eta(M)$

for manifolds which admit these sufficiently regular Morse functions. This means that we can express $\textbf{topological invariants}$ such as the Euler characteristic solely $\textbf{in terms of critical points}$ of height functions. In fact, a single Morse function, by giving rise to a cell complex structure, suffices in principle to classify manifolds up to homeomorphism type. What we will look at is the so called $\textbf{Morse homology}$ of a manifold, which is in fact just the homology of a cell complex obtained by a Morse function, from the classical point of view. But that is not how we are going to define it, as we will take a more abstract modern route. We will look at the so called $\textbf{Morse chain complex } \mathrm{CM}_\bullet(M;f)$ associated to a manifold with a Morse function $f$ on it. The $k$ -th chain groups will be free $\Z_2$ -modules generated by the critical points of index $k$ , and the boundary operator $\partial$ will be defined by counting solutions to a differential equation. Both approaches lead to the same result, however modern Morse homology has several advantages over classical Morse homology. One of them is that it lends itself better to our axiomatic treatment for showing its equivalence to singular homology. We will prove that Morse homology satisfies a number of axioms, called the $\textbf{Eilenberg-Steenrod-axioms}$ . Then we will invoke a uniqueness theorem that tells us that any homology theory satisfying the ES-axioms has to be isomorphic to singular homology. Let us now take a quick look at the structure of this thesis. The first chapter will focus on defining the Morse homology groups, and in particular show that it is well defined. We will extend our notion of index to general manifolds, and make concrete the expression “counting solutions to equations” with the so called trajectory spaces. Then the second chapter will deal with extending the groups to functors and show that the ES-axioms are satisfied, so that we can conclude via the uniqueness theorem that Morse homology is in fact isomorphic to singular homology. Finally, in the last chapter we will explores a number of theorems from algebraic topology in the spirit of Morse homology. We will cover Morse cohomology and Poincaré duality, which turns out is almost trivial to prove in Morse homology. To conclude, we will give a quick proof of the existence of an Eilenberg-Zilber chain map in by exploiting an additivity property of Morse functions. The rest of the thesis can be found here.

Shelah Product Tree Forcing

This forcing is used to prove the independence of the number of Ramsey ultrafilters from $\mathrm{ZFC}$ . We will give a brief overview of the results discussed in Halbeisen’s book. To be precise:

Theorem: Let $0 \le \kappa \le \omega_2$ be any cardinal. Then there is a model of $\mathrm{ZFC}$ with exactly $\kappa$ Ramsey ultrafilters.

The idea behind the proof is to start with a lot of Ramsey ultrafilters, say $\omega_2$ many, for instance by starting in a model where $\mathrm{CH} + 2^{\mathfrak{c}} = \omega_2$ holds. Then we have that $\mathrm{CH}$ implies $\mathrm{MA}$ , which in turn can be used to construct $\omega_2 = 2^\mathfrak{c}$ RUF. Next, we choose $\kappa$ of them to preserve, and destroy all of the others. This can be done with an iteration of some forcing notion whose job it is to destroy exactly one RUF. One then has to make sure of course that:

The iterated forcing notions don’t interfere with each other.
No new RUF are added at limit stages.

This forcing notion which will be responsible for eliminating a RUF at a time is Shelah’s product tree forcing. First some preliminary definitions.

$T_n^\otimes = \prod_{0\le l \le n} {}^l2 \quad T^\otimes = \bigcap_{n\in \omega} T_n^\otimes$

We define for $\eta, \zeta \in T^\otimes$ that $\zeta$ extends $\eta$ if:

$\eta \prec \zeta \Leftrightarrow \eta \in T_n^\otimes \wedge \zeta \in T_m^\otimes \wedge n \le m \wedge \zeta|_{n+1} = \eta$

We say a subset $T \subset T^\otimes$ is a (product) tree if for all $\zeta\in T$ and $\eta \in T^\otimes$ such that $\eta \prec \zeta$ we have $\eta \in T$ , i.e. $T$ is closed under taking initial segments. A tree is called perfect, if it splits indefinitely, i.e. for each element, we can find two different extensions. Product trees are a little more complicated than the usual trees encountered in Sacks or Miller forcing, since at level $n$ , they potentially split into $2^n$ branches. Let’s formalize that idea.

