Questions tagged [homology]
Homology is a general way of associating a sequence of algebraic objects such as abelian groups or modules to other mathematical objects such as topological spaces.
330 questions
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Induced homology map zero implies zero in cobordism?
I had asked this in math stackexchange, but got no reply. Hence, I'm asking here.
[I'm no expert in (co)bordism theory, and I've been struggling with it for the past few weeks. Any good references on ...
- 21
3
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50
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Hat knot Floer Homology with Z coefficients calculation
I would like to ask for recommended references which carry out the calculation of the hat knot Floer homology of a knot with $\mathbb{Z}$ coefficients, i.e., $\widehat{\operatorname{HFK}}(K;\mathbb{Z})...
- 171
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1
answer
215
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Compute the singular homology group modulo barycentric subdivision
Let $X$ be a topological space, and let $C(X)$ denote its singular chain complex with boundary operator $\partial$ and $n$-th chain group $C_n$. We know there exists a barycentric subdivision operator ...
- 807
1
vote
1
answer
101
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Cohomology of "symplectically self-dual" chain complex
Let $G,H$ be abelian groups (denoted additively) with their Pontryagin duals denoted as $G^*$ and $H^*$. (The cases I'm interested in are products of $\mathbb Z_n$, $\mathbb Z$, $\mathbb R$, and $\...
- 3,001
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1
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incidence coefficients in homological integration theory
I originally asked this on MSE but received no answers so I'm asking here.
I haven't found any reference on this after lots of looking since I first asked that question in February, so I think that ...
- 840
2
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1
answer
200
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Identifying $d_1$ in the Atiyah-Hirzebruch-Serre spectral sequence
In A Primer on Spectral Sequences (also later published in More Concise Algebraic Topology), J. Peter May describes the Serre Spectral Sequence for any homology theory. To recap, suppose $p\colon E\...
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107
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rational homology of SO(2,1) over number fields
Let $\mathrm{SO}(2,1)$ be the special orthogonal group defined by the quadratic from $q(x,y,z)=x^2+y^2-z^2$.
This is a connected non-simpy connected algebraic group.
Now, let $F$ be a number field, ...
5
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1
answer
588
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Was homology influenced by Euler's polyhedron formula?
First of all, Alama - Formal proofs and refutations contains information about Euler's Polyhedron Formula. If you look at pp. 47–49, you will see that Poincaré proved Euler's Polyhedron Formula, and ...
- 245
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Reference request: inverse image in singular homology as in Chow groups
I come from algebraic geometry and I have trouble finding a reference to check the construction of the inverse image in singular homology, analogous to that of the Chow groups. Let me be more precise:
...
- 2,871
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138
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Shub Conjecture and polynomial entropy
The Shub conjecture on topological entropy $h(f)$ of self map f on manifold M says that the topological entropy is greater (or equal) than (to) the log of maximum absolute values of the ...
- 356
5
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158
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Representing some odd multiples of integral homology classes by embedded submanifolds
Consider an $m$-dimensional compact closed orientable smooth manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]$ on $M$, with $1 \le n \le m-1$. Then does there exist an odd integer ...
- 587
13
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1
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517
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Impossibility of realizing codimension 1 homology classes by embedded non-orientable hypersurfaces
Suppose we have an $n+1$-dimensional compact closed oriented manifold $M$ and an $n$-dimensional integral homology class $[\Sigma]\in H_n(M,\mathbb{Z})$ on $M.$ Then is it true that $[\Sigma]$ mod $2$ ...
- 587
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271
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Reference to a definition of a graph homology
Let $G$ be a graph, and define $C_k$ to be the free abelian group on the homomorphisms from graphs $H$ such that $K_k$ is a minor of $H$ without needing to do any vertex deletions, only edge ...
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1
answer
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When are homologous embedded surfaces in 3-manifolds related by embedded cobordisms?
Let $M$ be an orientable closed 3-manifold and suppose $A$ and $B$ are embedded incompressible closed orientable surfaces in $M$ with $[A] = [B]$ in $H_2(M,\mathbb{Z})$.
In general, there are a ...
4
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2
answers
342
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On a generalized homotopy transfer theorem
In the book of Loday and Vallette "Algebraic Operads" a necessary condition for the Homotopy Transfer Theorem is that the starting operad is Koszul. I am interested in a generalization of ...
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1
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555
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Betti numbers of non-orientable $3$-manifolds
Let $M^3$ be a compact $3$-manifold with boundary $\partial M$.
If $M$ is orientable, then it is known (see Lemma 3.5 here) that $2\dim(\ker(H_1(\partial M,\mathbb{Q})\rightarrow H_1(M,\mathbb{Q})))=\...
- 193
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488
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Cohomology of finite symmetric products of manifolds
Let $M$ be a closed, orientable manifold of dimension $k$. I am looking for results to determine explicitly the (co)homology groups and/or cohomology ring structure (in integer or rational ...
