Imaging cone

In the mathematical subfield of topology , the mapping cone is a construction that assigns a third such space to a continuous function between two topological spaces . It is closely related to the concept of the cone over a topological space; just like this, the mapping cone is mainly considered in algebraic topology . More generally, there is the mapping cone of chain mappings between chain complexes in homological algebra .

definition

The imaging cone

Let two topological spaces and be a continuous function between them, let the cone over . ${\displaystyle X,Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle CX=(X\times [0;1])/(X\times \{0\})}$ ${\displaystyle X}$

The mapping cone is now obtained (as indicated in the drawing) by bonding of and in virtue . ${\displaystyle Cf}$ ${\displaystyle CX}$ ${\displaystyle Y}$ ${\displaystyle f}$

More precisely this means:

One identified in the disjunctive association each with each , the result is an implicit equivalence relation . ${\displaystyle CX\sqcup Y}$ ${\displaystyle [(x,1)]\in CX}$ ${\displaystyle f(x)\in Y}$ ${\displaystyle x\in X}$ ${\displaystyle \sim _{f}}$

The mapping cone is then the factor space provided with the quotient topology with respect to the canonical projection . ${\displaystyle (CX\sqcup Y)/\sim _{f}}$ ${\displaystyle CX\sqcup Y\to Cf;z\mapsto [z]_{\sim _{f}}}$

Reduced imaging cone

In the category of dotted topological spaces  - that is, dotted and applies  - one usually considers the reduced imaging cone . This is created by also  identifying the interval in the imaging cone  - more precisely, its image under the projection . ${\displaystyle (X;x_{0});(Y;y_{o})}$ ${\displaystyle f(x_{0})=y_{0}}$ ${\displaystyle C_{*}f}$ ${\displaystyle Cf}$ ${\displaystyle x_{0}\times [0,1]}$ ${\displaystyle X\times [0;1]\to CX}$

Analogously, in the above construction of the imaging cone , the reduced cone can also be assumed. ${\displaystyle C_{*}X}$

properties

• The space is naturally a subspace of , since each of its points is preserved under the projection . ${\displaystyle Y}$ ${\displaystyle Cf}$ ${\displaystyle CX\sqcup Y\to Cf}$
• Is injective and relatively open , so a homeomorphism to its image, so too , and thus to contain. ${\displaystyle f}$ ${\displaystyle CX}$ ${\displaystyle X}$ ${\displaystyle Cf}$
• If one looks at identity , homeomorphism applies . ${\displaystyle id\colon X\rightarrow X;x\mapsto x}$ ${\displaystyle C\,id\cong CX}$

All of the above relationships also apply the reduced mapping cone in the case of punctured spaces and and basispunkterhaltendem , may be required to ensure the reduced cone are gone. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f}$ ${\displaystyle C_{*}X}$

• If the image adhering to the skeleton is in a CW complex , the imaging cone is homeomorphic to the skeleton . ${\displaystyle \textstyle f\colon \coprod _{i\in I}S_{i}^{n}\rightarrow X_{n}}$ ${\displaystyle X}$ ${\displaystyle n}$ ${\displaystyle X_{n}}$ ${\displaystyle Cf}$ ${\displaystyle (n+1)}$ ${\displaystyle X_{n+1}}$

This is one of the main uses of the mapping cone in algebraic topology. The following also applies specifically to the reduced imaging cone:

• If dotted and constant , the following applies , where the reduced suspension of and denotes the wedge product . ${\displaystyle X;Y}$ ${\displaystyle f\equiv y_{0}}$ ${\displaystyle C_{*}f\cong \Sigma X\vee Y}$ ${\displaystyle \Sigma X}$ ${\displaystyle X}$ ${\displaystyle \vee }$
• For a well-dotted space , the reduced imaging cone is homotopy-equivalent to the normal imaging cone.

• A mapping induces an isomorphism for a homology theory if and only if . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle f_{*}\colon H_{*}(X)\rightarrow H_{*}(Y)}$ ${\displaystyle H_{*}}$ ${\displaystyle H_{*}(C_{f},*)=0}$

Role in homotopy theory

If two continuous maps are homotopic , then their mapping cones and are homotopy equivalent. ${\displaystyle f,g\colon X\rightarrow Y}$ ${\displaystyle Cf}$ ${\displaystyle Cg}$

