Zp Actions Research Articles

Hazel Leaf Polyphenol Extract Alleviated Cisplatin-Induced Acute Kidney Injury by Reducing Ferroptosis through Inhibiting Hippo Signaling.

Derived from hazelnuts, hazel leaf has been utilized in traditional folk medicine for centuries in countries such as Portugal, Sweden, and Iran. In our previous investigations, we conducted a preliminary assessment of the hazel leaf polyphenol extract (referred to as ZP) and identified nine compounds, such as kaempferol and chlorogenic acid, in its composition. ZP has shown promising properties as an antioxidant and anti-inflammatory agent. Our research has revealed that ZP has protective effects against cisplatin-induced acute kidney injury (AKI). We conducted a comprehensive examination of both the pathological and ultrastructural aspects and found that ZP effectively ameliorated renal tissue lesions and mitigated mitochondrial damage. Moreover, ZP significantly suppressed malondialdehyde levels while increasing glutathione and catalase concentrations in the kidneys of AKI-induced mice. ZP decreased the number of apoptotic cells and decreased pro-apoptotic protein expression in the kidneys of mice and human renal tubular epithelial cells (HK-2). Furthermore, treatment with ZP increased the levels of proteins marking anti-ferroptosis, such as GPX4, FTH1, and FSP1, in experiments both in vivo and in vitro. We elucidated the underlying mechanisms of ZP's actions, revealing its inhibitory effect on Yap phosphorylation and its regulation of Lats expression, which exert a protective influence on the kidneys. Furthermore, we found that inhibiting the Hippo pathway compromised ZP's nephroprotective effects in both in vitro and in vivo studies. In summary, this research shows that ZP exhibits renoprotective properties, effectively reducing oxidative damage, apoptosis, and ferroptosis in the kidneys by targeting the Hippo pathway.

Quantum lens spaces and graph algebras

We construct the C*-algebra C(L q (p; m 1 ,...,m n )) of continuous functions on the quantum lens space as the fixed point algebra for a suitable action of Zp on the algebra C(S 2n-1 q), corresponding to the quantum odd dimensional sphere. We show that C(L q (p; m 1 ,..., m n )) is isomorphic to the graph algebra C* (L( p;m 1 ,...,m n ) 2n-1). This allows us to determine the ideal structure and, at least in principle, calculate the K-groups of C(L q (p; m 1 ,..., m n )). Passing to the limit with natural imbeddings of the quantum lens spaces we construct the quantum infinite lens space, or the quantum Eilenberg-MacLane space of type (Z p ,1).

The Eichler Trace of ℤp Actions on Riemann Surfaces

AbstractWe study ℤp actions on compact connected Riemann surfaces via their associated Eichler traces.We determine the set of possible Eichler traces and determine the relationship between 2 actions if they have the same trace.

A converse to the P. A. Smith theorem for nonunitary homology spheres

If p is an odd prime and F is the fixed point set of a smooth Zp action on Sn or Dn, then F is a smooth manifold with a unitary structure. Conversely; most Zp homology disks or spheres with unitary structures are fixed point sets of smooth Zp actions on Dn or Sn for suitable n. The results of this paper show that an arbitrary oriented mod p homology disk or sphere is the fixed point set of a smooth Zp action on some Z[l/2]-homology disk or sphere. This result is in general the best possible.

On invariant knots

It was first proved by R. Lashof in [4], using the work of S. Cappell and J. Shaneson on four-dimensional surgeryu (see [1]), that there exist locally flat topological knots S3 ∪ S5 which are not smoothable. In [2] (compare also [6]) S. Cappell and J. Shaneson have constructed infinitely many non-smoothable locally fat topological knots as the fixed points of locally nice (= locally smoothable) Zp actions on S5, therefore giving non-trivial examples of locally smoothable but equivariantly non-smoothable actions of Zp on S5.