- 506
13
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1
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385
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Realizing integral homology classes on non-orientable manifolds by embedded orientable submanifolds
Let $M^m$ denote a compact, non-orientable smooth manifold and $\nu$ an integral homology class of dimension $n$. I am interested in understanding the representability of $\nu$ by embedded, orientable ...
- 587
1
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0
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Homology groups of moduli of parabolic bundles with fixed determinant
I am looking for the Homology groups of the moduli space of stable parabolic bundles over a smooth projective curve with fixed determinant.
In particular, what is the second homology group of such ...
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Amending flawed "proof" that homology groups are zero
I am trying to prove a certain statement that seems true based on computational data, and there is a nice argument that proves it, assuming all cycles are the simplest ones (e.g., when the only 1-...
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0
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85
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Projective dimension and certain subideals
My question is related to this one. I thought mine could be very elementary but I'm not sure how to look into it.
Let $J$ be an ideal of $R=k[\mathbf{x}]$ where $\mathbf{x}=\{x_1,\dots,x_n\}$. Let $\{...
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2
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899
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Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
Let $M$ be an $m$-dimensional compact closed smooth manifold and $z\in H_n(M,\mathbb{Z})$ an $n$-dimensional integral homology class, with $m>n.$ Does there exist a pair of $M$ and $z$ so that $z$ ...
- 587
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146
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Weaker condition for the excision axiom
This comes from a question I asked on mathstackexchange (link: here)
The excision axiom in homology states that if $\overline Z\subseteq\operatorname{Int}A$, then $h_n(X\setminus Z,A\setminus Z)$ is ...
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1
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217
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Integral homology classes of which no multiples admit embedded representatives with trivial normal bundle
Let $M$ be a closed smooth manifold of dimension $n$ and $z\in H_l(M,\mathbb{Z})$ a $k$-dimensional integral homology class. Theorem II.4 of Thom's classical 1954 paper states that for $l< n/2$ or $...
- 587
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127
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Triple insersection number of a surface in three-manifolds
I heard something about the triple intersection number $\text{mod}(2)$ (but maybe also $\text{mod}(n)$) of a surface in an orientable three-manifold but I couldn't find a precise definition. My guess ...
- 813
3
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3
answers
423
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Pairing between cohomology and the image of the Hurewicz homomorphism
Let $X$ be a compact manifold of dimension $\geq k$. Denote by
\begin{equation}
h: \pi _k(X) \rightarrow H_k(X,\mathbb{Z})
\end{equation}
be Hurewicz homomorphism and by $\Gamma _k(X)\subset H_k(X,\...
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3
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1
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Linking form for homology with general coefficients
For integral homology groups there is the notion of linking form (http://www.map.mpim-bonn.mpg.de/Linking_form)
$$
Tor(H_{l}(X,\mathbb{Z}))\times Tor(H_{n-l-1}(X,\mathbb{Z}))\rightarrow \mathbb{Q}/\...
- 813
3
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2
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677
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Does the cohomology Bockstein homomorphism map to the homology Bockstein homomorphism under Poincarè duality?
Given a manifold $X$ and short exact sequence of abelian groups
$$
1\rightarrow A_1\overset{\iota}{\rightarrow} A_2\overset{\pi}{\rightarrow} A_3\rightarrow 1
$$
we get the Bockstein map in cohomology ...
- 813
5
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1
answer
392
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Computation of the linking invariant on Lens spaces
Let $L_n(p)$ be the $2n+1$ dimensional Lens space
$$
S^{2n+1}/\mathbb{Z}_p
$$
where the action is given as $z_i\rightarrow e^{\frac{2\pi}{p}}z_i$, $i=1,...,n+1$, with $z_i$ the coordinates of $\mathbb{...
- 813
4
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1
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238
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Euler class of vertical tangent bundle of the surface bundle over circle
Suppose $\Sigma$ is an oriented genus $g>1$ surface and $h:\Sigma\to \Sigma$ is a diffeomorphism preserving a point $p$. Let $M$ be the surface bundle over $S^1$ obtained by gluing $\Sigma\times I$ ...
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314
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Reference for Künneth Theorem in (co)homology with local coefficients
Is there a discussion in the literature of Künneth-type theorems for (co)homology with local coefficients? The sources I know of that discuss local coefficients (Whitehead's Elements of Homotopy ...
- 8,803
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0
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147
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Trivial homology groups for p-torsion groups
Let $G$ be a group where each element has a $p$-power order.
Let $M$ be a $G$-module without $p$-torsion.
Here $G$ is a discrete infinite subgroup of a complete group. Then, it cannot be assumed pro-...
2
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0
answers
96
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Arf invariant for characteristic surfaces in closed 4-manifolds depends on homeomorphism type
Let $X$ be a closed smooth oriented 4-manifold with $H_1(X;\Bbb Z)=0$. Then for a smoothly embedded orientable surface $F\subset X$ which is characteristic, there is a well-defined invariant $\text{...
