Manifolds - Perspectives

#### **Chapter 1**

## Free Actions of Compact Lie Groups on Manifolds

*Thales Fernando Vilamaior Paiva*

#### **Abstract**

If a compact Lie group *G* acts freely on a manifold *X*, the resulting orbit space *X=G* is itself a manifold. This text is concerned with the existence of such actions as well as the cohomological classification of the respective orbit spaces by using some known tools of equivariant cohomology theory and spectral sequences.

**Keywords:** free actions, manifolds, orbit spaces, cohomology, spectral sequences

#### **1. Introduction**

When a topological group *G* acts on a manifold *X*, we can define the orbit space *X=G*, that does not necessarily have the structure of a manifold. However, when *G* is a compact Lie group and we impose the condition that the action be free, which means that the isotropy subgroup *Gx* contains only the trivial element 1 of *G*, for any *x*∈ *X*, then we can construct on *X=G* a manifold structure.

The general situation above can be illustrated by the construction of the projective spaces *kPn* , for *k* ¼ , or , that are orbit spaces of certain free actions of the groups 2, *S*<sup>1</sup> , and *S*<sup>3</sup> on spheres *S<sup>n</sup>*, *S*<sup>2</sup>*n*þ<sup>1</sup> , and *S*<sup>4</sup>*n*þ<sup>3</sup> , respectively. Such spaces, as we know, appear in different contexts and for this reason there is an interest in obtaining certain algebraic and geometric invariants that characterize them, such as their homotopic and cohomological classification.

Thus, given *X* and *G*, we can naturally consider the general problem of classifying the space of orbits *X=G*, through the existence or not of a free action of *G* on *X*, which is a typical transformation group problem associated with this data.

The interest in this type of problem, particularly when *G* is a finite group, has become greater since the publication of work [1], by H. Hopf in 1926, in which one formalizes the purpose of classification of all manifolds whose universal covering is homeomorphic to a sphere *Sn:* This problem, as we know, is equivalent to the classification of all finite groups that can act freely on *S<sup>n</sup>:*

However, we realize that to get a homotopic classification of such spaces can become extremely complicated, even when the space *X* already has a known classification. For example, let *Sn* be the *<sup>n</sup>*�sphere seen as the one-point compactification of euclidean space *<sup>n</sup>*, which does not have a complete classification of its homotopy groups. As a result, instead of a homotopic classification, we can consider a cohomological classification of these orbit spaces.

In this direction, we realize that the difficulty in computing the cohomology of the orbit space *X=G*, by direct methods, becomes evident when *X* has nontrivial cohomology on several levels. On the other hand, many such results have been obtained by using some tools of equivariant cohomology theory. This is due to the fact that as long as *G* is a compact Lie group acting freely on a space *X*, there is a homotopy equivalence between the orbit space *X=G* and the Borel space *XG*, so that we can use the so-called Leray-Serre spectral sequence associated with the Borel fibration:

$$X \longleftarrow X\_{\mathbb{G}} \longrightarrow B\_{\mathbb{G}},\tag{1}$$

where *BG* denotes the classifying space for group *G*, to investigate the cohomology ring *<sup>H</sup>*<sup>∗</sup> ð Þffi *XG*; *<sup>R</sup> <sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*; *<sup>R</sup> :*

In this text, we briefly deal with these tools and then we present some applications regarding the existence of free actions of certain compact Lie groups in some classes of smooth manifolds, as well as the cohomological classification of the respective orbit spaces.

#### **2. Preliminary concepts**

#### **2.1 Group actions and classifying spaces**

Let *μ* : *G* � *X* ! *X* be an action of a topological group *G* on a topological space *X*, i. e. *μ* is a continuous map such that

$$
\mu(\mathfrak{g}, \mu(h, \,\,\,\pi)) = \mu(gh, \,\,\pi),
\tag{2}
$$

$$
\mu(e, \mathfrak{x}) = \mathfrak{x},
\tag{3}
$$

for any *g*,*h*∈ *G* and for any *x*∈*X*, where *e* indicates the neutral element of *G*. In this case, we say that *G* acts on *X* and *X* is a *G*�space.

As it is usual, we denote by *μ*ð Þ¼ *g*, *x g x*ð Þ or simply *μ*ð Þ¼ *g*, *x gx* to indicate the action of the element *g* of *G* on *x*∈*X:*

For each *x*∈ *X*, the subspace *G x*ð Þ¼ f g *gx*; *g* ∈ *G* is called the orbit of the element *x:* It is a simple task to show that for any two orbits *G x*ð Þ and *G y*ð Þ, then *G x*ð Þ ∩ *G y*ð Þ¼ ∅ or *G x*ð Þ¼ *G y*ð Þ*:* Therefore, we can define the orbit space:

$$X/G = \{ G(\mathfrak{x}); x \in X \},\tag{4}$$

which is provided with the quotient topology induced by the natural map *q* : *X* ! *X=G*, given by *q x*ð Þ¼ *G x*ð Þ, which is called orbit map.

**Example 2.1.1.** Any group *G* acts on itself by multiplication. Precisely, we can define *μ* : *G* � *G* ! *G* by *μ*ð Þ¼ *g*, *h gh:*

An action *μ* of *G* on *X* induces a group homomorphism Γ*<sup>μ</sup>* : *G* ! Homeoð Þ *X* , such that, for each *g* ∈ *G*, we define Γ*μ*ð Þ¼ *g Lg*, where

$$L\_{\mathfrak{g}}: X \to X, L\_{\mathfrak{g}}(\mathfrak{x}) = \mathfrak{g}\mathfrak{x}.\tag{5}$$

The action *μ* is called effective when the kernel of the homomorphism Γ*<sup>μ</sup>* contains only the trivial element *e*∈ *G* and is called trivial when kerΓ*<sup>μ</sup>* ¼ *G:*

For each *x*∈*X*, we call the isotropy subgroup at *x* the following subgroup of *G*:

$$\mathbf{G}\_{\mathfrak{x}} = \{ \mathbf{g} \in G; \mathbf{g}\mathfrak{x} = \mathfrak{x} \}. \tag{6}$$

When *Gx* ¼ f g*e* , for any point *x*∈*X* then the action is called free, and *X* is said to be a free *<sup>G</sup>*�space. The set *<sup>X</sup><sup>G</sup>* <sup>¼</sup> f g *<sup>x</sup>*∈*X*; *gx* <sup>¼</sup> *<sup>x</sup>* is called the fixed point set of the action.

**Remark 2.1.1.** If *X* is a haursdorff space and *G* is a compact space, it is well known that any action *μ* : *G* � *X* ! *X* is a closed map, according to Theorem 1.2 of [2]. Furthermore, in this case, the subspace *μ*ð Þ *G* � *A* ⊆ *X* is closed (resp. compact) if *A* is closed (resp. compact).

Let *X* and *Y* be *G*�spaces. If the map *f* : *X* ! *Y* is equivariant, i. e. *f gx* ð Þ¼ *gf x*ð Þ, for any *g* ∈ *G* and any *x*∈ *X*, then we can define the map:

$$
\overline{f}: \mathbf{X}/\mathbf{G} \to \mathbf{Y}/\mathbf{G}, \; \overline{f}(\mathbf{G}(\mathbf{x})) = \mathbf{G}(f(\mathbf{x})).\tag{7}
$$

**Remark 2.1.2.** Even though it is possible to investigate actions in arbitrary topological spaces, we are interested in observing certain structures both in *X* and in the orbit space *X=G*, so that we will assume, from now on, that *X* is a manifold (smooth or not) and *G* is a Lie group.

We recall that a Lie group *G* is a topological group that is also a (real) finitedimensional smooth manifold, in which the multiplication operation *g*1, *g*<sup>2</sup> <sup>↦</sup> *<sup>g</sup>*1*g*<sup>2</sup> and the inversion map *g* ↦ *g*�<sup>1</sup> are smooth.

**Example 2.1.2.** The matrix groups *GL n*ð Þ , , *GL n*ð Þ , , of the invertible *n* � *n*�matrices with entries in or , respectively, are standard examples of Lie groups, along with respective subgroups (special linear groups) *SL n*ð Þ , and *SL n*ð Þ , *:*

**Example 2.1.3.** Let *O n*ð Þ⊂ *GL n*ð Þ , be the subgroup of the orthogonal matrices, i. e. those in which *AAt* <sup>¼</sup> *Id* and let *U n*ð Þ be the subgroup of *SL n*ð Þ , of the unitary matrices *AAt* <sup>¼</sup> *Id:* We can define the special orthogonal group by *SO n*ð Þ¼ *O n*ð Þ ∩ *SL n*ð Þ , and the special unitary group by *SU n*ð Þ¼ *U n*ð Þ ∩ *SL n*ð Þ , *:*

We know that *SU*ð Þ<sup>2</sup> is isomorphic to *<sup>S</sup>*<sup>3</sup> identified as the subgroup of the unitary quaternions. Also we know that the isomorphic groups *U*ð Þ1 and *SO*ð Þ2 are isomorphic to the circle group *S*<sup>1</sup> *:*

For each integer *m*, let *<sup>m</sup>* ¼ *=m* be the group of the integer modulo *m*, which can be identified with the subgroup of *<sup>S</sup>*<sup>1</sup> of all *<sup>m</sup>*�th roots of unity. In particular, we have the following chain of Lie (sub)groups:

$$\mathbb{Z}\_m \subset \mathbb{S}^1 \cong U(\mathbb{1}) \cong \mathbb{S}O(2) \subset \mathrm{SU}(2) \cong \mathbb{S}^3. \tag{8}$$

**Example 2.1.4.** (Free actions on spheres) Let *<sup>X</sup>* be the *<sup>n</sup>*�sphere *Sn* <sup>⊂</sup> *<sup>n</sup>*þ<sup>1</sup> and *<sup>G</sup>* be the finite group 2*:* Then, *G* acts freely on *X* by the antipodal map *μ*ð Þ¼ 1, *x A x*ð Þ¼ �*x*, and in this case we have *<sup>X</sup>=<sup>G</sup>* <sup>¼</sup> *Pn* the real projective space.

