Spaces of polynomials with constrained divisors as Grassmanians for traversing flows

This paper is the second in a series, which is inspired by the works of Arnold [Ar], [Ar1], and Vassiliev [V]. It is a direct continuation of [K9], where many ideas of this article are present. Here we apply these ideas to traversing flows on manifolds with boundary. As [K9], this article relies heavily on computations from [KSW1], [KSW2]. While [K9] is studying immersions of compact $n$-dimensional manifolds into products $\mathbb{R}\times Y$, where $Y$ is a compact $n$-dimensional manifold, the present paper deals with special traversing flows on compact $(n+1)$-dimensional manifolds $X$, the flows that admit a “virtual global section”. The existence of such sections allows to establish a transparent correspondence between the universe of immersions into the products and the universe of traversing flows with virtual sections. In fact, with this correspondence in place, many results of the present article are just reformulations of the corresponding results from [K9], thus motivating and justifying them. However, we prove also many propositions with no analogues in [K9] (for example, Proposition 4.1, Theorem 4.1, Corollary 4.4, Proposition 4.2, and Theorem 4.8).

Let us describe here our results informally, in a manner that clarifies their nature, but does not involve their most general forms, which carry the burden of combinatorial decorations.

As we have mentioned above, we study traversing vector flows $v$ (see Definition 4.1 and [K1], [K2]) on smooth compact $(n+1)$-dimensional manifolds $X$ with boundary. Every traversing flow admits a Lyapunov function, and this fact may serve as a working definition of such flows. We are interested in the combinatorial patterns $\omega$ that describe how the $v$-trajectories are tangent to the boundary $\partial X$. Specifically, we are concerned with the traversing vector flows whose tangency patterns $\omega$ do not belong to a given closed poset $\Theta$. Depending on $\Theta$, the very existence of such flows puts strong restrictions on the topology of $X$ (see [K1]-[K8], [K10]). For example, in Fig.1, $\Theta$ includes the cubic and the double tangencies of $\partial X$ to the family of $u$-directed lines. In other words, in this figure, the tangency patterns $\omega=(3)$ and $\omega=(22)$ are forbidden.

In order to simplify the investigation of such flows, we “envelop” them into, so called, convex pseudo-envelops $(\hat{X},\hat{v})$. A convex pseudo-envelop consists of: (i) a compact smooth $(n+1)$-manifold $\hat{X}$, (ii) a traversing vector field $\hat{v}$ on it such that the boundary $\partial\hat{X}$ is convex (see Definition 4.5) with respect to the $\hat{v}$-flow, (iii) a submersion $\alpha:X\to\hat{X}$ such that $\alpha(\partial X)$ avoids a forbidden tangency patterns $\Theta$ to the $\hat{v}$-trajectories, and (iv) the pull-back $\alpha^{\dagger}(\hat{v})$ of $\hat{v}$, coinciding with the given vector field $v$ (see Fig.1, in which $\hat{X}$ is a cylinder).

The existence of a convex pseudo-envelop provides the $v$-flow on $X$ with a “virtual global section”, a significantly effective tool. Its existence allows us to transfer the results from [K9] about immersions/embeddings $\{\beta:M\to\mathbb{R}\times Y\}$, where $M$ and $Y$ are compact smooth $n$-manifolds, to the parallel results about convex envelops of traversing flows.

Next, we organize convex pseudo-envelops $\{\alpha:(X,v)\to(\hat{X},\hat{v})\}$ with the fixed target $(\hat{X},\hat{v})$ into the quasitopy equivalence classes $\mathcal{QT}_{d}(\hat{X},\hat{v};\mathbf{c}\Theta)$, where $d$ is a given even natural number. For a typical $\hat{v}$-trajectory $\hat{\gamma}\subset\hat{X}$, $d$ gives an upper bound of the cardinality of the set $\hat{\gamma}\cap\alpha(\partial X)$. Fig. 3 may give some impression of the nature of quasitopy; it depicts a pair of submersions $\alpha_{0},\alpha_{1}$, linked by a kind of cobordism with a strict combinatorial control of its tangent patterns.