$T_n = T \cap T_n^\otimes$

is called the restriction of $T$ up to level $n$ . If we have an element $\eta \in T_n$ and some $t\in {}^n2$ , then we can concatenate them to get an element $\eta ^\frown t \in T_{n+1}$ . Let $\eta \in T$ be given. Then:

$\Theta_\eta = \{t \in {}^n2: \eta \frown t \in T_{n+1} \}$

Now we are ready to define the $n$ -th meta level.

$T \llbracket n\rrbracket = \{ \Theta_\eta : \eta \in T_n\}$

Next, we say a binary tree $t \subset {}^n2$ is a $k$ -tree, if it is a full binary tree when restricted to its first $k$ levels. We can then look at how “branching” the levels of $T$ are by examining:

$\text{fbt}_k(T) = \{ n \in \omega : \forall \Theta_\eta \in T \llbracket n\rrbracket \forall t \in\Theta_\eta (t \text{ is a k-tree})\}$

So in other words: $n \in\text{fbt}_k(T)$ means that every node on the $n$ -th meta level branches into at least $2^k$ nodes on the next meta-level. We are now finally in good shape to define the Shelah tree forcing. Consider a fixed P-family $\mathcal{V}$ . The conditions $T_\mathcal{V}^\otimes$ of the Shelah tree forcing restricted to $\mathcal{V}$ are then given by perfect product trees, for which $\text{fbt}_n(T) \in \mathcal{V}$ for every $n \in \omega$ . The order is given by reverse inclusion. Together we denote them by $\mathbb{T}_\mathcal{V}^\otimes$ .

Let us now take a look at the properties of this forcing notion.

Proposition: $\mathbb{T}_\mathcak{V}^\otimes$ is proper and ${}^\omega\omega$ -bounding.

Lemma: Suppose $\mathcal{U}$ is a Ramsey ultrafilter in the ground model and let $\mathcal{V}$ be a P-point in the ground model. Then if:

$\mathtt{0} \Vdash_{\mathbb{T}_\mathcal{V}^\otimes} "\mathcal{U} \text{ does not generate an ultrafilter}"$

We must have $\mathcal{U} \le_{RK} \mathcal{V}$ . In other words, the Shelah tree forcing destroys only RUF with are below $\mathcal{V}$ in the Rudin-Keisler order of ultrafilters.

Lemma: Let $\mathcal{V}$ be a P-point and $\mathbb{T}_\mathcak{V}^\otimes$ the corresponding Shelah tree forcing. If we have an $\omega_2$ -iteration $\P_{\omega_2}$ of other Shelah tree forcings $\mathbb{T}_\mathcak{V_\alpha}^\otimes$ for some P-points $V_\alpha$ , then they do not intere with each other in the following sense:

Let $G$ be $\mathbb{T}_\mathcak{V}^\otimes$ -generic over $\textbf{V}$ and $G_{\omega_2}$ be $\P_{\omega_2}$ -generic over $\textbf{V}[G]$ , then $\mathcal{V}$ does not generate an ultrafilter in $\textbf{V}[G][G_{\omega_2}]$ . In other words, $\mathbb{T}_\mathcak{V}^\otimes$ killed the ultrafilter $\mathcal{V}$ , and no $\omega_2$ -iterations of Shelah forcing can revive it.

With these properties, it is possible to prove that for instance the existence of just $27$ RUF is consistent with $\mathrm{ZFC}$ . The proof of these properties is rather complicated, and one still needs to prove things such that

a RUF will generate a RUF in an extension, if the forcing notion is proper, ${}^\omega\omega$ -bounding, and preserves it as a P-point.
non isomorphic RUF stay non isomorphic in ${}^\omega\omega$ -bounding forcing extensions.

Some Properties of Forcing Notions

Suppose $\mathbb{P}$ is a forcing notion in some model $\textbf{V}$ of $\mathrm{ZFC}$ , and let $G$ be a $\mathbb{P}$ -generic filter over $\textbf{V}$ . Then there are a few properties a $\mathbb{P}$ can have that are especially useful.

