Family Of Manifolds Research Articles

A Duality Theorem for the De Rham Theory of a Family of Manifolds

A Duality Theorem for the De Rham Theory of a Family of Manifolds

Actions of [formula omitted] on closed manifolds

We associate to a non-singular C r , r ⩾ 2, action of R p on a closed, orientable, connected m-manifold M a symmetric ( q + 1)-linear mapping α F of R p into H 2q+1 de R (M) , where q = m − p, called the characteristic mapping of F. We compute α F for several types of actions. Our main result says that, for a certain family of 3-manifolds, the non-degeneracy of α F , for an action F of R 2 , implies the existence and stability of a compact orbit.

Non embedding in sphere bundles

In order to get results of non-embedding of flag-manifolds in the total space of sphere bundles we consider algebraic properties of these two families of manifolds.

Matrices, Eigenvalues, and Complex Projective Space

(1978). Matrices, Eigenvalues, and Complex Projective Space. The American Mathematical Monthly: Vol. 85, No. 9, pp. 727-733.

The topological structure of $4$ manifolds with effective torus actions (II)

Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of S4, S2 X S2, cp2 Cp-2, SI X S3, and three families of elementary 4-manifolds, R(m, n), T(m, n; m', n'), L(n; p, q; m) has been obtained (where m, n, m', n', p, and q are integers). In addition, a stable homeomorphic relation for the manifolds R(m, n) and T(m, n; m', n') can also be found in [9]. But the topological classification of R's, T's, and especially L's were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R's and T's, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds L(n;p, q; m) form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of L(n; p, q; m) is finite cyclic of order n. And it is proved that ,r(L(n;p,q;m)) and irj (L(n; p',q';m')) act identically on the second cohomology of their common universal covering space, (S2 X S2) # ... # (S2 X S2) (n 1 copies), even though L(n; p, q; m) and L(n; p', q'; m') are not homotopically equivalent for some (n;p',q'; m')'s. This family of manifolds is explicitly constructed and completely classified. In addition, a normal form is imposed on a connected sum decomposition mentioned above. These nromal forms are unique. Most of the material of this paper appeared first in the author's doctoral dissertation. The author would like to thank his thesis advisor Professor F. Raymond for his help and encouragement.

Group presentations corresponding to spines of 3-manifolds. II

Let ϕ = ⟨ a 1 , … , a n | R 1 , … , R m ⟩ \phi = \langle {a_1}, \ldots ,{a_n}|{R_1}, \ldots ,{R_m}\rangle denote a group presentation. Let K ϕ {K_\phi } denote the corresponding 2-complex. It is well known that every compact 3-manifold has a spine of the form K ϕ {K_\phi } for some ϕ \phi , but that not every K ϕ {K_\phi } is a spine of a compact 3-manifold. Neuwirth’s algorithm (Proc. Cambridge Philos. Soc. 64 (1968), 603-613) decides whether K ϕ {K_\phi } can be a spine of a compact 3-manifold. However, it is impractical for presentations of moderate length. In this paper a simple planar graph-like object, called a RR-system (railroad system), is defined. To each RR-system corresponds a whole family of compact orientable 3-manifolds with spines of the form K ϕ {K_\phi } , where ϕ \phi has a particular form (e.g., ⟨ a , b a m b n a p b n , a m b n a m b q ⟩ \langle a,b{a^m}{b^n}{a^p}{b^n},{a^m}{b^n}{a^m}{b^q}\rangle ), subject only to certain requirements of relative primeness of certain pairs of exponents. Conversely, every K ϕ {K_\phi } which is a spine of some compact orientable 3-manifold can be obtained in this way. An equivalence relation on RR-systems is defined so that equivalent RR-systems determine the same family of manifolds. Results of Zieschang are applied to show that the simplest spine of 3-manifolds arises from a particularly simple kind of RR-system called a reduced RR-system. Following Neuwirth, it is shown how to determine when a RR-system gives rise to a collection of closed 3-manifolds.

The topological structure of 4-manifolds with effective torus actions. I

Torus actions on orientable 4-manifolds have been studied by F. Raymond and P. Orlik [8] and [9]. The equivariant classification problem has been completely answered there. The problem then arose as to what can be said about the topological classification of these manifolds. Specifically, when are two manifolds homeomorphic if they are not equivariantly homeomorphic? In some cases this problem was answered in the above mentioned papers. For example, if the only nontrivial stability groups are finite cyclic, then the manifolds are essentially classified by their fundamental groups. In the presence of fixed points, a connected sum decomposition in terms of S 4 , S 2 × S 2 , C P 2 , − C P − 2 , S 1 × S 3 {S^4},{S^2} \times {S^2},C{P^2}, - C{P^{ - 2}},{S^1} \times {S^3} , and three families of elementary 4-manifolds, R ( m , n ) , T ( m , n ; m ′ , n ′ ) , L ( n ; p , q ; m ) R(m,n),T(m,n;m’,n’),L(n;p,q;m) has been obtained (where m, n, m ′ , n ′ m’,n’ , p, and q are integers). In addition, a stable homeomorphic relation for the manifolds R ( m , n ) R(m,n) and T ( m , n ; m ′ , n ′ ) T(m,n;m’,n’) can also be found in [9]. But the topological classification of R’s, T’s, and especially L’s were still unsolved problems. Furthermore, the connected sum decomposition of a manifold with fixed points, even in the simply connected case, is not unique. In this paper, we completely classify the manifolds with fixed points. For the manifolds R’s and T’s, the above mentioned stable homeomorphic relation is proved to be the topological classification. The manifolds L ( n ; p , q ; m ) L(n;p,q;m) form a very interesting family of 4-manifolds. They behave similarly to lens spaces. For example, the fundamental group of L ( n ; p , q ; m ) L(n;p,q;m) is finite cyclic of order n. And it is proved that π 1 ( L ( n ; p , q ; m ) ) {\pi _1}(L(n;p,q;m)) and π 1 ( L ( n ; p ′ , q ′ ; m ′ ) ) {\pi _1}(L(n;p’,q’;m’)) act identically on the second cohomology of their common universal covering space, ( S 2 × S 2 ) # ⋯ # ( S 2 × S 2 ) ({S^2} \times {S^2})\# \cdots \# \;({S^2} \times {S^2}) ( n − 1 n - 1 copies), even though L ( n ; p , q ; m ) L(n;p,q;m) and L ( n ; p ′ , q ′ ; m ′ ) L(n;p’,q’;m’) are not homotopically equivalent for some ( n ; p ′ , q ′ ; m ′ ) (n;p’,q’;m’) ’s. This family of manifolds is explicitly constructed and completely classified. In addition, a normal form is imposed on a connected sum decomposition mentioned above. These nromal forms are unique. Most of the material of this paper appeared first in the author’s doctoral dissertation. The author would like to thank his thesis advisor Professor F. Raymond for his help and encouragement.