For *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>2</sup>*n*�<sup>1</sup> <sup>⊂</sup> <sup>2</sup>*<sup>n</sup>* ffi *<sup>n</sup>* and *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> seen as a subgroup of the complex plane, we can consider the free action induced by complex multiplication:

$$
\mu: G \times X \to X, \mu(z, \ (z\_1, \ \cdots, \ z\_n)) = (zx\_1, \ \cdots, zx\_n), \tag{9}
$$

and it follows that *<sup>X</sup>=<sup>G</sup>* <sup>¼</sup> *P<sup>n</sup>*�<sup>1</sup> is the complex projective space.

Let *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup>*4*n*�<sup>1</sup> <sup>⊂</sup> <sup>4</sup>*<sup>n</sup>* ffi *<sup>n</sup>* and *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> identified with the group of the unitary quaternions *SU*ð Þ2 , where denotes the quaternion algebra:

$$\mathbb{H} = \left\{ \begin{bmatrix} a & -\overline{\beta} \\ \beta & \overline{a} \end{bmatrix} ; a, \beta \in \mathbb{C} \right\}. \tag{10}$$

Similar to the previous case, we can define the free action *μ* : *G* � *X* ! *X*, induced by the multiplication, such that *<sup>X</sup>=<sup>G</sup>* <sup>¼</sup> *Pn*�<sup>1</sup> the quaternionic projective space.

**Example 2.1.5.** (Free involutions on projective spaces) An involution<sup>1</sup> on a space *X* is a continuous action of the group <sup>2</sup> on *<sup>X</sup>:* Let ½ � *<sup>x</sup>*1, *<sup>x</sup>*2, <sup>⋯</sup>, *<sup>x</sup>*2*n*�1, *<sup>x</sup>*2*<sup>n</sup>* <sup>∈</sup> *P*2*n*�<sup>1</sup> <sup>¼</sup> *S*2*n*�<sup>1</sup> *=*<sup>2</sup> be an arbitrary element. Its easy to see that the map:

$$T([\mathbf{x}\_1, \ \mathbf{x}\_2, \ \cdots, \ \mathbf{x}\_{2n-1}, \ \mathbf{x}\_{2n}]) = [-\mathbf{x}\_2, \mathbf{x}\_1, \ \cdots, -\mathbf{x}\_{2n}, \mathbf{x}\_{2n-1}] \tag{11}$$

defines a free involution on *P*<sup>2</sup>*n*�<sup>1</sup>

Similarly, for ½ � *<sup>z</sup>*1, *<sup>z</sup>*2, <sup>⋯</sup>, *zm*, *zm*þ<sup>1</sup> an arbitrary element in *P<sup>m</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>2</sup>*m*þ<sup>1</sup> *=S*<sup>1</sup> , if *m* > 1 is odd then we can define the free involution:

*:*

$$S([z\_1, z\_2, \dots, z\_m, \, z\_{m+1}]) = [-\overline{z}\_2, \overline{z}\_1, \dots, -\overline{z}\_{m+1}, \overline{z}\_m] \tag{12}$$

$$\mathbb{C}P^{\mathfrak{m}}.\tag{13}$$

When a group *G* acts on a manifold *X*, in general, we can consider the orbit space *X=G* with no other additional structure. However, when the action is free and proper, then this orbit space can be seen as a manifold, according to the Quotient Manifold Theorem:

**Theorem 2.1.1.** (Quotient Manifold Theorem) [4]. Let *G* be a compact Lie group acting freely (and smoothly) on a smooth manifold *X:* If the action is free and proper, then *X=G* is also a smooth manifold of dimension dimð Þ� *X* dimð Þ *G* , such that the quotient map *X* ! *X=G* is a principal *G*�bundle and a smooth submersion.

**Remark 2.1.3.** As an immediate consequence of the previous theorem, for every cohomology functor *H*<sup>∗</sup> we have *H<sup>j</sup>* ð Þ¼ *X=G*; *R* f g0 , for all *j*>dim*X* � dim*G*, for any commutative ring with unity *R:*

Recall that given any compact Lie group *G* we can construct the universal *G*�bundle *pG* : *EG* ! *BG*, with fiber space *G*, where the total space *EG* is the *G*�space defined as the join operation2 of infinite copies of *G*, and the base is the quotient space (by diagonal action) *BG* ¼ *EG=G*, which is called the classifying space for *G* and *pG* is the projection.

**Example 2.1.6.** (Classifying spaces for 2,*S*<sup>1</sup> and *S*<sup>3</sup> ) For *G* ¼ 2, we can see that *BG* <sup>¼</sup> *EG=<sup>G</sup>* ffi *P*<sup>∞</sup>*:* Consequently, the mod 2 cohomology of the classifying space *BG* is given by *<sup>H</sup>*<sup>∗</sup> ð Þ¼ *BG*; <sup>2</sup> 2½ �*<sup>t</sup>* , where deg*<sup>t</sup>* <sup>¼</sup> <sup>1</sup>*:* Regarding to *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> , since *BG* ¼ *EG=<sup>G</sup>* ffi *P*<sup>∞</sup>, then *<sup>π</sup>*1ð Þ¼ *BG* 1 and the mod 2 cohomology is give by *<sup>H</sup>*<sup>∗</sup> ð Þ¼ *BG*; <sup>2</sup> 2½ � *<sup>τ</sup>* , where deg*<sup>τ</sup>* <sup>¼</sup> <sup>2</sup>*:* With respect to the group *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> , it follows that *BG* ¼ *EG=G* ffi *P*<sup>∞</sup>*:* Since<sup>3</sup> *<sup>π</sup>i*ð Þffi *BG <sup>π</sup><sup>i</sup>*�<sup>1</sup>ð Þ *<sup>G</sup>* , therefore *<sup>π</sup>*1ð Þffi *BG* 1 and the mod 2 cohomology of *BG* is given by *<sup>H</sup>*<sup>∗</sup> ð Þ¼ *BG*; <sup>2</sup> 2½ � *<sup>τ</sup>* , where deg*<sup>τ</sup>* <sup>¼</sup> <sup>4</sup>*:*

<sup>1</sup> Formally, an involution is a map *<sup>T</sup>* : *<sup>X</sup>* ! *<sup>X</sup>* such tha *<sup>T</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup>*:* For an extensive treatment of involutions on manifolds, see [3].

<sup>2</sup> For more details on Milnor's construction of classifying spaces, see [5] and Section 4.11 of [6].

<sup>3</sup> For more details, see Corollary 8.13 of [7].

#### **2.2 The Leray-Serre spectral sequence of a Borel fibration**

Let *X* be a free *G*�space, *q* : *X* ! *X=G* the orbit map, and *ϖ* : *EG* ! *BG* the universal *G*�bundle. The group *G* acts freely on the product *X* � *EG*, by the diagonal action *g x*ð Þ¼ , *y* ð Þ *gx*, *gy* , where do we get

$$
\rho: X \times E\_G \to (X \times E\_G)/G = X\_G \tag{14}
$$

the respective orbit map. The quotient space *XG* is also know as the Borel space. Since the projections *proj*<sup>1</sup> : *X* � *EG* ! *X* and *proj*<sup>2</sup> : *X* � *EG* ! *EG* are

*G*�equivariant, they induce the fibrations *π* and *p*, respectively, according to the diagram below:

$$\begin{array}{c} E\_G \xleftarrow{\text{proj}\_2} X \times E\_G \xrightarrow{\text{proj}\_1} X\\ \bigcup\_{G\_G} \xleftarrow{\text{proj}\_1} X\_G \xrightarrow{\text{proj}\_2} X/G \end{array} \tag{15}$$

where *π* is called the Borel fibration with fiber *X*, and *p* is a principal *G*�bundle. Moreover, under the above hypothesis, the fiber *EG* of *p* is contractible and therefore *p* is a homotopy equivalence<sup>4</sup> , which induces a natural isomorphism *p*<sup>∗</sup> : *<sup>H</sup>*<sup>∗</sup> ð Þ! *<sup>X</sup>=G*; *<sup>R</sup> <sup>H</sup>*<sup>∗</sup> ð Þ *XG*; *<sup>R</sup>* , for any commutative ring with unit *<sup>R</sup>:*

By Theorem 5.2 of [9], there is a first quadrant cohomological spectral sequence *E*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* converging to *<sup>H</sup>*<sup>∗</sup> ð Þffi *XG*; *<sup>R</sup> <sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*; *<sup>R</sup>* , as an algebra, such that the *<sup>E</sup>*2�page *<sup>E</sup><sup>p</sup>*,*<sup>q</sup>* <sup>2</sup> , is isomorphic to

$$E\_2^{p,q} \cong H^p(B\_G; \mathcal{H}^q(X; R)),\tag{16}$$

where the symbol ℋ*<sup>q</sup>* ð Þ *X*; *R* indicates a system of local coefficients twisted by the action of the fundamental group *π*1ð Þ *BG* on the cohomology ring of *X:*

When *<sup>π</sup>*1ð Þ *BG* acts trivially on *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>*; *<sup>R</sup>* , the system of local coefficients <sup>ℋ</sup>*<sup>q</sup>* ð Þ *X*; *R* is simple and, according to Proposition 5.6 of [9], the *E*2�page as in (13) takes the form:

$$E\_2^{p,q} \cong H^p(B\_G; R) \otimes\_R H^q(X; R),\tag{17}$$

what happens, in particular, when *π*1ð Þ¼ *BG* 1*:* Moreover, by Theorem 5.9 of [9], the homomorphisms

$$H^{q}(B\_{G};R) = E\_{2}^{q,0} \twoheadrightarrow \cdots \twoheadrightarrow E\_{q}^{q,0} \twoheadrightarrow E\_{q+1}^{q,0} = E\_{\Leftrightarrow}^{q,0} \sqsubset H^{q}(X\_{G};R) \tag{18}$$

and

$$H^q(X\_G; R) \twoheadrightarrow E^{0,q}\_{\llcorner \llcorner} = E^{0,q}\_{q+1} \subseteq E^{0,q}\_q \subseteq \cdots \subseteq E^{0,q}\_2 = H^q(X; R) \tag{19}$$

coincide with the homomorphisms *π* <sup>∗</sup> : *H<sup>q</sup>* ð Þ! *BG*; *<sup>R</sup> <sup>H</sup><sup>q</sup>* ð Þ *XG*; *<sup>R</sup>* and *<sup>i</sup>* <sup>∗</sup> : *Hq* ð Þ! *XG*; *<sup>R</sup> <sup>H</sup><sup>q</sup>* ð Þ *X*; *R* , respectively.