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1
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2
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182
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Lattices formed by unions of elements in an antichain
Let $A_1, \dots, A_k$ be incomparable subsets (of $\{1, \dots, n\}$) and consider the poset $P$ consisting of all possible unions of these under inclusion. Its not hard to see that this is a lattice, ...
- 43
1
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0
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109
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bott element in periodic cyclic homology
I am reading a paper by Thomason "Algebraic K-theory and étale cohomology", which deals with algebraic K theory localized by inverting the bott element $\beta \in K_2(\overline{k})^{\wedge}...
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0
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213
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pullback square in abelian category and derived categories
Let $\mathcal{A}$ be an Abelian category. Take objects $A,B,C$ and $D$ in $\mathcal{A}$, and morphisms $b:B\to A$, $b':B\to A$, $c:C\to A$, $e:D\to C$, $e':D\to C$ and $f:D\to B$ such that diagrams
$\...
- 519
5
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1
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351
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"Singular homology = simplicial homology" relative to a fibration
Let $p:E\to B$ be a fibration. Suppose $B$ has a simplicial decomposition. For each $n\in\mathbb{Z}_{\ge0}$, let $C_n$ be the free abelian group generated by the set of pairs $(\sigma,\tau)$ where $\...
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1
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395
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Two surfaces in a 4-manifold whose algebraic intersection number is zero
Suppose $X$ is a smooth closed oriented 4-manifold, and $\Sigma_1,\Sigma_2$ are smoothly embedded compact oriented surfaces in $X$. Suppose they intersect transversally at two points with different ...
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269
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Homology classes in connected sum of $\Bbb CP^2$'s that can be represented by smoothly embedded spheres
Let $h=[\Bbb CP^1]\in H_2(\Bbb CP^2;\Bbb Z)$. By a theorem of Kronheimer and Mrowka (Theorem 1 of this paper: https://people.math.harvard.edu/~kronheim/thomconj.pdf), a class $nh \in H_2(\Bbb CP^2;\...
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Is there a way to calculate the Froyshov $h$-invariant for Seifert homology spheres?
In 2002, by using Floer theory, Froyshov defined the $h$-invariant for intergal homology 3-spheres, which is a surjective group homomorphism $\Theta^3_{\Bbb Z}\to \Bbb Z$, where $\Theta^3_{\Bbb Z}$ is ...
- 335
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0
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181
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Bar constructions and pushouts
Suppose that $\mathsf S$ is a span of associative algebras (or, more generally, if you'd like, any type of object admitting a bar-cobar formalism) and let $A$ be its pushout.
Is there any hope of ...
- 1,554
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votes
1
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1k
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Most efficient way of getting a brief overview of the current active research areas in Algebraic Topology
I'd be applying for a Ph.D. at various grad schools in the U.S. in the coming months and while I know I'd like to pursue research in the field of Algebraic Topology, I am not knowledgeable enough yet ...
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374
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Singular homology using singular cubes
When singular homology is defined using cubes instead of simplices it is important to factor out the degenerate cubes in the course of building the singular chain complex. If you omit this step it is ...
- 1,638
3
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0
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135
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Homology groups of a certain simplicial complex
I've run across a simplicial complex which, according to Sage, seems to have a very easily-described homology. However, proving this fact has been rather difficult.
Fix $s\ge 2$ (though I would be ...
- 807
10
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219
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Are multiples of representable homology classes still representable by smooth submanifolds?
Recently the following question comes up in my research. Suppose we have a closed compact connected smooth manifold $M,$ of dimension $d+c$ and an integral homology class $[N]$ induced by a compact ...
- 587
3
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0
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164
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Chekanov-Eliashberg Legendrian DGA with positive grading?
I was just looking back to some notes that I took a few years ago, when I was reading Etnyre's notes on Legendrian Contact Homology in $\mathbb R^3$ and I happened upon the following question that I ...
- 1,225
2
votes
2
answers
380
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Spaces homotopy dominated by $S^2 \times S^2\times S^2$
We say that a topological space $A$ is homotopy dominated by a topological space $X$ if there exist continuous maps $f:A\to X$ and $g:X\to A$ such that $g\circ f\simeq 1_A$.
Let $X$ be $S^2 \times S^2 ...
- 1,182
2
votes
0
answers
122
views
Different definitions of p-fusion and Mislin's theorem
Currently, I am trying to understand and compute homology of finite groups with coefficients in a field of positive characteristic. So, I was searching for some results that could reduce this problem (...
4
votes
1
answer
233
views
Mayer–Vietoris sequence for coproduct of Hopf algebras
Is there a Mayer–Vietoris-type sequence for the homology of a coproduct of two Hopf algebras over an ideal? The definition of the coproduct can be found in Agore - Categorical Constructions for Hopf ...
- 1,374
1
vote
0
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109
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Modular cycles?
It is well known that cocycles (differential forms) and cycles share many properties through duality (e.g., de Rham). I've been reading about modular forms recently and I came with a very naive ...
- 221