Adding reals: A lot of forcing notions add reals to the ground model in some form or another. Prominent examples would be Cohen reals for Cohen forcing, Sacks reals for Sacks forcing, etc. However forcing with $\mathbb{P}$ doesn’t only add a single $\mathbb{P}$ -real, as reals definable trough that new real also get added. Hence it would be lovely to have a few properties that describe general reals added by forcing notions. Examples of such properties are:

Adding dominating reals: In every forcing extension $\textbf{V}[G]$ , there is a real $f\in {}^\omega\omega \cap \textbf{V}[G]$ that dominates $^\omega\omega \cap \textbf{V}$ .
Adding unbounded reals: In every forcing extension $\textbf{V}[G]$ , there is a real $f\in {}^\omega\omega \cap \textbf{V}[G]$ that is not bounded by any real in the base model.
Adding splitting reals: In every forcing extension $\textbf{V}[G]$ , there is a real $x \subset \omega$ that splits any real in the base model, i.e.
$\forall y \in \mathcal{P}(\omega) \cap \textbf{V}: |x \cap y| = \omega = |x \setminus y|$

It turns out that if $\mathbb{P}$ adds a dominating real, it also adds both unbounded reals (this is easy) as well as splitting reals (this is not). However none of these inclusions are reversible as shown by the following table, exposing which forcing notions add which types of reals:

Notion	Dominating reals	Unbounded reals	Splitting reals
Mathias
Cohen	✘
Sacks	✘	✘	✘
Silver-like	✘	✘
Miller	✘		✘

Chain conditions: Let $\kappa$ be a regular cardinal. We say that $\mathbb{P}$ satisfies the $\kappa$ -chain condition, if every maximal antichain in $\mathbb{P}$ has size strictly less than $\kappa$ . The case $\kappa = \omega_1$ is also called the countable chain condition or ccc for short. This condition is useful since we have the following

Proposition: If $\mathbb{P}$ satisfies the $\kappa$ -chain condition, then all the cardinals $\lambda \ge \kappa$ are preserved in any forcing extension with $\mathbb{P}$ .

This essentially follows from the fact that if we have a surjective function $f : \alpha \rightarrow \beta$ for some cardinals $\alpha \in \beta$ in the extension, we also have a $\mathbb{P}$ -name $\tilde{f}$ for it. Then for $\lambda \in \alpha$ the set :

$\mathcal{D}_\lambda = \{ p \in \mathbb{P} : \exists \gamma \in \beta \exists q \in \mathbb{P} (q \ge p \wedge q \Vdash_{\mathbb{P}} \tilde{f}(\lambda) = \gamma) \}$

is an antichain in $\mathbb{P}$ , and hence of cardinality strictly smaller than $\kappa$ . Now this set consists of all the conditions above which the value of $f(\lambda)$ will eventually be forced to be some value $\gamma$ . Since $|\mathcal{D}_\lambda| < \kappa$ , we know that number the possible choices for $f(\lambda)$ must also be $< \kappa$ , since every such choice has to be forced by some condition in the filter, by the fundamental theorem of forcing. By the defineability of forcing, we can define another surjective function in the base model:

$g: \bigsqcup_{\lambda \in \alpha} |\mathcal{D}_{\lambda}| \times \alpha \rightarrow \beta$

where the parameter is used to sweep all the possible choices of $\tilde{f}(\lambda)[G]$ . Hence for $\beta > \kappa$ this implies that

$\kappa \times \alpha \ge \bigsqcup_{\lambda \in \alpha} |\mathcal{D}_{\lambda}| \times \alpha \ge \beta > \kappa$

So that $\alpha = \beta$ also in the base model. If $\beta = \kappa$ then if $\alpha<\kappa$ we would have by regularity of $\kappa$ :

$\bigsqcup_{\lambda \in \alpha}|\mathcal{D}_{\lambda}| < \kappa$

A contradiction to the surjectivity of $g$ . Hence in both cases, cardinals $\mu \ge \kappa$ are preserved.

Closedness Properties: Let $\kappa$ be any cardinal. Then $\mathbb{P}$ is called $\kappa$ -closed , if for any ascending sequence of conditions $\langle q_\lambda : \lambda \in \kappa \rangle$ there is a condition $q$ such that for all $\lambda \in \kappa : q \ge q_\lambda$ . For $\kappa = \omega$ , this is called $\sigma$ -closedness. This property complements the $\kappa$ -chain condition, in that the following hols:

Proposition: If $\mathbb{P}$ satisfies the $\kappa$ -closedness condition, then all the cardinals $\lambda \le \kappa$ are preserved in any forcing extension with $\mathbb{P}$ .