The quasitopies come in two flavors: $\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$, formed by regular embeddings $\alpha$, and $\mathcal{QT}_{d}^{\mathsf{sub}}(\hat{X},\hat{v};\mathbf{c}\Theta)$, formed by more general submersions $\alpha$.

Consider the locus $\partial_{1}^{+}\hat{X}(\hat{v})\subset\partial\hat{X}$, where $\hat{v}$ points inside of $\hat{X}$, and the locus $\partial_{2}^{-}\hat{X}(\hat{v})\subset\partial_{1}^{+}\hat{X}(\hat{v})$, where $\hat{v}$ is tangent to $\partial\hat{X}$. Then we construct two classifying maps

whose targets are sets of homotopy classes from the quotient space $\partial_{1}^{+}\hat{X}(\hat{v})\big{/}\partial_{2}^{-}\hat{X}(\hat{v})$ to the universal space $\mathcal{P}_{d}^{\mathbf{c}\Theta}$—the “Grassmanian”—, formed by real polynomials of degree $d$ whose real divisors avoid a given poset $\Theta$.

Then we prove (see Theorem 4.2) that $\Phi^{\mathsf{emb}}$ is a bijection and $\Phi^{\mathsf{sub}}$ is a split surjection.

For a constant vector field $\hat{v}$ on the standard ball $D^{n+1}$, the sets $\mathcal{QT}_{d}^{\mathsf{sub/emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ are groups; the group operation is induced by the boundary connected sum of pseudo-envelops/envelops. These groups are abelian for $n>1$. The group $\mathcal{QT}_{d}^{\mathsf{sub}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ is a split extension of $\mathcal{QT}_{d}^{\mathsf{emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ (see Theorem 4.4). We prove that

In the process, in the Subsection 4.2, we notice an interesting relation between the connected components of the of the space of convex traversing flows on the standard $(n+1)$-ball and the monoid $\mathbf{HS}_{n-1}$ of smooth types of integral homology $(n-1)$-spheres, being considered up to connected sums with smooth homotopy $(n-1)$-spheres (see Proposition 4.1).

The groups $\mathcal{QT}_{d}^{\mathsf{sub/emb}}(D^{n+1},\hat{v};\mathbf{c}\Theta)$ act on the quasitopies of convex pseudo-envelops $\mathcal{QT}_{d}^{\mathsf{sub/emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$, and the classifying maps $\Phi^{\mathsf{sub/emb}}$ in (1.1) are equivariant.

These classifying maps deliver a variety of (co)homotopy and (co)homology invariants of traversing flows which admit convex pseudo-envelops (see Theorem 4.7, Proposition 4.6, and Theorem 4.8).

We also establish several of results (Theorem 4.3, Corollary 4.6) about the stabilization of the sets $\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$ as $d\to\infty$.

Finally, we compute $\mathcal{QT}_{d}^{\mathsf{emb}}(\hat{X},\hat{v};\mathbf{c}\Theta)$ for may special cases of $\Theta$ and $(\hat{X},\hat{v})$.

For our reader convenience, we state here a number of results from [KSW1] and [KSW2] about the topology of spaces of real monic univariate polynomials with constrained real divisors. These results are crucial for the applications to follow. To make the present paper even partially self-contained, we are basically recycling Section 2 from [K9].

Let $\mathcal{P}_{d}$ denote the space of real monic univariate polynomials of degree $d$. Given a polynomial $P(u)=u^{d}+a_{d-1}u^{d-1}+\cdots+a_{0}$ with real coefficients, we consider its real divisor $D_{\mathbb{R}}(P)$. Let $\omega_{i}$ denotes the multiplicity of the $i$-th real root of $P$, the real roots being ordered by their magnitude. We call the ordered $\ell$-tuple of natural numbers $\omega=(\omega_{1},\ldots,\omega_{\ell})$ the real root multiplicity pattern of $P(u)$, or the multiplicity pattern for short.

