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Topology

Winding Around and Mountains

Just ... ...reading John Roe's book Winding Around. Here: For an extended riff on the idea of mathematics as mountaineering see "Into Thin Air" by Colin Adams, on Mathematical Intelligencer. How much about "mathematics as mountaineering"?

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Fundamental group of SO(3) and Dirac's scissors

Topology of $SO(3)$ and fixed-distance pairs of points in $S^2$. Consider the space $X\subset S^2\times S^2$ of all pairs $(A,B)$ of points such that $|A-B|=\sqrt 3$, where the distance is the euclidean distance of $S^2\subset \RR^3$, and $S^2$ is the unit sphere in $\RR^3$. (1) $X\approx SO(3) \approx \PP^3(\RR)$. Proof: The fact that $SO(3) \approx \PP^3(\RR)$ is a not-so-easy elementary exercise (cf. exercise (11.27) and exercise (7.30) of my Geometry Notes, protected by a single-character password: the number of non-zero digits appearing in the password itself; the problem might be ill-posed).

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Homology of (simplicial) spheres and applications

Nullhomotopic odd maps, Borsuk, Lusternik and Schnirelmann Let $\Delta^n$ the standard simplex of dimension $n$ in $\RR^{n+1}$, which is homeomorphic to a $n$-dimensional disk, and $\Sigma^{n}\subset \Delta^{n+1}$ the simplicial $n$-sphere, i.e. the boundary of $\Delta^{n+1}$ (the union of its $n+2$ $n$-dimensional faces), which is homeomorphic to an $n$-dimensional sphere. The homeomorphisms can be more explicit: let $B\in \Delta^{n+1}$ be its barycenter, and $\varphi\from \Sigma^n\subset \Delta^{n+1} \to \RR^{n+2}$ the map defined as $\varphi(x) = \dfrac{x-B}{|x-B|}$.

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Mesh, Barycentric Subdivision and Simplicial Approximation Theorem

Mesh of a euclidean simplicial complex. Let $|K| \subset \RR^k$ an euclidean simplicial complex (i.e., an abstract simplicial complex, where the set of vertices actually is a set of points in $\RR^k$, such that $|K|$ is the disjoint union of the interiors of the convex hulls of the sets of vertices ranging over the set of simplices). A vertex $P$ of $K$ is therefore a point of $\RR^k$. For each simplex $\sigma \subset K$, the diameter of $\sigma$ (or $|\sigma|$ more precisely) is the maximum distance $|x-y|$ as $x,y$ vary in $|\sigma|$.

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Joins and Products of polyhedra

Join $K*L$ of two simplicial complexes Let $K = (X_ K, \Phi_ K)$ and $L = (X_ L, \Phi_ L)$ be two simplicial complexes ($X_ K$ denotes the set of vertices of $K$ and $\Phi_ K$ the set of simplices). Then the join $K*L$ is the simplicial complex defined as follows. Its set of vertices $X=X_ K \cup X_ L$ is the union of the sets of vertices of $K$ and of $L$, and a non-empty subset of $X$ is a simplex of $K*L$ if and only if it is of type $\sigma \cup \tau$, where $\sigma \in \Phi_ K \cup \{ \emptyset \}$ and $\tau \in \Phi_ L \cup \{\emptyset \}$.

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Ranks, Tensors and Abelian Groups

Abelian Groups Given $a\in A$, the homomirphism $i_ a \from \ZZ \to A$ defined by $i_ a(k) = ka$ is mono (injective) iff $a$ has infinite order. Of $S$ is a set, the free Abelian group generated by $S$ is $$ \ZZ[S] = \ZZ S = \bigoplus_ {s \in S} \ZZ, $$ and it is the group of all finite linear combinations (with integer coefficients) of elements in $S$. The elements of $S$ are a basis (base) of $\ZZ[S]$.

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Graphs, trees and simplicial complexes

Graphs and simplicial complexes A one-dimensional simplicial complex is a graph. That is, $K=(X_ K,\Phi_ K)$ where $V(K) = X_ K$ is the set of vertices and $\Phi_ K$ is the set of simplexes, with $\Phi_ K = \Phi^1_ K \cup \Phi^0_ K$, where $E(K) = \Phi^1_ K$ are the 1-dimensional simplexes (termed edges of $K$) and $\Phi^0_ K = \{ \{P\} : P \in X_ K \}$ are the $0$-dimensional simplexes (sometimes termed again vertices of $K$).

