Generalized Dold Spaces in Topology

• Generalized Dold spaces are quotient spaces formed by free involutions on product spaces, extending classical Dold manifolds and projective product spaces.
• They exhibit detailed cohomology and characteristic class structures, with explicit formulas derived via spectral sequences and CW decompositions.
• Their applications span torus, flag, and projective manifolds, influencing topics like cobordism, stable parallelizability, and vector field analyses.

Generalized Dold spaces form a broad class of quotient spaces and manifolds constructed via free involutions on products of spaces, generalizing the classical Dold manifolds and encompassing projective product spaces, flag bundles, toric and small cover settings. Their algebraic and topological structure, characteristic classes, and equivariant cobordism reflect complex interactions between involutive automorphisms, fiber bundle structure, and underlying group actions.

1. Definitions and Constructions

A generalized Dold space is defined as the quotient $P(S,X) = (S \times X)/\sim$, where $S$ is a topological space equipped with a fixed-point-free involution $\alpha$, and $X$ is a space with an involution $\sigma:X \to X$ such that $\mathrm{Fix}(\sigma)\neq \emptyset$. The equivalence relation is $(v,x)\sim (\alpha(v),\sigma(x))$ (Mandal et al., 2021, Sarkar et al., 2020). When $S = \mathbb{S}^m$ with antipodal involution and $X$ a smooth almost complex manifold with complex-conjugation involution, the resulting space $P(m,X)$ is called a generalized Dold manifold (Nath et al., 2017).

In the smooth setting, $P(m,X)$ inherits a manifold structure of dimension $m + 2d$ if $X$ is $2d$-dimensional and $\sigma$ is a conjugation, i.e., $d\sigma \circ J = - J \circ d\sigma$ for the almost complex structure $J$ (Nath et al., 2020). The canonical projection $P(m,X) \to \mathbb{RP}^m$ exhibits $P(m,X)$ as an $X$-bundle over real projective space, with sections provided by fixed points of $\sigma$.

Generalized Dold spaces further include projective product spaces $P(M,N)$ formed from closed manifolds $M$, $N$ with involutions via diagonal $\mathbb{Z}_2$-quotients. These admit iterated sphere bundle, toric bundle, and small cover structures, respectively (Sarkar et al., 2020).

2. Cohomology, Homology, and Characteristic Classes

The cohomological properties of generalized Dold spaces fundamentally reflect the interplay between the involutive structure and bundle-type topology.

For $P(m,X)$, as a fiber bundle with fiber $X$ over $\mathbb{RP}^m$, and under the assumption $H^1(X;\mathbb{Z}_2)=0$, the total Stiefel-Whitney polynomial is given by:

$$w(P(m,X);t) = (1+xt)^{m+1}\sum_{j=0}^{d}\tilde{c}_j(X)(1+xt)^{d-j}t^{2j}$$

where $x$ is the first Stiefel-Whitney class of the Hopf bundle and $\tilde{c}_j(X)$ are the lifted Chern classes from $X$ (Nath et al., 2017).

Under further CW and involution compatibility hypotheses, $P(S,X)$ inherits a CW structure, and the mod $2$ cohomology algebra splits as $$H^*(P(S,X);\mathbb{Z}_2)\cong H^*(Y;\mathbb{Z}_2)\otimes H^*(X;\mathbb{Z}_2)$$, with $Y = S/\langle \alpha \rangle$ (Mandal et al., 2021, Sarkar et al., 2020). In the case where $X$ is a torus manifold or complex flag manifold, this product structure persists and is reflected in the specific relations among cohomological generators (e.g., Stanley-Reisner presentations for toric cases).

The characteristic classes of vector bundles induced from $X$ by the construction obey a universal Stiefel-Whitney formula:

$$w(\hat{\omega}) = \sum_{j=0}^r (1+x)^{r-j}x^j \tilde{c}_j(\omega)$$

for any $\sigma$-conjugate bundle $\omega$ over $X$ (Mandal et al., 2021).

Singular homology groups and Betti numbers can be determined via cell decompositions adapted to involution, leading to vanishing odd-degree Betti numbers for even $m$ and explicit $2$-torsion in degrees dependent on real locus Euler characteristic and flag data (Mandal et al., 2024).

3. Stable Parallelizability, Cobordism, and Vector Field Problems

Stable parallelizability of $P(m,X)$ is characterized as follows: if $P(m,X)$ is stably parallelizable then $X$ must be stably parallelizable and $2^{\varphi(m)}$ divides $m+1+d$, where $\varphi(m)$ is the Adams-Hurwitz invariant associated with the Hopf bundle over $\mathbb{RP}^m$ (Nath et al., 2017). For even $m$, full parallelizability further requires vanishing Euler class.