This follows from the fact, that if $f: \lambda \rightarrow X$ is a function in $\textbf{V}[G]$ with $\lambda \le \kappa$ , then $f$ is already in the ground-model. Hence no cardinals less than $\kappa$ can be collapsed. Now why is this the case? $f$ has a $\mathbb{P}$ -name $\tilde{f}$ in $\mathbf{V}$ . Start with any condition $p$ . Define the condition $q_0\ge p$ by:

$q_0 \Vdash_{\mathbb{P}} \tilde{f}(0) = x_0$

for some $x_0\in X$ . This is possible, since the conditions that define $\tilde{f}(0)[G]$ are open dense in $P$ . Again by open density find $q_1 \ge q_0$ such that for some $x_1 \in X$ with:

$q_1 \Vdash_{\mathbb{P}} \tilde{f}(1) = x_1$

Continue in this manner to obtain an ascending sequence $q_0 \le q_1 \le q_2 \le \cdots$ . Now at the limit stage, utilize the $\kappa$ -closedness to find $q_\omega \ge q_k$ for any $k \in \omega$ . Continue this process further until we have a condition $q_\lambda$ such that for all $\alpha \in \lambda$ :

$q_\lambda \Vdash_{\mathbb{P}} \tilde{f}(\alpha) = x_\alpha$

So above each $p$ , there is a $q_\lambda$ which essentially says that $f$ is definable in $\textbf{V}$ , hence no matter which generic filter we choose, there will be some $q_\lambda \in G$ , and hence $f$ is already a function in $\textbf{V}$ .

Properness: A generalization of the $ccc$ . Remember that $ccc$ assures that no uncountable cardinals get collapsed. Also $ccc$ gets preserved under finite support iterations. However this is often too much to ask from a forcing notion, hence the notion of properness, for which the following hold:

Proposition 1: If $\mathbb{P}$ is proper, then it does not collapse $\omega_1$ .

Proposition 2: If $\mathbb{P}_\alpha$ is a countable support iteration of $\langle \tilde{\mathbb{Q}_\beta : \beta \in \alpha}\rangle$ , and for each $\beta \in \alpha$ we have $\mathtt{0}_\beta \Vdash_{\mathbb{P}} "\tilde{\mathbb{Q}}_\beta \text{ is proper }"$ , then $\mathbb{P}_\alpha$ is proper.

Proposition 3: If $\mathbb{P}\alpha$ is a countable support iteration of proper forcing notions which satisfy one of the following properties

Laver Property
Preservation of P-points
${}^\omega\omega$ -boundedness

Then $\mathbb{P}\alpha$ is proper and has the respective property.

So now to the definition of properness. For a regular limit cardinal $\chi$ let:

$\textbf{H}_\chi = \{ x \in \textbf{V}_\chi : |\text{TC}(x)| < \chi\}$

These are all the sets which are hereditarily of cardinality $< \chi$ . It turns out $\textbf{H}_\chi$ is a model of $\mathrm{ZFC}$ minus the power set axiom. For the definition of properness, we need $\chi$ large enough. Assume $\chi > \beth^+_\omega$ in the following, this will suffice. Let $\textbf{N} = (N, \in)$ be an elementary submodel of $(\textbf{H}_\chi, \in)$ , i.e. any statement about elements of $N$ is true in $(\textbf{H}_\chi, \in)$ iff the corresponding statement (where all the quantifiers have to be relativised) is true in $textbf{N}$ .

Let $(P, \le) \in N$ . We say that $G \subset P$ is $\textbf{N}$ -generic if it has the following property: If $D \in N$ and $\textbf{N} \vDash "G \text{ is an open dense subset of P}"$ . So if the model $\textbf{N}$ says that a subset $D \subset P$ is open dense, $G$ should intersect it, similarly to genericity over $\textbf{V}$ . Next, a condition $q \in P$ , which does not need to be in $N$ , is $\textbf{N}$ -generic if

$\textbf{V} \vDash q \Vdash_{\mathbb{P}} "\dot{G} \text{ is } \textbf{N}\text{-generic}"$

where $\dot{G}$ is the canonical $\mathbb{P}$ -name for the $\mathbb{P}$ -generic filter in $\textbf{V}$ .