<sup>4</sup> According to ([8], p. 180.)

#### **3. Some applications**

Supposing that there is a free action of a group *G* on *X*, there is an interest in classifying the orbit space *X=G*, in the same way as it happens on the real, complex, and quaternionic projective spaces.

In order to do this, we will cite some results that use these tools presented in the previous sections to obtain cohomological classifications for certain known spaces.

However, we observe that in general way, the computation of the cohomology of the orbit space *X=G* can be a difficult task when the space *X* is not a sphere or, more generally, when *X* has nontrivial cohomology *H<sup>j</sup>* ð Þ *X*; *R* 6¼ f g0 on several levels 0<*j*<dim*X*

Remark 3.1. Since all these results use only the cohomological struture of space *X*, we observe that the same conclusions can be obtained replacing *X* by any finitistic space that has same *C ^* ech cohomology algebra. Recall that a finitistic space is a paracompact Hausdorff space whose every open covering has a finite-dimensional open refinement, where the dimension of a covering is one less than the maximum number of members of the covering which intersect nontrivially. It is known [10, 11] that if *G* is a compact Lie group acting continuously on *X*, then *X* is finitistic if and only if the orbit space *X=G* is finitistic. Therefore, we can consider the problem of cohomology classification of the orbit spaces up to finitistic spaces of isomorphic cohomology to initial space *X:*

#### **3.1 Free actions on spheres and projective spaces**

In 1926, Hopf posed the general problem of classifying all groups that present freely in *S<sup>n</sup>:* Posteriorly, in 1957, J. Milnor provided some answers for this problem by showing, among other things, that the symmetric group *S*<sup>3</sup> cannot act freely on *S<sup>n</sup>:*

Even considering this classification only on the category of compact Lie groups, this problem still does not have a complete solution for a arbitrary sphere *Sn:* However, if *n* is even, the only finite group that acts freely on *S<sup>n</sup>* is the group 2*:*

In fact, if *<sup>G</sup>* acts freely on *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>2</sup>*<sup>k</sup>* , then the quotient map *X* ! *X=G* is a covering projection; therefore, with *χ*ð Þ� the Euler characteristic, it follows that

$$\mathcal{Q} = \chi \text{(}\text{S}^{2k}\text{)} = |\mathbf{G}| \cdot \chi \text{(}\text{S}^{2k}/\text{G}\text{)},\tag{20}$$

which implies that ∣*G*∣ ¼ 1 or ∣*G*∣ ¼ 2*:* Since the action is free, the only possibility is <sup>∣</sup>*G*<sup>∣</sup> <sup>¼</sup> 2 and then *<sup>G</sup>* <sup>¼</sup> 2*:* Furthermore, the resulting orbit space *<sup>S</sup><sup>n</sup>=*<sup>2</sup> has the same homotopy type of real projective space *Pn:*

For *n* odd, the Section 3.8 of [2] contains a compilation of results related to the existence of free *<sup>G</sup>*�actions on spheres *<sup>S</sup><sup>n</sup>:* In particular, with respect to groups of positive dimension, the Theorem 8.5 of [2] states that a group *G* that act freely on *S<sup>n</sup>* must be isomorphic to *S*<sup>3</sup> , *S*<sup>1</sup> , or *N S*<sup>1</sup> *:*

Suppose that *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> and let *<sup>X</sup>* <sup>¼</sup> *Sn* and *XG* ! *BG* the Borel fibration. For *E*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* the associated Leray-Serre spectral sequence, we have

$$E\_2^{p,q} = H^p(B\_G; \mathbb{Z}\_2) \otimes H^q(X; \mathbb{Z}\_2) = \begin{cases} \mathbb{Z}\_2 \otimes \mathbb{Z}\_2, \text{ if } p \text{ is odd and } q = 0, \text{ or} \\ \{0\}, \text{ otherwise.} \end{cases}$$

Since this sequence converges to *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*;<sup>2</sup> , it follows that it cannot collapse on *<sup>E</sup>*2�page, which means that there is a nontrivial differential *<sup>d</sup>*2*<sup>k</sup>* : *<sup>E</sup>*0,*<sup>n</sup>* <sup>2</sup>*<sup>k</sup>* ! *<sup>E</sup>*2*k*,0 <sup>2</sup>*<sup>k</sup>* , for *k*∈ such that *n* ¼ 2*k* � 1*:*

Therefore, the sequence collapse on *E*2*k*þ1�page is *E*<sup>∞</sup> ¼ *E*2*k*þ1, whose only nonzero row is *E*even,0 <sup>∞</sup> and the total complex is isomorphic to the graded cohomology ring *H*<sup>∗</sup> *Pn*�<sup>1</sup> ; <sup>2</sup> *:*

Proceeding in a similar way, we can show that *H*<sup>∗</sup> *S*4*n*�<sup>3</sup> *=S*<sup>3</sup> ;<sup>2</sup> <sup>¼</sup> *<sup>H</sup>*<sup>∗</sup> *Pn*�<sup>1</sup> ; <sup>2</sup> *:*

In order to generalize this type of problem, we can consider *X* a product of spheres. For example, L. W. Cusick [12] showed that if a finite group G acts freely on a product of spheres of even dimensions, *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup>*2*n*<sup>1</sup> � <sup>⋯</sup> � *<sup>S</sup>*2*nk* , then then *<sup>G</sup>* must be isomorphic to a group of the type *<sup>r</sup>* 2, for some *r*≤*k:*

Concerning on free actions of a finite group *p*, *p* prime, and the circle group *S*<sup>1</sup> on a product of spheres *<sup>S</sup><sup>m</sup>* � *<sup>S</sup><sup>n</sup>*, Dotzel et al. [13] showed the following classification results according to Theorems 3.1.1, 3.1.2, and 3.1.3.

**Theorem 3.1.1.** Let *p*, *<sup>p</sup>* an odd prime, act freely on *<sup>X</sup>* <sup>¼</sup> *Sm* � *<sup>S</sup><sup>n</sup>*, 0 <sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup> *<sup>n</sup>:* Then, *H*<sup>∗</sup> *X=p*; *<sup>p</sup>* is isomorphic to *p*½ � *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup> <sup>=</sup>ϕ*ð Þ *<sup>x</sup>*, *<sup>y</sup>*, *<sup>z</sup>* , as a graded commutative algebra, where *ϕ*ð Þ *x*, *y*, *z* is one of the following ideals:


**Theorem 3.1.20**. Let <sup>2</sup> act freely on *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup><sup>m</sup>* � *Sn*, 0<sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup>*n:* Then, *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=*2; <sup>2</sup> is isomorphic to *p*½ � *y*, *z =ψ*ð Þ *y*, *z* , as a graded commutative algebra, where *ψ*ð Þ *y*, *z* is one of the following ideals:


**Theorem 3.1.3.** Let *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> act freely on *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup><sup>m</sup>* � *Sn*, 0<sup>&</sup>lt; *<sup>m</sup>* <sup>≤</sup>*n:* Then, *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*; is isomorphic to ½ � *<sup>y</sup>*, *<sup>z</sup> <sup>=</sup>ψ*ð Þ *<sup>y</sup>*, *<sup>z</sup>* , as a graded commutative algebra, where *ψ*ð Þ *y*, *z* is one of the following ideals:


iii. *<sup>y</sup>*ð Þ *<sup>n</sup>*þ<sup>1</sup> *<sup>=</sup>*2, *<sup>z</sup>*<sup>2</sup> � *bym* , *<sup>n</sup>* odd, deg*<sup>y</sup>* <sup>¼</sup> 2, deg*<sup>z</sup>* <sup>¼</sup> *<sup>m</sup>*, and *<sup>b</sup>* 6¼ 0 only when *<sup>m</sup>* is even and 2*m* <*n:*

Using the same techniques, it is shown in [14] similar results regarding the action of groups *S*<sup>1</sup> and *S*<sup>3</sup> on the product of spheres, considering both rational and mod 2 coefficients.