Cobordism criteria are intimately connected to those of $X$. Specifically, for $m\equiv d \bmod 2$, the unoriented cobordism class $[P(m,X)]$ vanishes if and only if $[X]$ vanishes (Nath et al., 2017). This holds in the equivariant case as well: if $G$ acts smoothly on $X$ commuting with $\sigma$ and $X^G$ is finite, then $[P(m,X),D\times G]=0$ if and only if $[X,G]=0$, where $D$ is the diagonal subgroup of $O(m+1)$ acting on $\mathbb{S}^m$ (Nath et al., 2020).

The tangent bundle decomposes naturally at fixed points:

$$T_{\pi(e_j,x)}P(m,X) \cong T_{[e_j]}\mathbb{RP}^m \,\oplus\, T_x X \,\oplus\, (E_j \otimes_\mathbb{R} T_x X)$$

where $E_j$ is the real line indexed by the $j$-th coordinate of $D$.

The presence of nontrivial vector fields is constrained by the Euler characteristics of the factors, rank conditions, and immersion data. The minimum number of independent vector fields on $P(M,N)$ is at least that of $M/\mathbb{Z}_2$; for sphere-product cases, explicit lower bounds can be derived from equivariant vector field data and immersion dimensions (Sarkar et al., 2020).

4. Equivariant Actions and Representation Theory

Generalized Dold spaces accommodate a wide spectrum of group actions—most notably finite abelian 2-groups acting diagonally and commuting with involutions (notably $\mathbb{Z}_2^s$). In the flag manifold case, one analyzes the induced action by mapping $G \subset U(n)$ into the diagonal subgroup, yielding explicit enumeration and decomposition of the $G$-module structure at coordinate flags:

$$T_{L(\mathbf A)}X \cong \bigoplus_{i<j} \operatorname{Hom}_{\mathbb{C}}(L_i, L_j)$$

The equivariant cobordism classification is determined by pairing tangent representations at fixed points and employing the injectivity of the Stong–Conner–Floyd map into the representation ring $R(G)$ (Nath et al., 2020).

5. Examples: Torus Manifolds, Flag Manifolds, and Projective Product Spaces

The paradigm examples include:

• Complex Flag Manifolds: For $$X=\mathrm{CG}(\nu)=U(n)/U(n_1)\times \cdots \times U(n_s)$$, the associated $P(m,\nu)$ is a smooth flag bundle over $\mathbb{RP}^m$ (Mandal et al., 2024). Its cohomology is calculated via Schubert cell decompositions, and ring structures are generated by canonical Chern classes subject to flag relations. The K-theory admits an explicit presentation via the Atiyah-Hirzebruch spectral sequence.
• Torus Manifolds: Applying the Leray–Hirsch theorem, $P(S,X)$ for $X$ a torus manifold admits a Stanley-Reisner cohomology algebra structure inherited from the orbit polytope $Q$ and characteristic subrings, extended linearly by $H^*(Y)$ from the $S$-factor (Mandal et al., 2021, Sarkar et al., 2020).
• Projective Product Spaces: Sphere-product and iterated sphere bundle constructions generalize Davis’ projective products, featuring explicit cell, cohomological, and tangent bundle calculations, as well as vector field bounds in terms of ranks and partial reflection parameters (Sarkar et al., 2020).

The classical case $P(m,n)=P(\mathbb{S}^m,\mathbb{CP}^n)$ admits the standard description $$H^*(P(m,n);\mathbb{Z}_2)=\mathbb{Z}_2[a,u]/(a^{m+1},u^{n+1})$$.

6. Spectral Sequences, Fibration Splittings, and Structural Results

The cell structure and spectral sequence techniques enable a full description of homological and K-theoretical invariants. Vanishing of boundary differentials due to even cell dimensions and compatible CW structures yields direct sums in homology. For K-theory, the Atiyah-Hirzebruch spectral sequence collapses additively, so $K^*(P(m,\nu))$ is determined up to finite 2-torsion, with a large canonical subring generated by line bundle pulls and flag-bundle classes having finite index in the full ring (Mandal et al., 2024).

The Gysin sequence and existence of canonical sections allow for splitting off summands associated with the base $\mathbb{RP}^m$ and structural control over the rest of the invariants.

7. Generalizations and Implications

Generalized Dold spaces provide a rich framework for analyzing equivariant topology, stable tangential structures, and immersion theory, unifying constructions across manifold theory, algebraic topology, and transformation groups. The extension to arbitrary CW complexes $S$ and $X$ (with compatible involutions) yields new families with explicit cellular, cohomological, and homological invariants, and sharp criteria for cobordism and parallelizability. The interplay between fixed-point sets, tangent bundle representations, and algebraic invariants under involution underpin their significance in geometric topology (Mandal et al., 2021, Nath et al., 2020, Sarkar et al., 2020, Nath et al., 2017, Mandal et al., 2024).