Anders BjSrner, Topological methods, Handbook of Combinatorics (R. Graham, M. GrStschel, and L. Lovsz, eds.), vol. 2, Elsevier Science B.V., 1995, pp. 1819-1872.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....are homotopic in the classical sense if and only if they are equal, and all homotopies are degenerate. However, studying homotopies of paths (not just loops) certainly remains interesting. In particular, the geometry of preorders and lattices viewed through their order complexes is a rich domain [8]. 4.3. Components The last section closes the case for non negative dimensions. In dimension 1, recall that there are two extremal ways to build an augmentation of a simplicial set (see e.g. 14] One, exemplified by the nerve construction for augmented simplicial sets, builds the augmentation ....

Anders BjSrner, Topological methods, Handbook of Combinatorics (R. Graham, M. GrStschel, and L. Lovsz, eds.), vol. 2, Elsevier Science B.V., 1995, pp. 1819-1872.

....1g is homeomorphic to the suspension of Q. It is easy to show that both h ffi g and the identity map on P are carried by the following contractible carrier on the order complex D(P) of P. C : D(P) 2 D(P) s 7 D P (iffi f ) mins) P ( jffi f ) maxs) P Thus, by the Carrier Lemma [3], the map h ffi g is homotopic to the identity on P, and g and h are homotopy inverses to each other. We now prove that the assumptions of the Suspension Lemma are satis Thetaed by the following set of data. P = S(n;d) Q = S(n Gamma 1;d) green(S(n;d) fT 2 S(n;d) fn Gamma d; ....

A. Bj#orner, Topological methods, Handbook of Combinatorics (R. Graham, M. Gr#otschel, and L. Lov#asz, eds.), North Holland, Amsterdam, 1995, pp. 1819#1872.

....are homotopic in the classical sense if and only if they are equal, and all homotopies are degenerate. However, studying homotopies of paths (not just loops) certainly remains interesting. In particular, the geometry of preorders and lattices through their order complexes is a rich domain [8]. 4.3. Components This closes the case for non negative dimensions. In dimension 1, recall that there are two extremal ways to build an augmentation of a simplicial set (see e.g. 14] One, exempli ed by the nerve construction for augmented simplicial sets, builds the augmentation of the ....

Anders Bjorner, Topological methods, Handbook of Combinatorics (R. Graham, M. Grotschel, and L. Lovasz, eds.), vol. 2, Elsevier Science B.V., 1995, pp. 1819-1872.

....simplicial complex on the vertex set E = f1; mg with vertextransitive action by some group G, then it is the standard (m 1) simplex m 1 . To be non evasive is a strong topological requirement. The following sequence of implications holds for nite simplicial complexes (cf. [3], 10] and [18] non evasive ) collapsible ) contractible ) Z acyclic ) Q acyclic ) 0 and leads to further generalizations of the above conjecture if we replace non evasive with the respective weaker requirements ( denotes the reduced Euler characteristic of a simplicial complex) ....

....duality observations. 2.1 The Nerve Operation Let K be a ( nite abstract) simplicial complex, F = F j ) j2J its collection of maximal faces (facets) and J the corresponding index set. We call the covering of K by its facets F the standard covering of K. Theorem 4 (Nerve Theorem, Borsuk, cf. [3]) Let N (K) be the nerve complex of K (with respect to the standard covering F = F j ) j2J ) that is, N (K) is the simplicial complex on the vertex set J such that J is a simplex of N (K) if and only if T j2 F j 6= Then K and N : N (K) are homotopy equivalent. The nerve complex of a ....

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A. Bjorner. Topological methods. Handbook of Combinatorics (R. Graham, M. Grotschel, and L. Lovasz, eds.), 1819-1872. Elsevier, Amsterdam, 1995.

....Terai [18] and Mustat a [16] gave a similar formula for H i I (S) Our result, Theorem 3.2, is a generalization of both formulas. Most results of Mustat a [16] for H i I (S) remain true for Gorenstein k[ In Theorem 4. 9, we prove that Ext i k[ k[ J; k[ Ext i k[ k[ J [2] ; k[ H i J (k[ and these inclusions are well regulated if we consider the Z n grading, where J [j] is the j th Frobenius power of J (if k[ is not Gorenstein, a somewhat weaker result holds) A general commutative ring R and an ideal I usually do not have such a ....

....is not Gorenstein, a somewhat weaker result holds) A general commutative ring R and an ideal I usually do not have such a simple property. There is one big di erence between [16] and our case. In [16] it is proved that Ext i S (S=J; S) determines H i J (S) In our case, Ext i k[ k[ J [2] ; k[ determines H i J (k[ but Ext i k[ k[ J; k[ cannot. The reason is that H i J (S) is a straight module (after suitable degree shifting) but H i J (k[ is only quasi straight; see Section 2. The paper is structured as follows. In Section 2, we prove some basic ....

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A. Bjorner, Topological methods, Handbook of combinatorics, Vol. 1, 2, 1819-1872, Elsevier, Amsterdam, 1995.

....[Yuz98] respectively. Since our theorem can be easily achieved by the same techniques as in Yuzvinsky s article [Yuz99] plugging in our stronger Theorem 5.2, we omit the proof. It is essentially based on the understanding of the homology of geometric lattices and the cross cut complex (cf. [Bj94]) Recall that a subset of atoms of a geometric lattice L is called independent if rk W = j j, otherwise it is called dependent. 5.5. Theorem. Let A be a ( 2) arrangement in W with geometric intersection lattice L. Fix an arbitrary linear order on the set of atoms of L. Then the integral ....

Anders Bjrner, Topological methods, Handbook of Combinatorics (R. Graham, M. Grtschel, and L. Lovsz, eds.), North Holland, Amsterdam, 1994, pp. 1819 1872.

....to the suspension of Q. It is easy to show that both h ffi g and the identity map on P are carried by the following contractible carrier on the order complex D(P) of P. C : ae D(P) 2 D(P) s 7 D Gamma P (iffi f ) mins) P ( jffi f ) maxs) P Delta : Thus, by the Carrier Lemma [3], the map h ffi g is homotopic to the identity on P, and g and h are homotopy inverses to each other. We now prove that the assumptions of the Suspension Lemma are satis Thetaed by the following set of data. ON SUBDIVISION POSETS OF CYCLIC POLYTOPES 7 P = S(n;d) Q = S(n Gamma 1;d) ....

A. Bj#orner, Topological methods, Handbook of Combinatorics (R. Graham, M. Gr#otschel, and L. Lov#asz, eds.), North Holland, Amsterdam, 1995, pp. 1819#1872.

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Anders Bj#orner, Topological methods, Handbook of Combinatorics (Ronald L. Graham, Martin Gr#otschel, and L#aszl#o Lov#asz, eds.), North Holland, Amsterdam, 1995, pp. 1819#1872.

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Anders Björner, Topological methods, Handbook of Combinatorics (Ronald L. Graham, Martin Grötschel, and László Lovász, eds.), North Holland, Amsterdam, 1995, pp. 1819--1872.

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Anders Björner, Topological methods, Handbook of Combinatorics (Ronald L. Graham, Martin Grötschel, and László Lovász, eds.), North Holland, Amsterdam, 1995, pp. 1819--1872.

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Anders Björner, Topological methods, Handbook of Combinatorics (Ronald L. Graham, Martin Grötschel, and László Lovász, eds.), North Holland, Amsterdam, 1995, pp. 1819--1872.

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