*:*

**Theorem 3.1.4.** The group *<sup>S</sup>*<sup>3</sup> cannot act freely on a *<sup>n</sup>*�torus *<sup>X</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> *<sup>n</sup>*

*Proof.* Let *<sup>X</sup>* be the *<sup>n</sup>*�torus *<sup>S</sup>*<sup>1</sup> *<sup>n</sup>* and suppose that *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> act freely on *<sup>X</sup>*, with *n*≥ 3*:* Let *x*1,⋯,*xn* ∈ *H*<sup>1</sup> ð Þ *X*, <sup>2</sup> be the generators. By Quotient Manifold Theorem, the spectral sequence *E*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* associated with the Borel fibration *XG* ! *BG* does not collapse on the *<sup>E</sup>*2�term. Therefore, there must exist some nontrivial differential *dp*,*<sup>q</sup> <sup>r</sup>* , for a certain *r*≥2, such that

$$E\_r^{p,q} \cong E\_{r-1}^{p,q} \cong \cdots \cong E\_2^{p,q} = H^p(B\_G; \mathbb{Z}\_2) \otimes H^q(X; \mathbb{Z}\_2),$$

and it is clear that this is only possible when *r*≥ 4*k*, for some *k*∈ℕ*:*

Let us suppose that *<sup>r</sup>* <sup>¼</sup> 4 and let *<sup>y</sup>* <sup>¼</sup> *xi*1*xi*2*xi*<sup>3</sup> <sup>∈</sup> *<sup>H</sup>*<sup>3</sup> ð Þ *X*; <sup>2</sup> be an element for which *d*0,3 <sup>4</sup> ð Þ¼ <sup>1</sup> <sup>⊗</sup> *<sup>y</sup> <sup>τ</sup>* <sup>⊗</sup> <sup>1</sup>*:* By dimensional reasons, *<sup>d</sup>*0,1 <sup>4</sup> ð Þ¼ 1 ⊗ *xi* 0 for all 1≤*i* ≤*n*; therefore, it follows that

$$\pi \otimes \mathbf{1} = d\_4^{0,3}(\mathbf{1} \otimes \mathbf{y}) = (\mathbf{1} \otimes \boldsymbol{\varkappa}\_{i\_1})(\mathbf{1} \otimes \boldsymbol{\varkappa}\_{i\_2})d\_4^{0,1}(\mathbf{1} \otimes \boldsymbol{\varkappa}\_{i\_3}) = \mathbf{0},$$

for *<sup>τ</sup>* the generator of *<sup>H</sup>*<sup>∗</sup> ð Þ *BG*; <sup>2</sup> , which is a contradiction. Since this argument works for any *r*≥4 and for any *y*∈ *H<sup>j</sup>* ð Þ *X*; <sup>2</sup> , *j*≥3, it follows that *G* cannot act freely on *<sup>X</sup>:* □

Let *p*≥ 2 be a positive integer and *q*1,⋯,*qm* be integers coprime to *p*, where *m* ≥ 1. Then the action of *<sup>p</sup>* on *S*<sup>2</sup>*m*�<sup>1</sup> ⊂ *<sup>m</sup>* defined by:

$$\left(e^{2\pi i q\_1/p}, \cdots, e^{2\pi i q\_m/p}\right) \* \left(z\_1, \cdots, z\_m\right) = \left(e^{2\pi i q\_1/p}z\_1, \cdots, e^{2\pi i q\_m/p}z\_m\right)$$

is itself free. Therefore, the resulting orbit space is a compact Hausdorff orientable manifold of dimension 2*m* � 1, which is called lens space and it is denoted by:

$$\mathbb{S}^{2m-1}/\mathbb{Z}\_p = L\_p^{2m-1}(q\_1,\cdots,q\_m) = L\_p^{2m-1}(q)\dots$$

**Theorem 31.5.** [15] Let *<sup>G</sup>* <sup>¼</sup> <sup>2</sup> act freely on *<sup>X</sup>* <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>*m*�<sup>1</sup> *<sup>p</sup>* ð Þ*<sup>q</sup> :* Then, *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*;<sup>2</sup> is isomorphic to one of the following graded commutative algebras:


*Free Actions of Compact Lie Groups on Manifolds DOI: http://dx.doi.org/10.5772/intechopen.106404*

Related to actions of <sup>2</sup> on the product of projective spaces, both real and complex, we can mention the work [16], which provides a list of possible cohomology algebras for the respective orbit spaces. In [17], the authors showed that the group *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> cannot act freely on the real projective space of any dimension.

**Theorem 3.1.6.** The group *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> cannot act freely on *<sup>X</sup>* <sup>¼</sup> *Pn*,*Pn*, for any *<sup>n</sup>* <sup>&</sup>gt;0*: Proof.* Let us suppose that the group *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> acts freely on *<sup>X</sup>* <sup>¼</sup> *Pn:* Then, the spectral sequence *E*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* associated with the Borel fibration *<sup>X</sup>*↣*XG* ! *BG*, which has the *<sup>E</sup>*2�term given by *Ep*,*<sup>q</sup>* <sup>2</sup> <sup>¼</sup> *<sup>H</sup>p*ð Þ *BG*; <sup>2</sup> <sup>⊗</sup> *<sup>H</sup><sup>q</sup>* ð Þ *<sup>X</sup>*; <sup>2</sup> , converges to *<sup>H</sup>*<sup>∗</sup> ð Þffi *XG*; <sup>2</sup> *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*;<sup>2</sup> , as an algebra. By the cohomology structures of *BG* ffi *P*<sup>∞</sup> and *<sup>X</sup>* <sup>¼</sup> *Pn*, it follows that

$$E\_2^{p,q} \cong \begin{cases} \mathbb{Z}\_2, & \text{if } p = 4i \text{ and } q = 2j, \text{ for all } i, j \ge 0, \\ \{0\}, & \text{otherwise.} \end{cases}$$

Therefore, a differential *d<sup>p</sup>*,*<sup>q</sup> <sup>r</sup>* : *Ep*,*<sup>q</sup> <sup>r</sup>* ! *Ep*þ*r*,*q*þ1�*<sup>r</sup> <sup>r</sup>* with bidegree ð Þ *<sup>r</sup>*, 1 � *<sup>r</sup>* , is nontrivial only if *p* ¼ 4*i* and *q* ¼ 2*j*≤ 2*n*, for some positive integers *i* and *j:* In this case, we have the following equality involving the bidegrees: 4ð *i* þ *r*, 2*j* þ 1 � *r*Þ ¼ ð Þ 4*k*, 2*l* , for certain integers *k*,*l*>0, that is, these numbers must satisfy the linear system:

$$\begin{cases} 4i + r = 4k, \\ 2j + 1 - r = 2l, \end{cases}$$

that clearly has no integer solution; therefore, we conclude that all differentials *d*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* are trivial, for all *r*≥2*:* This implies that the sequence collapses on its *Er* ffi *E*2�term and contradicts the Quotient Manifold Theorem.

Similarly, let us suppose that the group *<sup>S</sup>*<sup>1</sup> acts freely on *<sup>X</sup>* <sup>¼</sup> *P<sup>n</sup>*, and let us consider *E*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* the spectral sequence associated with the Borel fibration *XS*<sup>1</sup> ! *BS*<sup>1</sup> , whose *<sup>E</sup>*2�term is given by *Ep*,*<sup>q</sup>* <sup>2</sup> ffi *<sup>H</sup><sup>p</sup> BS*<sup>1</sup> ; <sup>2</sup> ⊗ *H<sup>q</sup>* ð Þ *X*;<sup>2</sup> *:*

Let *<sup>t</sup>* be the generator of *<sup>H</sup>*<sup>∗</sup> *P*<sup>∞</sup> ð Þffi ; <sup>2</sup> *<sup>H</sup>*<sup>∗</sup> *BS*<sup>1</sup> ; <sup>2</sup> and *τ* be the generator of *<sup>H</sup>*<sup>∗</sup> *P<sup>n</sup>* ð Þ ; <sup>2</sup> *:* Then,

$$E\_2^{p,q} \cong \begin{cases} \mathbb{Z}\_2, & \text{if } p = 2i \text{ and } q = 4j, \, i, j \ge 0, \\ \{0\}, & \text{otherwise.} \end{cases}$$

By Quotient Manifold Theorem, the spectral sequence does not collapse on it *<sup>E</sup>*2�term; therefore, there must exist some nontrivial differential *<sup>d</sup>*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup> :* If *r*≥2 is the smallest integer for which this happens, so that

$$E\_r^{p,q} \cong E\_{r-1}^{p,q} \cong \cdots \cong E\_2^{p,q},$$

for all *p*,*q*≥ 0, we see that this is only possible when the integers *r*,*i*,*j*, and *k* (which are obtained from the equality between the bidegrees involved) satisfy the linear system:

$$\begin{cases} r = 2i, \\ 4j + 1 - r = 4k. \end{cases}$$

But this system has no integer solution; therefore, the group *S*<sup>1</sup> cannot act freely on *X:* Since *S*<sup>1</sup> is a subgroup of *S*<sup>3</sup> , then *X* does not admit any free action of *S*<sup>3</sup> *:* □

## **3.2 Free actions on spaces of type** ð Þ *a***,** *b*

Let *X* be a finite CW complex. We say that *X* is a space of type ð Þ *a*, *b* , characterized by an integer *n*> 0, if

$$H^{j}(\mathbf{X};\mathbb{Z}) = \begin{cases} \mathbb{Z}, & \text{if } j = \mathbf{0}, n, \mathbf{2}n, \mathbf{3}n, \\ \{\mathbf{0}\}, & \text{otherwise}, \end{cases} \tag{21}$$

whose generators *ui* <sup>∈</sup> *<sup>H</sup>in*ð Þ *<sup>X</sup>*; satisfy the relations *au*<sup>2</sup> <sup>¼</sup> *<sup>u</sup>*<sup>2</sup> <sup>1</sup> and *bu*<sup>3</sup> ¼ *u*1*u*2, for certain integers *a* and *b:* By Universal Coefficient Theorem, the mod 2 cohomology of *<sup>X</sup>* is given by *<sup>H</sup>in*ð Þffi *<sup>X</sup>*; <sup>2</sup> *<sup>H</sup>in*ð Þ *<sup>X</sup>*; <sup>⊗</sup> <sup>2</sup> ffi 2, for *<sup>n</sup>* <sup>¼</sup> 0,1,2,3, and the relations come to depend only on the parity of the numbers *a* and *b:* In this case, we will use the same symbols to denote the generators, i. e.

$$\mathbb{Z}\_2 \cong \langle \mathfrak{u}\_i \rangle \cong H^{\text{in}}(\mathbf{X}) = H^{\text{in}}(\mathbf{X}; \mathbb{Z}\_2). \tag{22}$$

**Example 3.2.2**. The spaces of type (a, b) were first studied by James [18] and Toda [19]. Note that we can construct examples of these spaces by considering products or unions between certain known spaces, as spheres and projective spaces. Moreover, in Toda's work, it is shown that it is possible to construct a space of type ð Þ *a*, *b* for any choice of *a* and *b:* For example,


In 2010, Pergher et al. [20] investigated the existence of free actions of the groups <sup>2</sup> and *<sup>S</sup>*<sup>1</sup> on spaces of type ð Þ *<sup>a</sup>*, *<sup>b</sup>* , for *<sup>n</sup>* <sup>&</sup>gt;1, where they concluded that: **Theorem 3.2.1.** [20] Let *X* be a space of type ð Þ *a*, *b* , characterized by *n* >1*:*


$$\mathbb{Z}\_2[\mathbf{x}, \mathbf{z}] / \left\langle \mathbf{x}^{(3n+1)/2}, \mathbf{z}^2, \mathbf{z} \mathbf{x}^{(n+1)/2} \right\rangle, \text{ where } \text{degx} = 2 \text{ and } \text{degz} = n, \text{ }$$

or

$$\mathbb{Z}\_2[\mathbf{x}, z] / \left< \mathbf{x}^{(n+1)/2}, z^2 \right> \text{,} \text{where } \text{degx} = 2 \text{,} \text{degz} = 2n \text{ and } b \text{ is odd.}$$

*Free Actions of Compact Lie Groups on Manifolds DOI: http://dx.doi.org/10.5772/intechopen.106404*

In [21], Dotzel and Singh constructed a class of examples of free actions of *p*, *p* prime, on spaces of type 0, 0 ð Þ, by using some known topological operations.