Finally, a forcing notion $\mathbb{P}$ is called proper, if for all countable elementary submodels $\textbf{N}$ of $(\texbf{H}_\chi, \in)$ which contain $\mathbb{P}$ , and every condition $p \in N \cap P$ , there is a $\textbf{N}$ -generic condition $q \ge p$ .

Laver Property: A property that makes sure no Cohen reals are added while forcing. Let $\mathcal{F}$ be the family of all function $s: \omega \rightarrow \text{fin}(\omega)$ , such that $\forall n\in \omega: |s(n)| \le 2^n$ . The family $\mathcal{F}$ is restricting the growth of its member to exponential growth. A forcing notion $\mathbb{P}$ is said to have the Laver property, if for any function $f\in {}^\omega\omega$ in the ground model and any $\mathbb{P}$ -name $\tilde{g}$ of a function $g \in {}^\omega\omega \cap \textbf{V}[G]$ , such that $\mathtt{0} \Vdash_{\mathbb{P}} \forall n \in \omega: (\tilde{g}(n) < \dot{f}(n))$ we have:

$\mathtt{0} \Vdash_{\mathbb{P}} \exists s \in \dot{\mathcal{F}}\forall n \in \omega (\tilde{g}(n) \in s(n))$

Another way of looking at the Laver property is that if $\mathbb{P}$ satisfies the Laver property, the models $\textbf{V}$ and $\textbf{V}[G]$ cannot be too wildly different, since each bounded funtion in the extended model can at least be approximated in the ground model.

Preserving P-points: Let $\mathcal{U} \subset [\omega]^\omega$ be a P-point in $\textbf{V}$ . Then we say that $\mathbb{P}$ preserves $\mathcal{U}$ , if $\dot{\mathcal{U}}[G]$ generates an ultrafilter in any extension. Equivalently:

$\mathtt {0} \Vdash_{\mathbb{P}} "\mathcal{U} \text{ generates an ultrafilter }"$

Note, that we don’t require the ultrafilter generated in the extension to be a P-point. This is where the following statement comes in:

Proposition: Let $\mathbb{P}$ be a proper forcing notion and $\mathcal{U}$ be a P-point in the ground model $\textbf{V}$ . If the filter generated in $\textbf{V}[G]$ is an ultrafilter, then it is a P-point.

So nice enough forcing notions maintain P-points in this stronger sense.

Furthermore, the countable support iterations of P-point preserving proper forcing notions also preserves P-points.

${}^\omega\omega$ -bounding: A forcing notion $\mathbb{P}$ is called ${}^\omega\omega$ -bounding if there are no unbounded reals in any forcing extension.

${}^\omega\omega$ -boundedness of proper forcing notions is preserved under countable support iterations.

Some Forcing Notions

In this post I will give a few examples of forcing notions and explore their combinatorial properties. Most of this is directly taken from “combinatorial set theory with a gentle introduction to forcing” by Halbeisen and condensed down a bit.

Ultrafilter Forcing $\mathbb{U} := ([\omega]^\omega, \supseteq^*)$ :

What makes this one useful, is that a $\mathbb{U}$ -generic filter $G$ over $\textbf{V}$ is also a Ramsey Ultrafilter in the extension, different from all ultrafilters in the ground model. Hence, when forcing with this notion, one adds at least one Ramsey Ultrafilter to the ground model. There are a few things one needs to check to be able to conclude:

RUF’s $\math$ are characterised by the fact that for each coloring

$\pi : [\omega]^2 \rightarrow 2$

there is a $x \in \mathcal{U}$ such that $\pi|_{[x]^2}$ is constant, i.e. for each coloring, $\mathcal{U}$ contains a monochromatic set. We will conclude that the new filter $G$ has such a set for each coloring in the ground model. However one also needs to make sure that no new colorings get added, so that indeed the property holds true for all colorings in $\textbf{V}[G]$ . For this one checks that $\mathbb{U}$ is $\omega_1$ -closed, hence preserves all cardinals $\le \omega_1$ , and in particular all real numbers $r : \omega \rightarrow \omega$ . Since the reals are in one-to one correspondence with the two colorings of $[\omega]^2$ , there are no new colorings appearing in $\textbf{V}[G]$ .
$G$ is in fact an ultrafilter which is different from every ultrafilter in the ground model. For the first fact consider the open dense set:
$\mathcal{D}_x = \{ y \in [\omega]^\omega : x \supseteq^* y \vee \omega \setminus x \supseteq^* y\}$