Such sequences $\omega$ form a universal poset $(\mathbf{\Omega},\succ)$. The partial order “$\succ$” in $\mathbf{\Omega}$ is defined in terms of two types of elementary operations: merges $\{\mathsf{M}_{i}\}_{i}$ and inserts $\{\mathsf{I}_{i}\}_{i}$ The operation $\mathsf{M}_{i}$ merges a pair of adjacent entries $\omega_{i},\omega_{i+1}$ of $\omega=(\omega_{1},\dots,\omega_{i},\omega_{i+1},\dots,\omega_{q})$ into a single component $\tilde{\omega}_{i}=\omega_{i}+\omega_{i+1}$, thus forming a new shorter sequence $\mathsf{M}_{i}(\omega)=(\omega_{1},\dots,\tilde{\omega}_{i},\dots,\omega_{q})$. The insert operation $\mathsf{I}_{i}$ either insert $2$ in-between $\omega_{i}$ and $\omega_{i+1}$, thus forming a new longer sequence $\mathsf{I}_{i}(\omega)=(\dots,\omega_{i},2,\omega_{i+1},\dots)$, or, in the case of $\mathsf{I}_{0}$, appends $2$ before the sequence $\omega$, or, in the case $\mathsf{I}_{q}$, appends $2$ after the sequence $\omega$.

We define the order $\omega\succ\omega^{\prime}$, where $\omega,\omega^{\prime}\in\mathbf{\Omega}$, if one can produce $\omega^{\prime}$ from $\omega$ by applying a sequence of these elementary operations.

For a sequence $\omega=(\omega_{1},\omega_{2},\,\dots\,,\omega_{q})\in\mathbf{\Omega}$, we introduce the norm and the reduced norm of $\omega$ by the formulas:

Note that $q$, the cardinality of the support of $\omega$, is equal to $|\omega|-|\omega|^{\prime}$.

Let $\mathring{{\mathsf{R}}}^{\omega}_{d}$ be of the set of all polynomials with the real root multiplicity pattern $\omega$, and let ${\mathsf{R}}_{d}^{\omega}$ be its closure in $\mathcal{P}_{d}$.

For a given collection $\Theta$ of multiplicity patterns $\{\omega\}$, which share the pairity of their norms $\{|\omega|\}$ and is closed under the merge and insert operations, we consider the union $\mathcal{P}_{d}^{\Theta}$ of the subspaces $\mathring{{\mathsf{R}}}_{d}^{\omega}$, taken over all $\omega\in\Theta$ such that $|\omega|\leq d$ and $|\omega|\equiv d\mod 2$. We denote by $\mathcal{P}_{d}^{\mathbf{c}\Theta}$ its complement $\mathcal{P}_{d}\setminus\mathcal{P}^{\Theta}_{d}$.

Since $\mathcal{P}^{\Theta}_{d}$ is contractible, it makes more sense to consider its one-point compactification $\bar{\mathcal{P}}_{d}^{\Theta}$. If the set $\bar{\mathcal{P}}_{d}^{\Theta}$ is closed in $\bar{\mathcal{P}}_{d}$, by the Alexander duality on the sphere $\bar{\mathcal{P}}_{d}\cong S^{d}$, we get

This implies that the spaces $\mathcal{P}_{d}^{\mathbf{c}\Theta}$ and $\bar{\mathcal{P}}_{d}^{\Theta}$ carry the same (co)homological information. Let us describe it in pure combinatorial terms.

Let us consider the following domain in $\mathbb{R}^{d+1}$:

The pair $(\mathcal{E}_{d},\partial\mathcal{E}_{d})$ is of a fundamental importance for us.

