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- ...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
- ==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
- 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
- *(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
- *$ep:X\to X$ 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
- *$ep:X\to X$ 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
- ...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
- *[[boundary of a cubical cell|boundary of a cubical cell]] *[[cell approximation|cell approximation]]16 KB (1,773 words) - 00:41, 17 February 2016
- ...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
- 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 ho51 KB (8,919 words) - 01:58, 30 November 2015
- <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
- 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 comp29 KB (4,540 words) - 13:42, 14 March 2016
- #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 establish16 KB (2,139 words) - 23:01, 9 February 2015
- #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 establish16 KB (2,088 words) - 16:37, 29 November 2014
- ...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
- ...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
- #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
- 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
- ...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
- [[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
- ...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
- 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
- #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
- <!--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 be24 KB (4,382 words) - 15:52, 30 November 2015
- #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
- ...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
- ...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 $\partial \colon C(K) \rightarrow C(K)$ is2 KB (310 words) - 13:36, 19 July 2011