$G$ being $\mathbb{U}$ -generic means it intersects this set, and hence either $x \in G$ or $\omega \setminus x \in G$ . For the second fact let $\mathcal{V}$ be an ultrafilter in the ground model and consider the following open dense set:
$\mathcal{D}_{\mathcal{V}} = \{ y \in [\omega]^\omega : y \not\in \mathcal{V}\}$
Here, $G$ being $\mathbb{U}$ -generic implies, that for every ultrafilter $\mathcal{V}$ in the ground model, there is a set in $G$ that is not contained in $\mathcal{V}$ , hence in particular $\mathcal{V} \ne G$ .

Cohen Forcing $\mathbb{C}_\kappa = (\text{Fn}(\omega \times \kappa, 2), \subset)$ :

This forcing notion can be used to show that $\neg \textbf{CH}$ is consistent with $\textbf{ZFC}$ . Suppose $\textbf{V}$ was a model of $\textbf{ZFC}+\neg \textbf{CH}$ . Then forcing with $\mathbb{C}_\kappa$ adds $\kappa$ distinct new reals to the model, the so called Cohen reals, and preserves all cardinals. In fact if $G$ is $\mathbb{C}_\kappa$ -generic over $\textbf{V}$ , we get that $\bigcup G$ is a function $\omega \times \kappa \rightarrow 2$ . This can be concluded by looking at the open dense set:

$\mathcal{D}_\lambda = \{ p \in \text{Fn}(\omega \times \kappa, 2) : \alpha \in \text{dom}(p)\}$

which shows that $\bigcup G$ is defined for all $\lambda \in \kappa$ . Since $\mathfrak{c}$ counts exactly the number of reals, if we choose $\kappa > \omega_1$ we get:

$\mathfrac{c} \ge \kappa > \omega_1 .$

Hence we see that the continuum hypothesis no longer holds. However there are a few things one needs to check given a $\mathbb{C}_\kappa$ -generic filter $G$ over $\textbf{V}$ :

The added reals are indeed are all distinct and different from all existing reals. First, for $\lambda \in \kappa$ , we define the new reals by:
$r_\lambda = \left( \bigcup G \right)|_{\omega \times \{ \lambda\}}$

Then you look at:
$\mathcal{D}_{\lambda, \lambda'} = \{ p \in \text{Fn}(\omega \times \kappa, 2): \exists n \in \omega (p(\lambda, n) \ne p(\lambda', n))\}$

which exactly tells you that no two $r_\lambda$ ‘s are the same, and consider a similar open dense set to show that they also differ from all reals in the ground model.
The forcing doesn’t collapse any cardinals, since otherwise $\kappa$ might shrink down to $\omega_1$ again and we wouldn’t have proved anything. This is shown using the fact that $\mathbb{C}_\kappa$ is $ccc$ by a consequence of the $\Delta$ -system lemma, and hence preserves all infinite cardinals.

Collapsing Forcing $\mathbb{K}_{\kappa, \omega_1}$ :

The conditions are given by partial functions $\omega_1 \rightarrow \kappa$ with countable domain, and the order is the inclusion. As the name suggests, this forcing can be used to collapse a cardinal $\kappa$ to $\omega_1$ . In particular for $\kappa = \mathfrak{c}$ we have that for a $\mathbb{K}_{\kappa, \omega_1}$ -generic filter over $\mathbf{V}$ :

$\textbf{V}[G] \vDash \textbf{CH}$

One needs to show a couple of things for that:

No new reals get added. This can be assured by verifying the $\sigma$ -closedness of $\mathbb{K}_{\kappa, \omega_1}$ .
$\mathbb{K}_{\kappa, \omega_1}$ adds a surjective function $f : \omega_1 \twoheadrightarrow \kappa$ , so that
$\textbf{V}[G] \vDash \kappa \le \omega_1$