In particular, for *n* even, it is shown in [22] that the only group that can act freely on *X* is 2*:* In addiction, the authors construct a example of such action. If *n* is odd and *X* is of type 0, 1 ð Þ, then any finite group *G* which acts freely on *X* cannot contain the group *p*⊕*p*, for any *p* odd prime.

**Theorem 3.2.2.** Let *<sup>X</sup>* be a manifold<sup>5</sup> that is a space of type 0, ð Þ *<sup>b</sup>* , characterized by *<sup>n</sup>*<sup>&</sup>gt; <sup>1</sup>*:* If *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> acts freely on *<sup>X</sup>*, then *<sup>n</sup>* is an odd number of the form 4*<sup>k</sup>* � 1, for some *k*≥1 and *b* is odd. In this case, the cohomology algebra of the orbit space *X=G* is isomorphic to the graded polynomial algebra 2½ � *<sup>x</sup>*, *<sup>y</sup> <sup>=</sup> xk*, *<sup>y</sup>*<sup>2</sup> , where deg*<sup>x</sup>* <sup>¼</sup> 4 and deg*y* ¼ 2*n:*

*Proof.* Let us suppose know that the group *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> acts freely on a space *<sup>X</sup>* of type ð Þ 0, *<sup>b</sup>* and let *<sup>E</sup>*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* be the spectral sequence associated with the Borel fibration *XG* ! *BG*, with fiber *<sup>X</sup>*, such that *<sup>E</sup><sup>p</sup>*,*<sup>q</sup>* <sup>2</sup> <sup>¼</sup> *<sup>H</sup><sup>p</sup>*ð Þ *BG*; <sup>2</sup> <sup>⊗</sup> *<sup>H</sup><sup>q</sup>* ð Þ *X*; <sup>2</sup> , which converges to *<sup>H</sup>*<sup>∗</sup> ð Þffi *XG*;<sup>2</sup> *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*;<sup>2</sup> *:*

By Quotient Manifold Theorem, it follows that this sequence does not collapse on its *E*2�term. Then, there must exist some nontrivial differential *dri* , for some *ri* ≥ 2*:* If *r* ¼ min f g *ri* , then

$$E\_r^{p,q} \cong E\_{r-1}^{p,q} \cong \cdots \cong E\_2^{p,q},$$

and this is possible only if *r* ¼ 4*k* and *n* ¼ 4*k* � 1, for some *k*≥1*:* This provides the following possibilities for the action of the differentials *d* 4*l*,*q* <sup>4</sup>*<sup>k</sup>* , for *q* ¼ *n*,2*n*,3*n* :

$$\begin{aligned} \text{a. } &d\_r(\mathbf{1} \otimes u\_1) = 0, \, d\_r(\mathbf{1} \otimes u\_2) = \tau^k \otimes u\_1 \text{ and } d\_r(\mathbf{1} \otimes u\_3) = \tau^k \otimes u\_2, \\\\ \text{b. } &d\_r(\mathbf{1} \otimes u\_1) = \tau^k \otimes \mathbf{1} \, d\_r(\mathbf{1} \otimes u\_2) = \tau^k \otimes u\_1 \text{ and } d\_r(\mathbf{1} \otimes u\_3) = 0, \\\\ \text{c. } &d\_r(\mathbf{1} \otimes u\_1) = \tau^k \otimes \mathbf{1}, \, d\_r(\mathbf{1} \otimes u\_2) = 0 \text{ and } d\_r(\mathbf{1} \otimes u\_3) = \tau^k \otimes u\_2, \\\\ \text{d. } &d\_r(\mathbf{1} \otimes u\_1) = 0, \, d\_r(\mathbf{1} \otimes u\_2) = \tau^k \otimes u\_1 \text{ and } d\_r(\mathbf{1} \otimes u\_3) = \tau^k \otimes u\_2, \\\\ &\text{e. }d\_r(\mathbf{1} \otimes u\_1) = 0, \, d\_r(\mathbf{1} \otimes u\_2) = 0 \text{ and } d\_r(\mathbf{1} \otimes u\_3) = \tau^k \otimes u\_2, \\\\ &\text{f. }d\_r(\mathbf{1} \otimes u\_1) = \mathbf{0}, \, d\_r(\mathbf{1} \otimes u\_2) = \tau^k \otimes u\_1 \text{ and } d\_r(\mathbf{1} \otimes u\_3) = \mathbf{0}, \\\\ &\text{g. }d\_r(\mathbf{1} \otimes u\_1) = \tau^k \otimes \mathbf{1}, \, d\_r(\mathbf{1} \otimes u\_2) = \mathbf{0} \text{ and } d\_r(\mathbf{1} \otimes u\_3) = \mathbf{0}. \end{aligned}$$

We will divide the analysis of these cases according to the parity of *b:*

**Case** *b* **odd**: In this case, we have the relation *u*1*u*<sup>2</sup> ¼ *u*<sup>3</sup> and, by the multiplicative properties of the differentials, we have

<sup>5</sup> This proof works even if *<sup>X</sup>* is a more general finitistic space of type ð Þ *<sup>a</sup>*, *<sup>b</sup>* , and for that we only need an adaptation of the Quotient Manifold Theorem for a more general result concerning the cohomology of the quotient of a finitistic CW complex space. More precisely, it is possible to show that if *X* is a finitistic free *<sup>G</sup>*�space, where *<sup>G</sup>* <sup>¼</sup> 2,*S*<sup>1</sup> or *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>3</sup> , and if there is *n* >0 such that *H<sup>j</sup>* ð Þ¼ *X*; <sup>2</sup> f g0 for all *j*>*n* then *Hj* ð Þ¼ *X=G*; <sup>2</sup> f g0 for all *j* >*n:*

$$d\_r(\mathbf{1}\otimes u\_3) = (\mathbf{1}\otimes u\_1)d\_r(\mathbf{1}\otimes u\_2) + (\mathbf{1}\otimes u\_2)d\_r(\mathbf{1}\otimes u\_1).$$

So, if one of the cases ð Þ *b* ,ð Þ *d* ,ð Þ*e* , or ð Þ*g* occurred, it would lead to the contradiction <sup>00</sup> <sup>¼</sup> *<sup>τ</sup><sup>k</sup>* <sup>⊗</sup> *<sup>u</sup>*2*:*

If case ð Þ *<sup>a</sup>* occurred, then the differentials *<sup>d</sup>*4*i*,3*<sup>n</sup>* <sup>4</sup>*<sup>k</sup>* and *<sup>d</sup>*4*i*,2*<sup>n</sup>* <sup>4</sup>*<sup>k</sup>* would be isomorphisms, whence it would follow that

$$\text{im}d\_{4k}^{4i,3n} \cong \mathbb{Z}\_2 \nsubseteq \{0\} = \text{ker}d\_{4k}^{4(i+k),2n} \text{\textquotedblleft}$$

which is a contradiction.

If case ð Þ*f* occurred, then the sequence would collapse on its *E*4*k*þ1�term, with the lines *E*<sup>∗</sup> ,0 <sup>4</sup>*k*þ<sup>1</sup> and *<sup>E</sup>*<sup>∗</sup> ,3*<sup>n</sup>* <sup>4</sup>*k*þ<sup>1</sup> containing an infinite number of nonzero elements. This would contradict the Quotient Manifold Theorem.

Therefore, ð Þ*c* is the only possible case, and it produces the following pattern:

$$E\_{4k+1}^{p,q} = \begin{cases} \mathbb{Z}\_2, & \text{if } p = 0, 4, \cdots, 4(k-1) \text{ and } q = 2n, \\ \{0\}, & \text{otherwise.} \end{cases}$$

Then, the sequence collapses on its *<sup>E</sup>*4*k*þ<sup>1</sup>�term, and *<sup>E</sup><sup>p</sup>*,*<sup>q</sup>* <sup>∞</sup> ffi *Ep*,*<sup>q</sup>* <sup>4</sup>*k*þ<sup>1</sup>, for all *<sup>p</sup>*,*q*<sup>≥</sup> <sup>0</sup>*:* So, *H<sup>j</sup>* ð Þffi *<sup>X</sup>=G*;<sup>2</sup> Tot*<sup>j</sup>* ð Þ *E*<sup>∞</sup> *:*

The elements *τ* ⊗ 1 and 1 ⊗ *u*<sup>2</sup> are the only permanent co-cycles, so they determine the nonzero elements *x* and *y* in *E*4,0 <sup>∞</sup> and *<sup>E</sup>*0,2*<sup>n</sup>* <sup>∞</sup> , respectively. By (15), we have *<sup>π</sup>* <sup>∗</sup> ð Þ¼ *<sup>τ</sup> <sup>x</sup>*; then, 0 <sup>¼</sup> *<sup>π</sup>* <sup>∗</sup> *<sup>τ</sup><sup>j</sup>* <sup>¼</sup> *xj* for all *<sup>j</sup>*≥*k:* By the structure of the *<sup>E</sup>*∞�term, it follows that *<sup>y</sup>*<sup>2</sup> <sup>¼</sup> 0; therefore, *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*; <sup>2</sup> is isomorphic to the graded polynomial algebra 2½ � *<sup>x</sup>*, *<sup>y</sup> <sup>=</sup> <sup>x</sup><sup>k</sup>*, *<sup>y</sup>*<sup>2</sup> , where deg*<sup>x</sup>* <sup>¼</sup> 4 and deg*<sup>y</sup>* <sup>¼</sup> <sup>2</sup>*n:*