For a subposet $\Theta\subset{\mathbf{\Omega}}$ and natural numbers $d,k$, we introduce the following notations:

Assuming that $\Theta\subset{\mathbf{\Omega}}$ is a closed sub-poset, let

We denote by $\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$ the free $\mathbb{Z}$-module, generated by the elements of ${\mathbf{\Omega}}_{\langle d]}$. Using the merge operators ${\mathsf{M}}_{k}$ and the insert operators ${\mathsf{I}}_{k}$ from [KSW2] on ${\mathbf{\Omega}}$, we define homomorphisms $\partial_{{\mathsf{M}}}:\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]\to\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$, $\partial_{{\mathsf{I}}}:\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]\to\mathbb{Z}[{\mathbf{\Omega}}_{\langle d]}]$ by

where $s_{\omega}=_{\mathsf{def}}|\omega|-|\omega|^{\prime}$. In fact, $\partial_{{\mathsf{M}}}$ and $\partial_{{\mathsf{I}}}$ are anti-commuting differentials [KSW2]. Therefore, the sum

is a differential.

For a closed poset $\Theta$, the restrictions of the operators $\partial,\partial_{{\mathsf{M}}}$, and $\partial_{{\mathsf{I}}}$ to the free $\mathbb{Z}$-module $\mathbb{Z}[\Theta]$ are well-defined. Thus, for any closed subposet $\Theta_{\langle d]}\subset\mathbf{\Omega}_{\langle d]}$, we may consider the differential complex $\partial:\mathbb{Z}[\Theta_{\langle d]}]\to\mathbb{Z}[\Theta_{\langle d]}]$, whose $(d-j)$-grading is defined by the module $\mathbb{Z}[\Theta_{\langle d]}\cap\Theta_{|\sim|^{\prime}=j}]$. We denote by $\partial^{\ast}:\mathbb{Z}[\Theta_{\langle d]}]^{\ast}\to\mathbb{Z}[\Theta_{\langle d]}]^{\ast}$ its dual operator, where $\mathbb{Z}[\Theta_{\langle d]}]^{\ast}=_{\mathsf{def}}\mathsf{Hom}(\mathbb{Z}[\Theta_{\langle d]}],\mathbb{Z})$.

Then we consider the quotient set $\mathbf{\Theta}^{\#}_{\langle d]}:=\mathbf{\Omega}_{\langle d]}\big{/}\Theta_{\langle d]}$. For the closed subposet $\Theta_{\langle d]}$, the partial order in $\mathbf{\Omega}_{\langle d]}$ induces a partial order in the quotient $\mathbf{\Theta}^{\#}_{\langle d]}$.

Finally, we introduce a new differential complex $(\mathbb{Z}[\mathbf{\Theta}^{\#}_{\langle d]}],\partial^{\#})$ by including it in the short exact sequence of differential complexes:

We will rely on the following result from [KSW2], which reduces the computation of the reduced cohomology $\tilde{H}^{\ast}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})$ to Algebra and Combinatorics.

Consider the embedding $\epsilon_{d,d+2}:\mathcal{P}_{d}\to\mathcal{P}_{d+2}$, defined by the formula

It preserves the ${\mathbf{\Omega}}$-stratifications of the two spaces by the combinatorial types $\omega$ of real divisors. The embedding $\epsilon_{d,d+2}$ makes it possible to talk about stabilization of the homology/cohomology of the spaces $\bar{\mathcal{P}}_{d}^{\Theta}$ and $\mathcal{P}_{d}^{\mathbf{c}\Theta}$, as $d\to\infty$.

For a closed poset $\Theta\subset{\mathbf{\Omega}}$, consider the closed finite poset $\Theta_{\langle d]}={\mathbf{\Omega}}_{\langle d]}\cap\Theta$. It is generated by some maximal elements $\omega_{\star}^{(1)},\dots,\omega_{\star}^{(\ell)}$, where $\ell=\ell(d)$. We introduce two useful quantities:

Note that

where both summands, the “codimension” $(d-|\omega_{\star}^{(i)}|^{\prime})$ and the “support” $(|\omega_{\star}^{(i)}|-|\omega_{\star}^{(i)}|^{\prime})$, are positive and each does not exceed $d$. At the same time, $\eta_{\Theta}(d)$ may be negative.

In the stabilization results about quasitopies, the quantity

plays the key role.

Now we are in position to state the main stabilization result from [KSW2]:

As a result, we may talk about the stable homology $H_{j}(\mathcal{P}_{\infty}^{\mathbf{c}\Theta};\mathbb{Z})$, the direct limit $\lim_{d\to\infty}H_{j}(\mathcal{P}_{d}^{\mathbf{c}\Theta};\mathbb{Z})$.

