Search results

Create the page "Collapse cell" on this wiki! See also the search results found.

• Cell maps
  ...ow the theory of simplicial maps and their homology is extended to general cell complexes. ...esenting the spaces as cell complexes, we examine to what cell in $L$ each cell in $K$ is taken by $f$:
  42 KB (7,005 words) - 03:10, 30 November 2015
• Homology of maps
  ==Maps of cell complexes== ...ow the theory of simplicial maps and their homology is extended to general cell complexes.
  41 KB (6,926 words) - 02:14, 21 October 2015
• Combinatorial cell maps
  Below, we will see how the theory of simplicial maps is extended to general cell complexes. ...esenting the spaces as cell complexes, we examine to what cell in $L$ each cell in $K$ is taken by $f$:
  31 KB (5,330 words) - 22:14, 14 March 2016
• Chain maps
  *(A) $g$ maps each cell in $K$ to a cell in $L$ of the same dimension, otherwise it's $0$. $\\$ The condition corresponds to the clone/collapse condition for a simplicial map.
  47 KB (8,115 words) - 16:19, 20 July 2016
• Homotopy
  *$ep:X\to X$ is the collapse, or self-projection, of the cylinder on its bottom edge. ...center. Note here that, if done incrementally, this contraction becomes a collapse and can't be reversed.
  46 KB (7,846 words) - 02:47, 30 November 2015
• Homotopy and homotopy equivalence
  *$ep:X\to X$ is the collapse, or self-projection, of the cylinder on its bottom edge. ...center. Note here that, if done incrementally, this contraction becomes a collapse and can't be reversed.
  45 KB (7,738 words) - 15:18, 24 October 2015
• Computing homology
  ...ain]] is a "formal" [[linear combination]] of finitely many oriented $k$-[[cell]]s, such as $3a + 5b - 17c$ with, typically, integer coefficients. Then the ...{\bf Z}, \sigma_i {\rm \hspace{3pt} is \hspace{3pt} a \hspace{3pt}} k{\rm -cell \hspace{3pt} in \hspace{3pt}} K \right\}.$$
  26 KB (4,370 words) - 21:55, 10 January 2014
• Topology Illustrated -- Index
  *[[boundary of a cubical cell|boundary of a cubical cell]] *[[cell approximation|cell approximation]]
  16 KB (1,773 words) - 00:41, 17 February 2016
• Cohomology
  ...f homology theory and it starts with the concept of a $k$-''cochain'' on a cell complex $K$. It is any linear function from the module of $k$-chains to $R$ Recall that a cell complex $K$ is called acyclic if its chain complex is an ''exact sequence''
  45 KB (6,860 words) - 16:46, 30 November 2015
• Manifolds
  The $1$-cell is the segment $[0,1]$. Here, some of the neighborhoods are open intervals, The same happens in all dimensions. For an $n$-cell $\sigma$, the neighborhoods of the boundary points $\partial \sigma$ are ho
  51 KB (8,919 words) - 01:58, 30 November 2015
• Homology maps
  <center>(A) $g$ maps each cell in $X$ to a cell in $Y$ of the same dimension, or $0$.</center> ...ter option is reserved for cells that collapse (just like homology classes collapse as discussed in [[Maps and homology]]).
  13 KB (2,315 words) - 22:15, 15 July 2014
• Duality: forms as cochains
  A $k$-''cochain'' on a cell complex $K$ is any linear function from the module of $k$-chains to $R$: Recall that a cell complex $K$ is called acyclic<!--\index{acyclic space}--> if its chain comp
  29 KB (4,540 words) - 13:42, 14 March 2016
• Test
  #Zooming in beyond the pixels: [[cell decomposition of images]] Roughly, they are constant on each cell of a cubical complex. They are "discrete" in this sense. We don't establish
  16 KB (2,139 words) - 23:01, 9 February 2015
• Applied Topology and Geometry
  #Zooming in beyond the pixels: [[cell decomposition of images]] Roughly, they are constant on each cell of a cubical complex. They are "discrete" in this sense. We don't establish
  16 KB (2,088 words) - 16:37, 29 November 2014
• Simplicial maps
  ...allowing an edge to be taken to a node, by means of a “collapse”<!--\index{collapse}-->. Since graphs are $1$-dimensional simplicial complexes, we can rewrite It is the case of collapse that needs special attention. We will prove the general case.
  34 KB (5,929 words) - 03:31, 29 November 2015
• Simplicial maps and chain maps
  ...allowing an edge to be taken to a node, by means of a “collapse”<!--\index{collapse}-->. Since graphs are $1$-dimensional simplicial complexes, we can re-write It is the case of collapse that needs special attention. We will prove the general case.
  34 KB (5,897 words) - 16:05, 26 October 2015
• Chain operators
  #redirect[[cell maps]] <center>(A) $g$ maps each cell in $X$ to a cell in $Y$ of the same dimension, or $0$.</center>
  8 KB (1,361 words) - 20:48, 27 August 2015
• How to compute Betti numbers
  We summarize the procedure for computing the Betti numbers of a [[cell complex]], by hand. ===Step 1: present the cell complex===
  5 KB (890 words) - 14:47, 24 August 2014
• How to compute homology
  ...procedure for computing the [[homology as a vector space|homology]] of a [[cell complex]], by hand. ==Step 1: present the cell complex==
  6 KB (1,049 words) - 09:21, 3 September 2011
• Operations on functions
  [[image:collapse of axis with arrows.png| center]] The real line is shrunk to a single point; we can call this transformation ''collapse''.
  142 KB (23,566 words) - 02:01, 23 February 2019
• Maps of graphs
  ...to be taken to a node, the event that we will call a “collapse”<!--\index{collapse}-->: <!--150-->[[image:grid with function approximated by cell map.png|center]]
  29 KB (5,042 words) - 17:57, 28 November 2015
• Introductory algebraic topology: review exercises
  2. Represent the sphere as a cell complex with two $2$-cells, list all cells, and describe/sketch the gluing ...jective plane and represent the result as a cell complex with a single $2$-cell (standard diagram for surfaces). What is it?
  9 KB (1,553 words) - 20:10, 23 October 2012
• Homology classes under maps
  #redirect[[cell maps]] ...ightarrow Y$ is a constant map, all $k$-homology classes of $X$ with $k>0$ collapse or, algebraically speaking, are mapped to $0$ by $f_*$. Meanwhile, all $0$-
  4 KB (672 words) - 20:46, 27 August 2015
• Set-valued maps
  <!--150-->[[Image:torus collapse.png|center]] In the latter case, these cycles collapse to points and, therefore, their homology classes are lost. It remains to be
  24 KB (4,382 words) - 15:52, 30 November 2015
• Homology operator
  #redirect[[cell maps]] Given a [[cell map]] $f \colon K \rightarrow L$, the ''homology operator induced by f'',
  2 KB (388 words) - 20:49, 27 August 2015
• Algebra on graphs
  ...en to a node, the event that we will call a “collapse”<!--\index{collapsed cell}-->: We say that, in this case, the edge is ''cloned''<!--\index{cloned cell}-->.
  36 KB (6,177 words) - 02:47, 21 February 2016
• New complexes from old
  ...in examples are the following (given for [[cubical complex]]es, same for [[cell complex]]es): ...xes]]: Given a [[cubical complex]] $K$ and another cubical complex $L$ all cell of which also belong to $K$. If $\partial \colon C(K) \rightarrow C(K)$ is
  2 KB (310 words) - 13:36, 19 July 2011