**Case** *b* **even**: We will show that if *b* is even, then none of the cases can occur. By the relation *u*1*u*<sup>2</sup> ¼ 0, we have

$$0 = (\mathbf{1} \otimes \boldsymbol{\mu}\_1) d\_r(\mathbf{1} \otimes \boldsymbol{\mu}\_2) + (\mathbf{1} \otimes \boldsymbol{\mu}\_2) d\_r(\mathbf{1} \otimes \boldsymbol{\mu}\_1),$$

and this allows us to eliminate the cases ð Þ *b* ,ð Þ*c* , and ð Þ*g* , since they produce the contradiction 0 <sup>¼</sup> *<sup>τ</sup><sup>k</sup>* <sup>⊗</sup> *<sup>u</sup>*2*:*

By the same reason of the previous case (*b* odd), we can eliminate case ð Þ *a* ; that is, it implies that

$$\text{im}d\_{4k}^{4i,3n} \nsubseteq \text{ker}d\_{4k}^{4(i+k),2n}.$$

By a similar reason we can eliminate ð Þ *<sup>d</sup>* , since it implies that the differentials *<sup>d</sup>*<sup>4</sup>*i*,2*<sup>n</sup>* 4*k* and *d* 4*j*,3*n* <sup>4</sup>*<sup>k</sup>* are isomorphisms.

For case ð Þ*<sup>e</sup>* , the sequence would collapse on its *<sup>E</sup>*4*k*þ<sup>1</sup>�term, with the lines *<sup>E</sup>*<sup>∗</sup> ,0 <sup>4</sup>*<sup>k</sup>* and *E*<sup>∗</sup> ,*<sup>n</sup>* <sup>4</sup>*<sup>k</sup>* containing infinite nonzero elements, which would contradict the Quotient Manifold Theorem. Finally, by the same reason of the previous case, we can eliminate ð Þ*<sup>f</sup>* ; therefore, when *<sup>b</sup>* is even, the space *<sup>X</sup>* does not admit any free action of *<sup>G</sup>:* □

#### **3.3 Free actions on Dold, Wall, and Milnor manifolds**

The Dold manifolds *P m*ð Þ , *n* , as they came to be known, were defined by A. Dold [23] as orbit spaces of free actions of 2, or equivalently free involutions, on a product *Free Actions of Compact Lie Groups on Manifolds DOI: http://dx.doi.org/10.5772/intechopen.106404*

of the form *<sup>S</sup><sup>m</sup>* � *Pn:* Precisely, for each pair of nonnegative integers *<sup>m</sup>* and *<sup>n</sup>*, *P m*ð Þ , *<sup>n</sup>* is the orbit space *<sup>S</sup><sup>m</sup>* � *Pn=T*, where *T x*ð Þ¼ � , ½ � *<sup>z</sup>* ð Þ *<sup>x</sup>*, ½ � *<sup>z</sup> :*

Let *<sup>R</sup>* : *<sup>S</sup><sup>m</sup>* ! *Sm* be the involution defined by the reflection of the last coordinate *R x*ð Þ¼ 0, <sup>⋯</sup>, *xm* ð Þ *<sup>x</sup>*0, <sup>⋯</sup>, *xm*�1, �*xm* , and 1 : *Pn* ! *P<sup>n</sup>* be the identity map. Since the involution *<sup>R</sup>* � <sup>1</sup> : *<sup>S</sup><sup>m</sup>* � *P<sup>n</sup>* ! *<sup>S</sup><sup>m</sup>* � *P<sup>n</sup>* commutes with the involution *<sup>T</sup>*, it induces an involution *S* : *P m*ð Þ! , *n P m*ð Þ , *n :*

For each pair of nonnegative integers *<sup>m</sup>* and *<sup>n</sup>*, the Wall manifold<sup>6</sup> *Q m*ð Þ , *<sup>n</sup>* is defined as the mapping torus of the homeomorphism *S*, that is,

$$Q(m,n) = \frac{P(m,n) \times [0,1]}{([\mathbf{x}, \ \mathbf{z}], \ \mathbf{0}) \sim (\mathbf{S}[\mathbf{x}, \ \mathbf{z}], \ \mathbf{1})}.\tag{23}$$

Let *m*, *n* be integers, such that 0≤*n* ≤ *m:* It is called a (real) Milnor manifold<sup>7</sup> of dimensions *<sup>n</sup>* <sup>þ</sup> *<sup>m</sup>* � 1 to the smooth closed submanifold of codimension 1 in *P<sup>m</sup>* � *P<sup>n</sup>*, described in homogeneous coordinates as:

$$\mathbb{R}H\_{m,n} = \left\{ ([\mathbf{x}\_0, \cdots, \mathbf{x}\_m], [\mathbf{y}\_0, \cdots, \mathbf{y}\_n]) \in \mathbb{R}P^n \times \mathbb{R}P^t \, | \, \mathbf{x}\_0 \mathbf{y}\_0 + \cdots + \mathbf{x}\_n \mathbf{y}\_n = \mathbf{0} \right\}, \tag{24}$$

which is also denoted by *H m*ð Þ , *n :* Equivalently, *Hm*,*<sup>n</sup>* is the total space of the bundle:

$$\text{R}P^{\text{pre}-1} \xleftarrow{\colon} \text{R}P\\\text{H}\_{\text{W},\text{W}} \xleftarrow{\:} \text{R}(\text{R}P),\tag{25}$$

The manifolds *P m*ð Þ , *n* , *Q m*ð Þ , *n* , and *H m*ð Þ , *n* were constructed to provide representatives for generators in odd dimension to the unoriented cobordism ring ℜ<sup>∗</sup> , since we have the projective spaces as representatives in even dimensions. Precisely, the following sets are generator sets for ℜ<sup>∗</sup> :

$$\{ \left[ \mathbb{R}P^{2^i} \right], \left[ P(2^r - 1, \mathfrak{s} 2^r) \right]; i, r, s \ge 1 \}, \tag{26}$$

$$\{ \left[ \mathbb{R}P^{2i} \right], \left[ Q(\mathcal{T} - 2, \mathfrak{s} \mathcal{T}) \right]; i, r, s \ge 1 \}, \tag{27}$$

and

$$\{\left[\mathbb{R}P^{2i}\right], \left[H\left(2^k, 2t2^k\right)\right]; i, k, t \ge 1\}. \tag{28}$$

For these reasons, the analysis of certain structures and algebraic invariants related to the Dold, Milnor, and Wall manifolds is a relevant research topic, as it is done on the works [26–28] of Mukerjee. On the particular interest of investigating the existence of free actions of compact Lie groups on these spaces and also the cohomology classification of the respective orbit spaces, there are several results in the literature in which we will briefly discuss some of them below.

Regarding the existence of free actions of <sup>2</sup> on Dold manifolds, Morita et al. [29] partially solved the problem by considering free involutions on *P*ð Þ 1, *n* , for *n*≥ 1 an odd integer. Later this problem was completely solved by Dey [30], according to the following.

**Theorem 3.3.1.** [30] If *<sup>G</sup>* <sup>¼</sup> <sup>2</sup> acts freely on *<sup>X</sup>* <sup>¼</sup> *P m*ð Þ , *<sup>n</sup>* , then *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*; <sup>2</sup> is isomorphic to one of the following graded algebras:

<sup>6</sup> For more details on the construction of this manifolds, see [24].

<sup>7</sup> For more details on Milnor's manifold, see [25, 26].


Concerning on free involutions on Wall manifods *Q m*ð Þ , *n* , the work of Khare [31] shows that these manifolds bounds if and only if *n* is odd or *n* ¼ 0 and *m* odd. By Proposition 3.5 in [32], we can conclude that *X* ¼ *Q m*ð Þ , odd admit free involutions and about the orbit spaces *X=*<sup>2</sup> we have the following result, for some values of *m:*

**Theorem 3.3.2**. [32, 33] Let *X* ¼ *Q m*ð Þ , *n* , where *n*>0 is odd, equipped with a free action of the group *G* ¼ 2*:*

i. If *m* ¼ 1 and the induced action of <sup>2</sup> on the mod 2 cohomology is trivial, then

$$H^\*(\mathbf{X}/\mathbf{G}; \mathbb{Z}\_2) \cong \mathbb{Z}\_2[\mathbf{x}, \mathbf{y}, \mathbf{z}, w] / \left\langle \mathbf{x}^3, \mathbf{y}^3, \mathbf{z}^2, \mathbf{y}^2 + \mathbf{y}w, w^{(n+1)/2} \right\rangle,$$

where deg*x* ¼ deg*y* ¼ deg*z* ¼ 1, and deg*w* ¼ 4*:*

ii. If *m* is even and the induced action of <sup>2</sup> on the mod 2 cohomology is trivial, then *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=*<sup>2</sup> is isomorphic to one of the following graded polynomial algebras:

$$
\mathbb{Z}\_2[\mathfrak{x}, \mathfrak{y}, \mathfrak{z}, w] / \left< \mathfrak{x}^3, \mathfrak{y}^2, \mathfrak{z}^{m+1} + \mathfrak{z}^m \mathfrak{y}, \mathfrak{w}^{(n+1)/2} \right>,
$$

where deg*x* ¼ deg*y* ¼ deg*z* ¼ 1 and deg*w* ¼ 4, or

$$
\mathbb{Z}\_2[\mathfrak{x}, \mathfrak{y}, \mathfrak{z}, w] / \langle \mathfrak{x}^2, \mathfrak{y}^{m+1}, \mathfrak{y}^2 + \mathfrak{z}, w^{n+1} \rangle,
$$

where deg*x* ¼ deg*y* ¼ 1 and deg*z* ¼ deg*w* ¼ 2*:*

**Example 3.3.1.** (Free *S*<sup>1</sup> �actions on Dold manifolds) Let *<sup>G</sup>* be the group *<sup>S</sup>*<sup>1</sup> and *<sup>m</sup>*,*<sup>n</sup>* odd integers, where *<sup>m</sup>* <sup>¼</sup> <sup>2</sup>*<sup>k</sup>* � 1, for some *<sup>k</sup>*≥1*:* Considering *<sup>S</sup><sup>m</sup>* <sup>⊆</sup> *<sup>k</sup>* , we define a free action of *<sup>G</sup>* on *<sup>S</sup><sup>m</sup>* � *P<sup>n</sup>* by:

$$zz\*(w,v)\mapsto((zw\_1,\ \cdots,\ zw\_k),\ [zv\_0:\cdots:zv\_n]),\tag{29}$$

where *<sup>w</sup>* <sup>¼</sup> ð Þ *<sup>w</sup>*1, <sup>⋯</sup>, *wk* <sup>∈</sup>*S<sup>m</sup>* <sup>⊆</sup> *<sup>k</sup>* and *<sup>v</sup>* <sup>¼</sup> ½ � *<sup>v</sup>*<sup>0</sup> : <sup>⋯</sup> : *vn* <sup>∈</sup> *Pn:*