Actions on Invariant Spheres around Isolated Fixed Points of Actions of Cyclic Groups

Fix a prime number p and let Zp be a cyclic group of order p. We consider a pair (M, 0) consisting of a compact simply connected almost complex manifold M without boundary and a smooth Zp-action 0: ZPX M—>M preserving the almost complex structure of M. We suppose that M is given an invariant Riemannian metric. If a(eM) is an isolated fixed point, then the induced action of Zp on the tangent space at a gives a complex Zp-module Va which has no trivial irreducible factor. Let f : EZp—*BZp be a universal principal Zp-bundle and let f (Vtt) : EZpXzp Va —>BZP be the V0-bundle associated with ?. If a and b are isolated fixed points, we compare the cobordism Euler classes 0(f (Va)) and e(£ (Vb)) which belong to the complex cobordism group MU* (BZP) of the classifying space BZP of Zp. Let .PV be the universal formal group law over MU*9 and write

Companionship of knots and the Smith conjecture

This paper studies the Smith Conjecture in terms of H. Schubert’s theory of companionship of knots. Suppose J is a counterexample to the Smith Conjecture, i.e. is the fixed point set of an action of Z p {{\textbf {Z}}_p} on S 3 {S^3} . Theorem. Every essential torus in an invariant knot space C ( J ) C(J) of J is either invariant or disjoint from its translates. Since the companions of J correspond to the essential tori in C ( J ) C(J) , this often allows one to split the action among the companions and satellites of J. In particular: Theorem. If J is composite, then each prime factor of J is a counterexample, and conversely. Theorem. The Smith Conjecture is true for all cabled knots. Theorem. The Smith Conjecture is true for all doubled knots. Theorem. The Smith Conjecture is true for all cable braids. Theorem. The Smith Conjecture is true for all nonsimple knots with bridge number less than five. In addition we show: Theorem. If the Smith Conjecture is true for all simple fibered knots, then it is true for all fibered knots. Theorem. The Smith Conjecture is true for all nonfibered knots having a unique isotopy type of incompressible spanning surface.

Differentiable group actions on homotopy spheres

In several previous papers numerous results conserning smooth group actions on homotopy spheres not bounding parallelizable manifolds have been obtained [5, 13, 21, 23-26, 31, 32, 35, 39, 42-50]. This paper is a first step towards placing such results in a more unified setting; we shall concentrate our attention on semifiee actions to avoid book-keeping problems with various orbit types, postponing these considerations to later papers in the series. When combined with appropriate homotopy-theoretic machinery, this setting yields new proofs and substantial strengthenings of many known results. The basic underlying problem of this paper is the following: Given a homotopy (n +2k)-sphere 22, does it admit a semifree G-action with an n-dimensional fixed point set'? Our techniques show that the answer depends on the decomposition of the Pontrjagin-Thom invariants of 22 (which form a coset in the stable homotopy of spheres) in terms of lower-dimensional stable homotopy classes. In a subsequent paper we shall illustrate this by studying certain low-dimensional homotopy spheres in detail, obtained virtually complete information. On the other hand, the new non-existence theorems obtained in this paper also fall into this pattern indirectly; for if one knows that some specific composition operation (ordinary or higher order) on a stable homotopy class is nontriviat, such information often implies the class cannot be decomposed in the "right" way for an action to exist. Presumably some appropriate global information about the stable homotopy of spheres (perhaps resembling the Kahn-Priddy Theorem) is needed to fully answer questions such as the existence of smooth Zp actions on every homotopy sphere. The general viewpoint of this paper is derived from the work of Browder, Petrie, Rothenberg, and Sondow that classified certain semifree actions on homotopy spheres ([8, 10, 11, 39, 40]; also see [48]). A general motivating problem was the determination of the underlying differential structure on the ambient G-manifold from the classification information presented in the above papers; the insight needed to study this problem evolved from consideration of the examples in [45-47] where the differential structure was determined by entirely different methods. Our basic result (Theorem 3.4) establishes a strong relationship between the differentiable structure on 2" and an invariant that vaguely measures the knottedness of the fixed point set (the knot invariant of Section 2, which agrees with the internal normal invariant of [49]). This relationship exists because the

