Area-perimeter duality in polygon spaces

Two natural foliations, guided by area and perimeter, of the configurations spaces of planar polygons are considered and the topology of their leaves is investigated in some detail. In particular, the homology groups and the homotopy type of leaves are determined. The homology groups of the spaces of polygons with fixed area and perimeter are also determined. Besides, we extend the classical isoperimetric duality to all critical points. In conclusion a few general remarks on dual extremal problems in polygon spaces and beyond are given.

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Hurewicz fibrations, almost submetries and critical points of smooth maps

Forum Mathematicum ◽

10.1515/forum-2016-0009 ◽

2017 ◽

Vol 29 (4) ◽

Author(s):

Sergio Luigi Cacciatori ◽

Stefano Pigola

Keyword(s):

Homotopy Type ◽

Riemannian Manifolds ◽

Critical Points ◽

Smooth Maps ◽

Metric Properties ◽

Cw Complex

AbstractWe prove that the existence of a Hurewicz fibration between certain spaces with the homotopy type of a CW-complex implies some topological restrictions on their universal coverings. This result is used to deduce differentiable and metric properties of maps between compact Riemannian manifolds under curvature restrictions.

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Homology of Deleted Products of Contractible 2-Dimensional Polyhedra. II

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-081-7 ◽

1968 ◽

Vol 20 ◽

pp. 842-854

Author(s):

C. W. Patty

Keyword(s):

Homotopy Type ◽

Product Space ◽

Homology Groups ◽

Definition Of

The deleted product space X* of a space X is X × X — △. In (4), I computed the homology groups of the deleted product of a polyhedron in a subcollection (see §2 of this paper for the definition of ) of the finite, contractible, 2-dimensional polyhedra. In the present paper, I show that there is an infinite subcollection ( of such that the deleted product of each member of has the homotopy type of the 2-sphere.

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ALGEBRAIC CYCLES ON REAL VARIETIES AND ℤ/2-EQUIVARIANT HOMOTOPY THEORY

Proceedings of the London Mathematical Society ◽

10.1112/s002461150201376x ◽

2003 ◽

Vol 86 (2) ◽

pp. 513-544 ◽

Cited By ~ 6

Author(s):

PEDRO F. DOS SANTOS

Keyword(s):

Galois Group ◽

Homotopy Type ◽

Homotopy Theory ◽

Projective Variety ◽

Algebraic Cycles ◽

Subject Classification ◽

Equivariant Homotopy ◽

The Real ◽

Homology Groups ◽

Secondary 14C05

In this paper the spaces of algebraic cycles on a real projective variety $X$ are studied as $\mathbb{Z}/2$-spaces under the action of the Galois group ${\rm Gal}(\mathbb{C}/\mathbb{R})$. In particular, the equivariant homotopy type of the group of algebraic $p$-cycles $\mathcal{Z}_p(\mathbb{P}_{\mathbb{C}}^n)$ is computed. A version of Lawson homology for real varieties is proposed. The real Lawson homology groups are computed for a class of real varieties.2000 Mathematical Subject Classification: primary 55P91; secondary 14C05, 19L47, 55N91.

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On the homotopy type of the p-subgroup complex for finite solvable groups

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700002184 ◽

2000 ◽

Vol 69 (2) ◽

pp. 212-228 ◽

Cited By ~ 5

Author(s):

Jürgen Pulkus ◽

Volkmar Welker

Keyword(s):

Solvable Group ◽

Homotopy Type ◽

Solvable Groups ◽

Finite Solvable Group ◽

Module Structure ◽

Homology Groups

AbstractWe provide a wedge decomposition of the homotopy type of the p-subgroup complex in the case of a finite solvable group G. In particular, this includes a new proof of the result of Quillen which says that this complex is contractible if and only if there is a non-trivial normal p-subgroup in G. We also provide reduction formulas for the G-module structure of the homology groups. Our results are obtained with diagram-methods by gluing the p-subgroup complex of G along the p-subgroup complex of = G/N for a normal p′-subgroup of G.

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The relationship of the scleractinian corals to the rugose corals

Paleobiology ◽

10.1017/s0094837300025707 ◽

1980 ◽

Vol 6 (02) ◽

pp. 146-160 ◽

Cited By ~ 29

Author(s):

William A. Oliver

Keyword(s):

Critical Points ◽

Scleractinian Corals ◽

Convincing Evidence ◽

Early Triassic ◽

Rugose Corals ◽

History Of ◽

The Subject ◽

Relationship Of ◽

The Relationship ◽

Biologic Factors

The Mesozoic-Cenozoic coral Order Scleractinia has been suggested to have originated or evolved (1) by direct descent from the Paleozoic Order Rugosa or (2) by the development of a skeleton in members of one of the anemone groups that probably have existed throughout Phanerozoic time. In spite of much work on the subject, advocates of the direct descent hypothesis have failed to find convincing evidence of this relationship. Critical points are:(1) Rugosan septal insertion is serial; Scleractinian insertion is cyclic; no intermediate stages have been demonstrated. Apparent intermediates are Scleractinia having bilateral cyclic insertion or teratological Rugosa.(2) There is convincing evidence that the skeletons of many Rugosa were calcitic and none are known to be or to have been aragonitic. In contrast, the skeletons of all living Scleractinia are aragonitic and there is evidence that fossil Scleractinia were aragonitic also. The mineralogic difference is almost certainly due to intrinsic biologic factors.(3) No early Triassic corals of either group are known. This fact is not compelling (by itself) but is important in connection with points 1 and 2, because, given direct descent, both changes took place during this only stage in the history of the two groups in which there are no known corals.

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