Sacks Forcing $\mathbb{S}$ :

Here the conditions are given by:

$S = \{ T \subset {}^{<\omega}2 : T \text{ is a perfect tree}\}$

and $\mathbb{S} = (S, \supset)$ . A tree $T \subset {}^{<\omega}2 = \bigcup_{n \in \omega} {}^n2$ is a subset that is closed under taking initial segments, i.e. if $p \in T$ , then $p|_k \in T$ for all $k \in \omega$ . We also say that $t$ extends $s$ , written $s \prec t$ , if $s$ is an initial segment of $t$ . A perfect tree, is a tree that splits indefinitely, i.e. $\forall t \in T \exists t' \ne t''\in T: t \prec t' \wedge t \prec t''$ . Sacks forcing allows us to build a model $\textbf{V}$ [G], such that there is no “intermediate model” between $\textbf{V}$ and $\textbf{V}$ [G], meaning if $\textbf{V} \subset \textbf{W} \subset \textbf{V}[G]$ is a model of $\mathrm{ZFC}$ then either $\textbf{V} = \textbf{W}$ or $\textbf{W} = \textbf{V}[G]$ . Forcing with perfect trees was used by Gerald Enoch Sacks to produce a real a with minimal degree of constructibility.

Sacks Forcing adds neither splitting nor unbounded reals

Miller Forcing $\mathbb{M}$ :

Here the conditions are given by:

$M = \{ T \subset \text{seq}(\omega) : T \text{ is a superperfect tree} \}$

and $\mathbb{M} = (M, \supset)$ . Here superperfect means that for $s \in T$ :

$\text{next}(t) := \{ t\in T: \text{dom}(t) = \text{dom}(s)+1 \wedge s \prec t\}$

is infinite for all $t \in T$ .

It adds unbounded reals but no splitting reals.

Mathias Forcing $\mathbb{M}_{\mathcal{U}}$ :

where $\mathcal{U}$ is a fixed Ramsey family. The conditions are given by:

$M_{\mathcal{U}} = \{ (s,x) : s\in \text{fin}(\omega) \land x \in \mathcal{U}   \land \text{max}(s) < \text{min}(x)\}$

and the preorder is given by:

$(s,x) \le (t, y) \Leftrightarrow s \subset t \land y \subset x \land t \setminus s \subset x$

Mathias forcing can be used to construct a pseudointersection of an ultrafilter.

Restricted Laver Forcing $\mathbb{L}_{\mathcal{U}}$ :

Here $\mathcal{U} \subset [\omega]^\omega$ is a fixed ultrafilter. Now a Laver tree is a tree $T \subset \text{seq}(\omega)$ if it has a special element $s \in T$ , called the stem, such that for any $t \in T$ we have $t \prec s \vee s \prec t$ , and for every $t \in T$ with $s \prec t$ , we have that $\text{next}(t)$ is infinite. A Laver tree restricted to $\mathcal{U}$ is a Laver tree such that $\text{next}(t) \in \mathcal{U}$ . The conditions of Laver forcing are exactly the Laver trees restricted to $\mathcal{U}$ , with reverse inclusion.

This forcing is $\sigma$ -centred and adds a real which is almost homogenous for all colorings in the ground model.

Silver Forcing $\mathbb{S}_{\mathcal{E}}$ :

Here $\mathcal{E}$ is a fixed P-family. Define:

$S_\mathcal{E} = \bigcup\{ {}^x2: \omega \setminus x \in \mathcal{E}\}$

and for $p,q \in S_\mathcal{E}$ :

$p \le q \Leftrightarrow \text{dom}(p) \subset \text{dom}(q) \wedge q|_{\text{dom}(p)} = p$

Then $\mathbb{S}_{\mathcal{E}} = (S_\mathcal{E}, \le)$ . For $\mathcal{E} = [\omega]^\omega$ write $\mathbb{S}$ and just call it Silver forcing. Silver reals are given by:

$r = \bigcup G$

In general, Silver filters cannot be reconstructed from Silver reals, since multiple Silver filters yield the same real. However, if $\mathcal{E}$ is an ultrafilter we can always do the reconstruction.

Shelah’s Tree forcing $\mathbb{T}$ : See my other post here.