A W(Z 2 ) Invariant for Orientation Preserving Involutions

In this paper we calculate an invariant in W(Z2), the Witt ring of nonsingular, symmetric Z2-inner product spaces, for orientation-preserving involutions on compact, closed, connected 4n-dimensional manifolds M. This invariant with the Atiyah-Singer index theorem uniquely determines the orthogonal representation of Z on H2n(M; Z)/TOR. We also give an example to show that this invariant detects actions that the Atiyah-Singer theorem cannot. In this paper we calculate a torsion invariant for orientation-preserving involutions. This invariant and the Atiyah-Singer-Segal G-signature theorem allow one to compute precisely the element of W(Z; Z2) given by the orthogonal representation of Z2 on H2n(M; Z)/TOR. This cannot be done with the Atiyah-Singer-Segal theorem alone. We will return to this in the last section. In [4] Conner and Raymond define an invariant q(T, M) for orientationpreserving actions of Zp, p a prime, T a generator of Zp, on closed, compact, oriented manifolds M of dimension 4n. Briefly the action of Zp on M gives us an orthogonal representation of Z on H2n(M; Q). If we denote by w(T, M) and sgn(T, M) the rational Witt class and signature, respectively, of the inner product (x, y)p(x u y, [M]) on the subspace of fixed vectors then q(T, M) = w(T, M) sgn(T, M)1. In [1] the problem of expressing q(T, M) in terms of fixed point information was solved for P odd. In this paper we compute q(T, M) for p = 2 and give some applications. First some notation and background. W(R) will always denote the Witt group of nonsingular, symmetric, inner-product spaces over R, R a ring. W(Z2) is isomorphic to Z2 and the isomorphism is given by taking the rank of the inner product space mod 2. From now on we identify W(Z2) and Z2 by this isomorphism. If V is a disjoint union of a finite number of closed Received by the editors June 11, 1974. AMS (MOS) subject classifications (1970). Primary 57D85; Secondary IOC05.

Equivariant maps between representation spheres

Let G be a finite group, V and W be finite representations of G, S(V) and S(W) be the unit spheres in V and W. Suppose dim VH ^dim WH for every subgroup H of G. We seek to classify the G- equivariant homotopy classes of 6-equivariant maps from S(V) to S(W). Introduction. We wish to consider the following problem: Let G be a finite group, and V and W be finite dimensional orthogonal representations of G. Let 5(V) and S(W) denote the unit spheres of V and W respectively. Then S(V) and S(W) inherit G-actions. Classify the equivariant homotopy classes of G-maps from S(V) to S(W). The case where G = Zp and S(V) and S(W) have free Zp actions was done by Olum (6) and was used to give a classification of lens spaces up to homotopy equivalence. In this paper we generalize Olum's result. Our approach is to consider the behavior of an equivariant map restricted to the various fixed point sets. Explicity, if X is a space with a left action of the group G, and H is a subgroup of G, we denote by XH the set of points in X left fixed by each element of H. If S(V)-*S(W) is a G- equivariant map (i.e., f(gυ) = gf(v) for all g G G and veS(V)), then / induces maps /: S(VH->S(W)H for each subgroup H. Since V and W are linear representations, these fixed point sets S(V)H and S(W)H are again spheres, and we may choose an orientation for each S(V)H and S(W)H. If X is a manifold, denote by dimX the (real) dimension of X. When dimS( V)H = dimS(HOH, fH has a well-determine d degree, denoted by deg/ H. Our major theorem asserts that, under suitable hypotheses, the homotopy classes of the maps fH for all H determine the equivariant homotopy class of DEFINITION. Let G be a finite group. If H is a subgroup of G, denote by N(H) the normalizer of H in G. An orthogonal representa- tion V of G is completely orientable if for every subgroup H of G, the induced action of N(H) on S(V)H is orientation-p reserving.

BORDISMS WITH THE ACTION OF THE GROUPZpAND FIXED POINTS

A full description of the ring of unitary bordisms with the action of Zp and of the ring of all admissible collections of fixed submanifolds under the action of Zp is given in terms of generators and relations. The calculations are based on a special choice of polynomial generators in the ordinary ring of unitary bordisms and Poincare duality in bordism theory. Bibliography: 3 titles.