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Computing the Smith Normal Form of a matrix, and homology groups?

A simple algorithm for computing the Smith Normal Form of a matrix in $\ZZ$ The proof and the algorithm are the same. First, a few auxiliary functions. Given a matrix $M$, the follow two functions are self-explanatory. def dims(M): num_righe=len(M) num_colonne=len(M[0]) return (num_righe,num_colonne) def MinAij(M,s): num_righe, num_colonne=dims(M) ijmin=[s,s] valmin=max( max([abs(x) for x in M[j][s:]]) for j in range(s,num_righe) ) for i in (range(s,num_righe)): for j in (range(s,num_colonne)): if (M[i][j] != 0 ) and (abs(M[i][j]) <= valmin) : ijmin = [i,j] valmin = abs(M[i][j]) return ijmin def IdentityMatrix(n): res=[[0 for j in range(n)] for i in range(n)] for i in range(n): res[i][i] = 1 return res def display(M): r="" for x in M: r += "%s\n" % x return r +"" Then, one needs the elementary operations on rows and columns on the matrix $M$: swap (permute) two rows, add to a row an integer multiple of another row, and change sign of a row.

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Császár torus, python and sagemath

The simplicial complex Consider the (abstract) simplicial $T$ defined as follows: vertices are $$ X=\{ 0,1,2,3,4,5,6 \} = \{ j \mod 7 \}. $$ Now, for each $x\in X$ let $\Phi_ x$ denote the following subsets of $2^X$ : $$ \begin{aligned} H_ x &= \{ \{j,j+1\} : j =0 \ldots 6 \} = \{ \{j,j-1\} : j =0 \ldots 6 \} \\ V_ x &= \{ \{ j,j+x \} : j =0 \ldots 6 \} = \{ \{ j,j-x \} : j =0 \ldots 6 \} \\ D_ x &= \{ \{ j, j + x+1 \} : j = 0 \ldots 6 \} = \{ \{ j, j - x-1 \} : j = 0 \ldots 6 \} \\ \Phi_ x & = H_ x \cup V_ x \cup D_ x \end{aligned} $$ where all the integers $j$ are meant $\mod 7$.

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Categorical products and coproducts

Universal property of coproducts Let $X_1, X_2$ be objects of the category $\vC$. The coproduct of $X_1$ and $X_2$ (if it exists) is a space $X_1 \coprod X_2$ and two morphisms $j_1 \from X_1 \to X_1 \coprod X_2$, $j_2 \from X_2 \to X_1 \coprod X_2$ such that the following diagram commutes (the first is with MathJax AMScd, the second is an embedded SVG produced with xypic): $\require{AMScd}$ \begin{CD} X_1 @>{j_1}>> X_1 \coprod X_2 @<{j_2}<< X_2 \\ @| @VVhV @| \\ X_1 @>{h_1}>> Z @<{h_2}<< X_2 \\ \end{CD} Given a set $S$ of indices, the coproduct of a family $X _ {s}$, $s\in S$, of objects is (if it exists) the space $\coprod _ {s \in S} X _ s $, together with morphisms $j _ \alpha \from X _ \alpha \to \coprod _ {s \in S} X_ s$, such that for each $Z$ and each family of morphisms $h_ \alpha \from X_ \alpha \to Z$ there exists a unique $h\from \coprod_ {s \in S} X_ s \to Z$ such that for each $\alpha \in S$ on has $hj_ \alpha = h_ \alpha$, i.e.

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A counter-example for the exponentiation / adjoint of a function

The problem: Let $X$, $Y$, and $Z$ be topological spaces. Let $Y^X$ denote the space of all continuous functions $X\to Y$, with the compact-open topology generated by the elements of the sub-basis $W_{K,U} = \{\varphi \in Y^X : \varphi K \subset U\}$, where $K$ ranges over all compact subsets of $X$ and $U$ ranges over over all open subsets of $Y$. Finite intersections of $W_{K,U}$ are a basis for the C.O.

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