Let us describe a few special cases of stabilization from [KSW2]. For $k\geq 1$, $q\in[0,d]$, and $q\equiv d\mod 2$, let us consider the closed poset

Note that, for $\Theta={\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq k}^{(0)}=_{\mathsf{def}}{\mathbf{\Omega}}_{|\sim|^{\prime}\geq k}$, the space $\bar{\mathcal{P}}_{d}^{\Theta}$ is the entire $(d-k)$-skeleton of $\bar{\mathcal{P}}_{d}$.

the absolute value of the Euler number of the differential complex $\big{(}\mathbb{Z}\big{[}{\mathbf{\Omega}}_{|\sim|^{{}^{\prime}}\geq k}^{(q)}\big{]},\partial\big{)}$.

This section forms a bridge between the results from [K9] and our main results from Section 4. The reader may choose to surf Section 3 or to proceed directly to Section 4. Since all the results of this section are instant derivations of the similar results from [K9], we provide just an outline of their validations. Although we will not use directly the results of Section 3 in Section 4, this section may induce the “right mindset” for the reader.

Let $Y$ be a smooth compact $n$-manifold. Having in mind applications to traversing vector flows, we move away from immersions and embeddings $\beta:M\to\mathbb{R}\times Y$ of $n$-manifolds $M$, the topic of [K9], to submersions and regular embeddings $\alpha:X\to\mathbb{R}\times Y$ of compact smooth $(n+1)$-manifolds $X$ with boundary into the product $\mathbb{R}\times Y$. When $\dim(X)=\dim(Y)+1$, the submersions and immersions are the same notion. Moreover, the restriction of a submersion $\alpha$ to $\partial X$ is an immersion. Of cause, if $\alpha:X\to\mathbb{R}\times Y$ is an embedding, so is $\alpha^{\partial}:=\alpha|:\partial X\to\mathbb{R}\times Y$.

Therefore many constructions and notions from [K9], with the help of the correspondence $\alpha\leadsto\alpha^{\partial}$, apply instantly to submersions $\alpha:X\to\mathbb{R}\times Y$ such that $\dim X=\dim Y+1$.

Let us introduce the central to this paper notion of quasitopy for submersions $\alpha:X\to\mathbb{R}\times Y$, an analogue of Definition 3.7 from [K9]. We fix a natural number $d$ and consider a closed sub-poset $\Theta\subset\mathbf{\Omega}$ such that $(\emptyset)\notin\Theta$ (the, so called, $\Lambda$-condition (3.10) from [K9]). For topological reasons, we will consider only the case $d\equiv 0\mod 2$.

Let $\mathcal{L}$ be the $1$-foliation of $\mathbb{R}\times Y$ by the fibers of the obvious projection $\mathbb{R}\times Y\to Y$, and $\mathcal{L}^{\bullet}$ be the $1$-foliation of $\mathbb{R}\times Y\times[0,1]$ by the fibers of $\mathbb{R}\times Y\times[0,1]\to Y\times[0,1]$.

Let $X$ be a compact smooth $(n+1)$-dimensional manifold with boundary. Consider a smooth map $\alpha:X\to\mathbb{R}\times Y$ such that:

• •

  $\alpha$ is a submersion,

• •

  for each $y\in Y$, the total multiplicity $m_{\beta}(y)$ of $\alpha(\partial X)$ with respect to the foliation $\mathcal{L}$ (see [K9], formula (3.4)) is less than or equal to $d$ and $m_{\beta}(y)\equiv d\mod 2$,

• •

  for each $y\in Y$, the combinatorial tangency pattern $\omega^{\alpha}(y)\in\mathbf{\Omega}_{\langle d]}$ of $\alpha(\partial X)$ with respect to $\mathcal{L}$ does not belong to $\Theta$,

• •

  for each $y\in\partial Y$, $\omega^{\alpha}(y)=(\emptyset)$.