If a closed subspace and the inclusion is a cofiber , then homotopy is equivalent to the quotient space . It can also be shown that inclusion is always a closed cofibre. It is thus obtained that the imaging cone is equivalent to homotopy , where the suspension of denotes here. If one continues in the same way, it follows that the mapping cone of the inclusion of to results in the suspension of , etc. ${\displaystyle A\subseteq X}$ ${\displaystyle i\colon A\hookrightarrow X}$ ${\displaystyle C\,i}$ ${\displaystyle X/A}$ ${\displaystyle j\colon Y\hookrightarrow Cf}$ ${\displaystyle Cj}$ ${\displaystyle Cf/Y\cong SX}$ ${\displaystyle SX}$ ${\displaystyle X}$ ${\displaystyle Cf}$ ${\displaystyle SX}$ ${\displaystyle Y}$

If one also has a continuous in a topological space , then the composition is homotopic to a constant mapping if and only if it can be continued to a mapping . In the event that the result is a little clearer: a mapping is homotopic to a constant mapping if and only if it can be continued into a mapping . To construct the map , one simply uses the homotopy , which is constant on . ${\displaystyle h\colon Y\rightarrow Z}$ ${\displaystyle Z}$ ${\displaystyle h\circ f}$ ${\displaystyle h}$ ${\displaystyle h'\colon Cf\rightarrow Z}$ ${\displaystyle f=id\colon X\rightarrow X}$ ${\displaystyle h\colon X\rightarrow Z}$ ${\displaystyle h'\colon CX\rightarrow Z}$ ${\displaystyle h'}$ ${\displaystyle H\colon X\times [0,1]\rightarrow Z}$ ${\displaystyle X\times \{0\}}$

If one considers dotted spaces and base point preserving maps, it means that the following sequence is exact :

${\displaystyle \dots \rightarrow [\Sigma Y,Z]\rightarrow [\Sigma X,Z]\rightarrow [C_{*}f,Z]\rightarrow [Y,Z]\rightarrow [X,Z]}$

This exact sequence is also called a doll sequence .

Image cone of a chain image

Let two chain complexes with differentials d. H., ${\displaystyle A,B}$ ${\displaystyle d_{A},d_{B};}$

${\displaystyle A=\dots \to A^{n-1}{\xrightarrow {d_{A}^{n-1}}}A^{n}{\xrightarrow {d_{A}^{n}}}A^{n+1}\to \cdots }$

and accordingly for ${\displaystyle B.}$

For a chain mapping one defines the mapping cone or as the chain complex: ${\displaystyle f\colon A\to B,}$ ${\displaystyle \operatorname {Cone} (f)}$ ${\displaystyle C(f)}$

${\displaystyle C(f)=A[1]\oplus B=\dots \to A^{n}\oplus B^{n-1}\to A^{n+1}\oplus B^{n}\to A^{n+2}\oplus B^{n+1}\to \cdots }$

with differential

${\displaystyle d_{C(f)}={\begin{pmatrix}d_{A[1]}&0\\f[1]&d_{B}\end{pmatrix}}}$.

Here the chain complex is designated with and . The differential is explicitly calculated as follows: ${\displaystyle A[1]}$ ${\displaystyle A[1]^{n}=A^{n+1}}$ ${\displaystyle d_{A[1]}^{n}=-d_{A}^{n+1}}$

${\displaystyle {\begin{array}{ccl}d_{C(f)}^{n}(a^{n+1},b^{n})&=&{\begin{pmatrix}d_{A[1]}^{n}&0\\f[1]^{n}&d_{B}^{n}\end{pmatrix}}{\begin{pmatrix}a^{n+1}\\b^{n}\end{pmatrix}}\\&=&{\begin{pmatrix}-d_{A}^{n+1}&0\\f^{n+1}&d_{B}^{n}\end{pmatrix}}{\begin{pmatrix}a^{n+1}\\b^{n}\end{pmatrix}}\\&=&{\begin{pmatrix}-d_{A}^{n+1}(a^{n+1})\\f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})\end{pmatrix}}\\&=&\left(-d_{A}^{n+1}(a^{n+1}),f^{n+1}(a^{n+1})+d_{B}^{n}(b^{n})\right).\end{array}}}$

If there is a continuous mapping between topological spaces and the induced chain mapping between the singular chain complexes , then is ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle f_{*}\colon C_{*}(X)\rightarrow C_{*}(Y)}$

${\displaystyle C_{*}(Cf)=C(f_{*})}$.

See also

• Image torus
• Luffing
• Doll series

literature

• Glen E. Bredon : Topology and Geometry. Revised 3rd printing. Springer, New York a. a. 1997, ISBN 0-387-97926-3 ( Graduate Texts in Mathematics 139).
• Robert M. Switzer: Algebraic Topology - Homology and Homotopy. Reprint of the 1975 edition. Springer, Berlin a. a. 2002, ISBN 3-540-42750-3 ( Classics in Mathematics ).