Let us consider an arbitrary element ½ � *<sup>w</sup>*, *<sup>v</sup>* <sup>∈</sup>*P m*ð Þ¼ , *<sup>n</sup> Sm* � *Pn* ð Þ*=T*, and note that the isotropy subgroup *G*½ � *<sup>w</sup>*, *<sup>v</sup>* is trivial. Therefore, *z* must be equal to 1∈ *G*, that is, *G*½ � *<sup>w</sup>*, *<sup>v</sup>* ¼ f g1 , so the induced action on Dold manifold *P m*ð Þ , *n* is free.

If *n* is odd and *m* ¼ 2*k* is even, then we can consider

$$\mathcal{S}^{\mathfrak{m}} = \left\{ (w, \ t) \in \mathbb{C}^{k} \times \mathbb{R}; \ \|w\| + |t| = 1 \right\},\tag{30}$$

and the analogous free action of *<sup>G</sup>* on *<sup>S</sup><sup>m</sup>* � *P<sup>n</sup>* is defined by:

$$z\*( (w, \ t), v) \mapsto ((zw\_1, \ \cdots, \ zw\_k, \ t), [zv\_0 : \cdots : zv\_n]).\tag{31}$$

Since this action is free, it induces a free action of *G* on *P m*ð Þ , *n* , as in the previous case.

**Theorem 3.3.3**. There is no free action of *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> on *<sup>X</sup>* <sup>¼</sup> *Q m*ð Þ , *<sup>n</sup>* , for any *<sup>m</sup>*,*<sup>n</sup>* <sup>&</sup>gt;0*: Proof.* Recall that<sup>8</sup> *<sup>H</sup>*<sup>∗</sup> ð Þffi *<sup>X</sup>*;<sup>2</sup> 2½ � *<sup>x</sup>*, *<sup>c</sup>*, *<sup>d</sup> <sup>=</sup> <sup>x</sup>*2, *cm*þ<sup>1</sup> <sup>þ</sup> *cmx*, *dn*þ<sup>1</sup> , where deg*<sup>x</sup>* <sup>¼</sup> deg*<sup>c</sup>* <sup>¼</sup> 1 and deg*<sup>d</sup>* <sup>¼</sup> <sup>2</sup>*:* Suppose that there is a free action of *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> on *<sup>X</sup>* and let *<sup>X</sup>* ↪ *XG* ! *BG* be the associated Borel fibration, with *<sup>E</sup>*<sup>∗</sup> , <sup>∗</sup> *<sup>r</sup>* , *dr* , and such that *Ep*,*<sup>q</sup>* 2 ffi *<sup>H</sup>p*ð Þ *BG*; <sup>2</sup> <sup>⊗</sup> *<sup>H</sup><sup>q</sup>* ð Þ *<sup>X</sup>*;<sup>2</sup> , that converges, as an algebra, to *<sup>H</sup>*<sup>∗</sup> ð Þffi *XG*; <sup>2</sup> *<sup>H</sup>*<sup>∗</sup> ð Þ *<sup>X</sup>=G*; <sup>2</sup> *:* Since this sequence does not collapse on *E*2�page, there is some nontrivial differential *d*<sup>∗</sup> , <sup>∗</sup> <sup>2</sup> , according to the following cases.

**Case** *m* **odd**: In this case, we have *d*0,*m*þ<sup>1</sup> <sup>2</sup> <sup>1</sup> <sup>⊗</sup> *cm*þ<sup>1</sup> ð Þ¼ <sup>0</sup>*:* In fact, since *<sup>m</sup>* <sup>þ</sup> <sup>1</sup> <sup>¼</sup> <sup>2</sup>*r*, for some *r*>0, then *d*0,2*<sup>r</sup>* <sup>2</sup> <sup>1</sup> <sup>⊗</sup> *<sup>c</sup>*<sup>2</sup>*<sup>r</sup>* ð Þ¼ *<sup>d</sup>*0,2*<sup>r</sup>* <sup>2</sup> <sup>1</sup> <sup>⊗</sup> *<sup>c</sup><sup>r</sup>* ð Þ <sup>1</sup> <sup>⊗</sup> *<sup>c</sup><sup>r</sup>* <sup>ð</sup> ð ÞÞ ¼ <sup>0</sup>*:* However, by relation *<sup>c</sup><sup>m</sup>*þ<sup>1</sup> <sup>¼</sup> *<sup>c</sup>mx*, it follows that

$$d\_2^{0,m+1}(\mathbf{1}\otimes c^{m+1}) = (\mathbf{1}\otimes c^m)d\_2^{0,1}(\mathbf{1}\otimes \mathbf{x}) + (\mathbf{1}\otimes \mathbf{x})d\_2^{0,m}(\mathbf{1}\otimes c^m).$$

Therefore, *d*0,1 <sup>2</sup> ð Þ <sup>1</sup> <sup>⊗</sup> *<sup>x</sup>* 6¼ 0 and *<sup>d</sup>*0,1 <sup>2</sup> ð Þ 1 ⊗ *c* 6¼ 0, cannot occur simultaneously because in this case we will have 0 <sup>¼</sup> *<sup>τ</sup>* <sup>⊗</sup> *<sup>c</sup><sup>m</sup>*�<sup>1</sup>ð Þ *<sup>c</sup>* <sup>þ</sup> *<sup>x</sup>* , which is a contradiction.

Similarly, *d*0,1 <sup>2</sup> ð Þ¼ <sup>1</sup> <sup>⊗</sup> *<sup>x</sup>* 0 and *<sup>d</sup>*0,1 <sup>2</sup> ð Þ 1 ⊗ *c* 6¼ 0, cannot occur simultaneously. Therefore, it follows there are only the possibilities:

$$\begin{aligned} 1. \,d\_2^{0,1}(\mathbf{1}\otimes c) = d\_2^{0,1}(\mathbf{1}\otimes \boldsymbol{\pi}) = \mathbf{0} \text{ and } d\_2^{0,2}(\mathbf{1}\otimes d) = \boldsymbol{\pi}\otimes c, \\\\ 2. \,d\_2^{0,1}(\mathbf{1}\otimes c) = d\_2^{0,1}(\mathbf{1}\otimes \boldsymbol{\pi}) = \mathbf{0} \text{ and } d\_2^{0,2}(\mathbf{1}\otimes d) = \boldsymbol{\pi}\otimes \boldsymbol{\pi}. \end{aligned}$$

We claim that (1) and (2) cannot occur.

If (1) occurs, then for any *j*≥0, *k*∈f g 1⋯, *m* and *l*∈f g 1⋯, *n* , it follows that

$$d\_2^{i,2l} \left(\boldsymbol{\tau}^j \otimes d^l\right) = \begin{cases} 0, & \text{if } l \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{c} d^{l-1}, & \text{if } l \text{ is odd,} \end{cases}$$

$$d\_2^{i,k+2l} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{c}^k d^l\right) = \begin{cases} 0, & \text{if } l \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{c}^{k+1} d^{l-1}, & \text{if } l \text{ is odd,} \end{cases}$$

$$d\_2^{i,2l+1} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{\kappa} d^l\right) = \begin{cases} 0, & \text{if } l \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{\kappa} \boldsymbol{d}^{l-1}, & \text{if } l \text{ is odd,} \end{cases}$$

$$d\_2^{i,2l+k+1} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{\kappa} \boldsymbol{c}^k d^l\right) = \begin{cases} 0, & \text{if } l \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{\kappa} \boldsymbol{c}^{k+1} d^{l-1}, & \text{if } l \text{ is odd,} \end{cases}$$

therefore, we will have *Ep*,*<sup>q</sup>* <sup>3</sup> ffi f g0 , for all *p* odd or *q* � 2 mod4 ð Þ and *q*>0*:* We can see that the sequence collapses on *<sup>E</sup>*3�page, however *<sup>E</sup>*<sup>2</sup>*r*,*<sup>q</sup>* <sup>3</sup> 6¼ f g0 , for all *q* � *s*ð Þ mod4 , *s* ¼ 0,2,3, and *r*≥ 0, which contradicts the Quotient Manifold Theorem.

<sup>8</sup> See [24].