Note that the normal bundle $\nu^{\alpha}$ to $\alpha(\partial X)$ in $\mathbb{R}\times Y$ is trivial.

Let $X_{0},X_{1}$ be two compact smooth (oriented) $(n+1)$-dimensional manifolds with boundary. We consider a compact smooth (oriented) $(n+2)$-manifold $W$ with conners $\partial X_{0}\coprod\partial X_{1}$ such that $\partial W=(X_{0}\coprod X_{1})\bigcup_{\{\partial X_{0}\coprod\partial X_{1}\}}\delta W$, where $\delta W$ is a smooth (oriented) cobordism between $\partial X_{0}$ and $\partial X_{1}$. Let $Z=_{\mathsf{def}}\mathbb{R}\times Y$ and $Z^{\partial}=_{\mathsf{def}}\mathbb{R}\times\partial Y$.

Let $A:W\to\mathbb{R}\times Z$, where $\dim(W)=\dim(Z)+1$, be a submersion. In particular, $A:\delta W\to\mathbb{R}\times Z,\;A|_{\partial X_{0}}\to\mathbb{R}\times(Y\times\{0\})$, and $A|_{\partial X_{1}}\to\mathbb{R}\times(Y\times\{1\})$ are immersions.

The next two definitions lay down the foundation for notions of quasitopy of traversing vector fields, the main subject of Section 4 (see Fig. 2).

It is easy to check that the quasitopy of submersions is an equivalence relation. Recall that in [K9], Definition 3.7, we have introduced a similar notion of quasitopy for immersions/embeddings $\beta:(M,\partial M)\to(\mathbb{R}\times Y,\mathbb{R}\times\partial Y)$. There, we used the notation

for the set of equivalence classes of immersions/embeddings $\beta$ under the quasitopy relation.

As for immersions $\beta:M\to\mathbb{R}\times Y$, for any choice of connected components $\kappa_{1}\in\pi_{0}(\partial Y_{1}),\kappa_{2}\in\pi_{0}(\partial Y_{2})$, the connected sum operation (see [K9], formula (3.15))

is well-defined for the quasitopies of submersions.

It converts the set $\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(D^{n};\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$ into a group $\underline{\overline{\mathsf{H}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(n;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$ (abelian for $n>1$). This group acts, via the connected sum operation $\uplus$, on the set $\underline{\overline{\mathsf{QT}}}^{\mathsf{\,sub/emb}}_{\,d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$, provided that a connected component of $\partial Y$ is chosen. To get a better insight, compare this claim with Proposition 3.2 from [K9].

For two pairs $X_{1}\supset A_{1}$ and $X_{2}\supset A_{2}$ of topological spaces, we denote by $[(X_{1},A_{1}),\hfill\break(X_{2},A_{2})]$ the set of homotopy classes of continuous maps $g:X_{1}\to X_{2}$, where $g(A_{1})\subset A_{2}$.

Following the proof of Proposition 3.3 and Theorem 3.2 from [K9], we get Theorem 3.1, their analogue for submersions. It is crucial that here $d\equiv 0\mod 2$, which implies that any regular embedding $\beta:M\to\mathbb{R}\times Y$, such that all the multiplicities $\{m_{\beta}(y)\}_{y\in Y}$ are even, bounds a regular embedding $\alpha:X\to\mathbb{R}\times Y$, where $\partial X=M$ and $\alpha|_{\partial X}=\beta$.

For an $n$-dimensional $Y$, the next lemma reduces the computation of quasitopies $\underline{\overline{\mathsf{QT}}}^{\mathsf{\,emb}}_{\,d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$, based on regular embeddings $\alpha:X\to\mathbb{R}\times Y$ of $(n+1)$-dimensional manifolds $X$, to the computation of quasitopies $\mathsf{QT}^{\mathsf{emb}}_{d,d^{\prime}}(Y;\mathbf{c}\Theta;\mathbf{c}\Theta^{\prime})$, based on the regular embeddings $\beta:M\to\mathbb{R}\times Y$ of $n$-dimensional manifolds $M$.