Reduced homology

The given homologies are an insignificant modification of the homology theory, which makes it possible to formulate some statements of algebraic topology , such as Alexander duality , for example, without exception.

The homology and cohomology given is usually denoted by a wave. At the same time, the difference from the usual homology appears only in the zero dimension; i.e $H_{0}(X)={\tilde {H}}_{0}(X)\oplus \mathbb {Z}$ ${\ displaystyle H_ {0} (X) = {\ tilde {H}} _ {0} (X) \ oplus \ mathbb {Z}}$ ${\ displaystyle H_ {0} (X) = {\ tilde {H}} _ {0} (X) \ oplus \ mathbb {Z}}$ and $H_{n}(X)={\tilde {H}}_{n}(X)$ ${\ displaystyle H_ {n} (X) = {\ tilde {H}} _ {n} (X)}$ ${\ displaystyle H_ {n} (X) = {\ tilde {H}} _ {n} (X)}$ for all positive n .

Chain Complex

In the usual definition of the homology of space, is constructed by a chain complex

\dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\partial _{0}}{\longrightarrow \,}}0

{\ displaystyle \ dotsb {\ overset {\ partial _ {n + 1}} {\ longrightarrow \,}} C_ {n} {\ overset {\ partial _ {n}} {\ longrightarrow \,}} C_ {n -1} {\ overset {\ partial _ {n-1}} {\ longrightarrow \,}} \ dotsb {\ overset {\ partial _ {2}} {\ longrightarrow \,}} C_ {1} {\ overset {\ partial _ {1}} {\ longrightarrow \,} C_ {0} {\ overset {\ partial _ {0}} {\ longrightarrow \,}} 0}

and defined as factors $H_{n}(X)=\ker(\partial _{n})/\mathrm {im} (\partial _{n+1})$ ${\ displaystyle H_ {n} (X) = \ ker (\ partial _ {n}) / \ mathrm {im} (\ partial _ {n + 1})}$

To determine the given homology, one should use the same definition for an augmented chain complex.

\dotsb {\overset {\partial _{n+1}}{\longrightarrow \,}}C_{n}{\overset {\partial _{n}}{\longrightarrow \,}}C_{n-1}{\overset {\partial _{n-1}}{\longrightarrow \,}}\dotsb {\overset {\partial _{2}}{\longrightarrow \,}}C_{1}{\overset {\partial _{1}}{\longrightarrow \,}}C_{0}{\overset {\epsilon }{\longrightarrow \,}}\mathbb {Z} \to 0

{\ displaystyle \ dotsb {\ overset {\ partial _ {n + 1}} {\ longrightarrow \,}} C_ {n} {\ overset {\ partial _ {n}} {\ longrightarrow \,}} C_ {n -1} {\ overset {\ partial _ {n-1}} {\ longrightarrow \,}} \ dotsb {\ overset {\ partial _ {2}} {\ longrightarrow \,}} C_ {1} {\ overset {\ partial _ {1}} {\ longrightarrow \,} C_ {0} {\ overset {\ epsilon} {\ longrightarrow \,}} \ mathbb {Z} \ to 0}

Literature

Vic J. W. Homology Theory. Introduction to algebraic topology. - M .: MTSNMO , 2005
Dold A. Lectures on algebraic topology. - M .: Mir, 1976
Dubrovin B. A., Novikov S. P., Fomenko A. T. Modern Geometry: Methods of Homology Theory. - M .: Science, 1984
Seifert G., Trelfall V. Topology. - Izhevsk: RHD, 2001
Lefschetz S. Algebraic topology. - M .: IL, 1949
Novikov P.S. Topology. - 2 ed. corrected and add. - Izhevsk: Institute for Computer Studies, 2002
Prasolov VV Elements of homology theory. - M .: MTSNMO , 2006
Switzer R. M. Algebraic topology. - homotopy and homology. - M .: Science, 1985
Spanier E. Algebraic topology. - M .: World, 1971
N. Steenrod, S. Eilenberg. Foundations of algebraic topology. - M .: Fizmatgiz, 1958
Fomenko A. T., Fuchs D. B. The course of homotopy topology. - M .: Science, 1989

Reduced homology

Chain Complex

Literature

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