If the case (2) occurs, then for all *j*≥ 0, *k*∈f g 1⋯, *m* and *l* ∈f g 1⋯, *n* , we have

$$d\_2^{i,2l} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{d}^l\right) = \begin{cases} 0, & \text{if } l \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{\varkappa} \boldsymbol{d}^{l-1}, & \text{if } l \text{ is odd,} \end{cases}$$

$$d\_2^{i,2l+k} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{\varsigma}^k \boldsymbol{d}^l\right) = \begin{cases} 0, & \text{if } l \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{\varkappa} \boldsymbol{d}^k \boldsymbol{d}^{l-1}, & \text{if } l \text{ is odd,} \end{cases}$$

while *dj*,2*l*þ<sup>1</sup> <sup>2</sup> *<sup>τ</sup><sup>j</sup>* <sup>⊗</sup> *xd<sup>l</sup>* � � <sup>¼</sup> 0, since *<sup>x</sup>*<sup>2</sup> <sup>¼</sup> <sup>0</sup>*:* Therefore, analogous to case (1), we can conclude that this case is not possible either.

**Case** *m* **even**: In this case, we have *d*0,*<sup>m</sup>* <sup>1</sup> <sup>⊗</sup> *<sup>c</sup><sup>m</sup>* ð Þ¼ 0, so *<sup>d</sup>*0,*m*þ<sup>1</sup> <sup>1</sup> <sup>⊗</sup> *<sup>c</sup><sup>m</sup>* ð Þ¼ *<sup>x</sup>* <sup>⊗</sup> *<sup>c</sup><sup>m</sup>* ð Þ*d*0,1 ð Þ <sup>1</sup> <sup>⊗</sup> *<sup>x</sup>* , while, by relation *<sup>c</sup><sup>m</sup>*þ<sup>1</sup> <sup>¼</sup> *cmx*, *<sup>d</sup>*0,*m*þ<sup>1</sup> <sup>1</sup> <sup>⊗</sup> *<sup>c</sup><sup>m</sup>* ð Þ¼ *<sup>x</sup> d*0,*m*þ<sup>1</sup> <sup>1</sup> <sup>⊗</sup> *<sup>c</sup><sup>m</sup>*þ<sup>1</sup> ð Þ¼ <sup>1</sup> <sup>⊗</sup> *cm* ð Þ*d*0,1 ð Þ 1 ⊗ *c* , therefore, we should have necessarily *d*0,1 ð Þ¼ <sup>1</sup> <sup>⊗</sup> *<sup>c</sup> <sup>d</sup>*0,1 ð Þ 1 ⊗ *x :* If *d*0,1 ð Þ¼ <sup>1</sup> <sup>⊗</sup> *<sup>c</sup> <sup>d</sup>*0,1 ð Þ¼ <sup>1</sup> <sup>⊗</sup> *<sup>x</sup> <sup>τ</sup>* <sup>⊗</sup> 1, then *<sup>d</sup>*0,2 ð Þ¼ 1 ⊗ *d* 0, otherwise we will have

$$\text{im}d\_2^{0,2} = \left\langle d\_2^{0,2}(\mathbf{1}\otimes d) \right\rangle \nsubseteq \text{ker}d\_2^{2,1}\langle \mathfrak{r}\otimes (\mathfrak{c}+\mathfrak{x})\rangle,$$

which is a contradiction.

Let us suppose now that *d*0,1 <sup>2</sup> ð Þ <sup>1</sup> <sup>⊗</sup> *<sup>c</sup>* and *<sup>d</sup>*0,1 <sup>2</sup> ð Þ <sup>1</sup> <sup>⊗</sup> *<sup>x</sup>* are nontrivial, and *<sup>d</sup>*0,2 <sup>2</sup> ð Þ¼ 1 ⊗ *d τ* ⊗ *c:* Then, for example,

$$\mathsf{im}d\_2^{0,4} = \langle \mathfrak{r} \otimes cd \rangle \oplus \langle \mathfrak{r} \otimes c^2 \mathfrak{x} \rangle \nsubseteq \mathfrak{r} \,\mathsf{ker}d\_2^{2,3} = \langle \mathfrak{r} \otimes \mathsf{x}c^2 \rangle \oplus \langle \mathfrak{r} \otimes c^3 \rangle \lambda\_2$$

which is a contradiction. Therefore, for *m* even, we must consider only the cases:

$$\begin{aligned} \text{i.e. } d\_2^{0,1}(\mathbf{1} \otimes \boldsymbol{\varepsilon}) &= d\_2^{0,1}(\mathbf{1} \otimes \boldsymbol{\varkappa}) = \boldsymbol{\tau} \otimes \mathbf{1} \text{ and } d\_2^{0,2}(\mathbf{1} \otimes \boldsymbol{d}) = \mathbf{0};\\ \text{i.e. } d\_2^{0,1}(\mathbf{1} \otimes \boldsymbol{\varepsilon}) &= d\_2^{0,1}(\mathbf{1} \otimes \boldsymbol{\varkappa}) = \mathbf{0} \text{ and } d\_2^{0,2}(\mathbf{1} \otimes \boldsymbol{d}) = \boldsymbol{\tau} \otimes \boldsymbol{\varkappa}. \end{aligned}$$

We will show that both ð Þ*i* and ð Þ *ii* cannot occur.

If ð Þ*i* is true, then for all *j*≥ 0, *k*∈f g 1⋯, *m* and all *l*∈f g 1⋯, *n* , we have

$$d\_2^{j,k} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{c}^k\right) = \begin{cases} 0, & \text{if } k \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{c}^{k-1}, & \text{if } k \text{ is odd,} \end{cases}$$

$$d\_2^{j,k+1} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{c}^k \boldsymbol{x}\right) = \begin{cases} \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{c}^k, & \text{if } k \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes (\boldsymbol{c} + \boldsymbol{x}) \boldsymbol{c}^{k-1}, & \text{if } k \text{ is odd,} \end{cases}$$

$$d\_2^{j,2l+k} \left(\boldsymbol{\tau}^j \otimes \boldsymbol{c}^k \boldsymbol{d}^l\right) = \begin{cases} 0, & \text{if } k \text{ is even,} \\ \boldsymbol{\tau}^{j+1} \otimes \boldsymbol{c}^{k-1} \boldsymbol{d}^l, & \text{if } k \text{ is odd,} \end{cases}$$

and *d j*,2*l*þ1 <sup>2</sup> *<sup>τ</sup><sup>j</sup>* <sup>⊗</sup> *xd<sup>l</sup>* � � <sup>¼</sup> *<sup>τ</sup><sup>j</sup>*þ<sup>1</sup> <sup>⊗</sup> *dl :* Therefore, *Ep*,*<sup>q</sup>* <sup>3</sup> ffi f g0 , for all *q* �� 1 mod4 ð Þ, e *p*> 0*:* However, for *q* � 1 mod4 ð Þ, *q*≥5, we have

$$E\_3^{2j,q} \cong \left\langle \mathfrak{r}^j \otimes (\mathfrak{c} + \mathfrak{x}) d^{(q-1)/2} \right\rangle \neq \{0\}.$$

*Free Actions of Compact Lie Groups on Manifolds DOI: http://dx.doi.org/10.5772/intechopen.106404*

which contradicts the Quotient Manifold Theorem.

For ð Þ *iii* , note that it will result in a pattern similar to case (2); therefore, it cannot occur either due to the same arguments. □

In order to investigate the existence of free involutions on a Milnor manifold *X* ¼ *H m*ð Þ , *n* , Dey and Singh [34] showed that if *G* ¼ <sup>2</sup> acts freely on *X*, with 1< *n*< *m* and *m* �� 2 mod4 ð Þ, then necessarily *m* and *n* must be odd. Furthermore, they construct some examples of such free actions and, in this case, it follows that

$$H^\*\left(\mathbf{X}/\mathbf{G}; \mathbb{Z}\_2\right) \cong \mathbb{Z}\_2[\mathbf{x}, \mathbf{y}, \mathbf{z}, w]/I,\tag{32}$$

where

$$I = \langle \mathbf{z}^2, \mathbf{w}^2 - \chi\_1 \mathbf{z} \mathbf{w} - \chi\_2 \mathbf{x} - \chi\_3 \mathbf{y}, \mathbf{x}^{(n+1)/2} + a\_0 \mathbf{z} \mathbf{w} \mathbf{x}^{(n-1)/2} \mathbf{y} + \dots + a\_{\frac{n-1}{2}} \mathbf{z} \mathbf{w} \mathbf{y}^{(n-1)/2}, \mathbf{z} \rangle$$

$$(\boldsymbol{w} + \boldsymbol{\beta}\_0 \mathbf{z}) \mathbf{y}^{(m-1)/2} + (\boldsymbol{w} + \boldsymbol{\beta}\_1 \mathbf{z}) \mathbf{x} \mathbf{y}^{(m-3)/2} + \dots + \left(\boldsymbol{w} + \boldsymbol{\beta}\_{\frac{n-1}{2}} \mathbf{z}\right) \mathbf{x}^{(n-1)/2} \mathbf{y}^{(m-n)/2} \rangle,$$

with deg*x* ¼ deg*y* ¼ 2, deg*z* ¼ deg*w* ¼ 1, and *αi*,*βi*,*γ<sup>i</sup>* ∈2*:* If *<sup>G</sup>* <sup>¼</sup> *<sup>S</sup>*<sup>1</sup> act freely on *<sup>X</sup>* <sup>¼</sup> *H m*ð Þ , *<sup>n</sup>* , then *<sup>H</sup>*<sup>∗</sup> ð Þffi *<sup>X</sup>=G*; <sup>2</sup> 2½ � *<sup>x</sup>*, *<sup>y</sup>*, *<sup>w</sup> <sup>=</sup>I*, where

$$I = \left\langle x^{(n+1)/2}, wy^{(m-1)/2} + xwy^{(m-3)/2} + \dots + wx^{(n-1)/2}y^{(m-n)/2}, w^2 - a\mathbf{x} - \beta y \right\rangle\_{\mathbb{R}^n}$$

with deg*x* ¼ deg*y* ¼ 2, deg*w* ¼ 1, and *α*,*β* ∈ 2*:*

#### **Author details**

Thales Fernando Vilamaior Paiva Universidade Federal de Mato Grosso do Sul, Aquidauana, Brazil

\*Address all correspondence to: thales.paiva@ufms.br

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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#### **Chapter 2**
