Singular homology

Singular homology is a homology theory in which invariance and functoriality immediately become obvious, but the basic definition requires working with infinite-dimensional spaces.

Build

Let be $X$ ${\ displaystyle X}$ - any topological space .

Singular simplex dimension $k$ ${\ displaystyle k}$ - it's a couple $(\Delta ^{k},f)$ ${\ displaystyle (\ Delta ^ {k}, f)}$ Where $\Delta ^{k}$ ${\ displaystyle \ Delta ^ {k}}$ Is a standard simplex $\langle a_{0},a_{1},...~a_{k}\rangle$ ${\ displaystyle \ langle a_ {0}, a_ {1}, ... ~ a_ {k} \ rangle}$ , but $f$ ${\ displaystyle f}$ - its continuous mapping to $X$ ${\ displaystyle X}$ ; $f:\Delta ^{k}\to X$ ${\ displaystyle f: \ Delta ^ {k} \ to X}$ .

The group of singular chains is defined as the set of formal linear combinations:

c_{k}=\sum _{i}z_{i}(\Delta ^{k},f_{i})

{\ displaystyle c_ {k} = \ sum _ {i} z_ {i} (\ Delta ^ {k}, f_ {i})}

with integer (usually they are also considered limited) coefficients

z_{i}

{\ displaystyle z_ {i}}

.

For linear display $s_{\pi }:\Delta ^{k}\to \Delta ^{k}$ ${\ displaystyle s _ {\ pi}: \ Delta ^ {k} \ to \ Delta ^ {k}}$ determined by the permutation $\pi$ ${\ displaystyle \ pi}$ points $(a_{0},a_{1},...~a_{k})$ ${\ displaystyle (a_ {0}, a_ {1}, ... ~ a_ {k})}$ believe $(\Delta ^{k},f)=(-1)^{\pi }(\Delta ^{k},f\circ s_{\pi })$ ${\ displaystyle (\ Delta ^ {k}, f) = (- 1) ^ {\ pi} (\ Delta ^ {k}, f \ circ s _ {\ pi})}$ .

Boundary operator $\partial$ ${\ displaystyle \ partial}$ defined on a singular simplex $(\Delta _{k},f)$ ${\ displaystyle (\ Delta _ {k}, f)}$ So:

\partial (\Delta _{k},f)=\sum _{i}(-1)^{i}(\Delta _{k-1},f_{i})

{\ displaystyle \ partial (\ Delta _ {k}, f) = \ sum _ {i} (- 1) ^ {i} (\ Delta _ {k-1}, f_ {i})}

,

Where $\Delta _{k-1}$ ${\ displaystyle \ Delta _ {k-1}}$ standard $(k-1)$ ${\ displaystyle (k-1)}$ -dimensional simplex, and $f_{i}=f\circ \epsilon _{i}$ ${\ displaystyle f_ {i} = f \ circ \ epsilon _ {i}}$ where $\epsilon _{i}$ ${\ displaystyle \ epsilon _ {i}}$ - this is its mapping to $i$ ${\ displaystyle i}$ side of standard simplex $\Delta ^{k}(\langle a_{0},...~{\hat {a_{i}}},...~a_{k}\rangle )$ ${\ displaystyle \ Delta ^ {k} (\ langle a_ {0}, ... ~ {\ hat {a_ {i}}}, ... ~ a_ {k} \ rangle)}$ .

Similarly, simplicial homology proves that $\partial \partial =0$ ${\ displaystyle \ partial \ partial = 0}$ .

As before, the concepts of singular cycles are introduced - such chains $c_{k}$ ${\ displaystyle c_ {k}}$ , what $\partial {c_{k}}=0$ ${\ displaystyle \ partial {c_ {k}} = 0}$ and boundaries - chains $c_{k}=\partial {c_{k+1}}$ ${\ displaystyle c_ {k} = \ partial {c_ {k + 1}}}$ for some $c_{k+1}$ ${\ displaystyle c_ {k + 1}}$ .

Factor group of cycles on the group of boundaries $H_{k}=Z_{k}/B_{k}$ ${\ displaystyle H_ {k} = Z_ {k} / B_ {k}}$ is called a singular homology group .

Example

Let us find, for example, the singular homology of a space from one point $X=*$ {\ displaystyle X = *} .

For each dimension there is only one single mapping. $f^{k}:\Delta ^{k}\to *$ ${\ displaystyle f ^ {k}: \ Delta ^ {k} \ to *}$ .

Simplex Border $\partial _{k}(\Delta ^{k},f^{k})=\sum (-1)^{i}(\Delta ^{k-1},f_{i}^{k-1})$ ${\ displaystyle \ partial _ {k} (\ Delta ^ {k}, f ^ {k}) = \ sum (-1) ^ {i} (\ Delta ^ {k-1}, f_ {i} ^ { k-1})}$ where is everyone $f_{i}^{k-1}$ ${\ displaystyle f_ {i} ^ {k-1}}$ are equal since they display the simplex in one point (we denote $f^{k-1}$ ${\ displaystyle f ^ {k-1}}$ ).

So:

\partial (\Delta ^{k},f^{k})=0

{\ displaystyle \ partial (\ Delta ^ {k}, f ^ {k}) = 0}

, if a

k

{\ displaystyle k}

odd (the total number of members is even, and the signs alternate);

\partial (\Delta ^{k},f^{k})=(\Delta ^{k-1},f^{k-1})

{\ displaystyle \ partial (\ Delta ^ {k}, f ^ {k}) = (\ Delta ^ {k-1}, f ^ {k-1})}

, if a

k\not =0

{\ displaystyle k \ not = 0}

and even;

\partial (\Delta ^{k},f^{k})=0

{\ displaystyle \ partial (\ Delta ^ {k}, f ^ {k}) = 0}

, if a

k=0

{\ displaystyle k = 0}

.

From here we get for zero dimension: $Z_{0}=C_{0}=\mathbb {Z} ;\quad B_{0}=0;\quad H_{0}=\mathbb {Z} .$ ${\ displaystyle Z_ {0} = C_ {0} = \ mathbb {Z}; \ quad B_ {0} = 0; \ quad H_ {0} = \ mathbb {Z}.}$

For an odd dimension $k=2n-1:Z_{k}=C_{k}=\mathbb {Z} ;\quad B_{k}=\mathbb {Z} ;\quad H_{k}=0.$ ${\ displaystyle k = 2n-1: Z_ {k} = C_ {k} = \ mathbb {Z}; \ quad B_ {k} = \ mathbb {Z}; \ quad H_ {k} = 0.}$

For even dimension $k=2n\not =0:Z_{k}=0;\quad B_{k}=0;\quad H_{k}=0.$ ${\ displaystyle k = 2n \ not = 0: Z_ {k} = 0; \ quad B_ {k} = 0; \ quad H_ {k} = 0.}$

That is, the homology group is equal to $\mathbb {Z}$ ${\ displaystyle \ mathbb {Z}}$ for zero dimension and zero for all positive dimensions.

It is possible to prove that singular homology on a set of polyhedra coincides with previously defined simplicial ones.

History

Singular homology was introduced by Lefschetz .

Singular homology

Build

Example

History

More articles: