Homotopy equivalence

A homotopy equivalence is a central concept in the mathematical sub-area of topology : a continuous mapping that has a "continuous inverse mapping down to homotopy".

Two spaces are called homotopy equivalent if there is a homotopy equivalence between them. (One then also says that the two spaces have the same homotopy type .) Homotopy equivalence defines a weaker equivalence relation than homeomorphism . Topology is actually about properties that are invariant under homeomorphisms , but many topological invariants are also invariant under homotopy equivalence.

While one imagines a homeomorphism as stretching, compressing, bending, distorting, twisting (but not cutting), in the case of homotopy equivalences, thickening and squeezing are also permissible.

definition

A continuous mapping between topological spaces and is a homotopy equivalence if there is a continuous mapping so that the links and are each homotopic to the identity mappings of or . The mapping is called the homotopy inverse of , it is i. A. not clearly determined. ${\displaystyle f\colon X\to Y}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle g\colon Y\to X}$ ${\displaystyle g\circ f}$ ${\displaystyle f\circ g}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle g}$ ${\displaystyle f}$

Two topological spaces and are called homotopy equivalent if there is a homotopy equivalence . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$

Special cases

The black subspaces are each deformation retracts.

• Every homeomorphism is a homotopy equivalence.
• A homotopy equivalence between CW-complexes is called simple homotopy equivalence if it is homotopy to a sequence of elementary collapses and expansions.
• A subspace is a deformation retract of if the inclusion is a homotopy equivalence and there is a homotopy inverse with . ${\displaystyle A\subset X}$ ${\displaystyle X}$ ${\displaystyle i\colon A\rightarrow X}$ ${\displaystyle r\colon X\rightarrow A}$ ${\displaystyle r\circ i=id_{A}}$
• A topological space is contractible or contractible if it is homotopy equivalent to the point.

Weak homotopy equivalence

Be and topological spaces, and , and be ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle x\in X}$ ${\displaystyle y\in Y}$

${\displaystyle f\colon X\to Y}$

a continuous mapping with . Then for all n ≥ 0 one has a homomorphism of the homotopy groups ${\displaystyle f(x)=y}$

${\displaystyle f_{n}\colon \pi _{n}(X,x)\to \pi _{n}(Y,y).}$

${\displaystyle f}$is called weak homotopy equivalence if all are isomorphisms. Each homotopy equivalence is in particular a weak homotopy equivalence. ${\displaystyle f_{n}}$

Two topological spaces and are called weakly homotopy equivalent if there is weak homotopy equivalence . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$

A weak homotopy equivalence induces isomorphisms ${\displaystyle f\colon X\to Y}$

${\displaystyle f_{*}\colon H_{*}(X;G)\to H_{*}(Y;G)}$ and ${\displaystyle f^{*}\colon H^{*}(Y;G)\to H^{*}(X;G)}$

the homology and cohomology groups for all coefficient groups . ${\displaystyle G}$

Whitehead's theorem

JHC Whitehead proved the following theorem in 1949:

Any weak homotopy equivalency between related CW complexes is a homotopy equivalency.

However, it is not true that there is always a (weak) homotopy equivalence between spaces with isomorphic homotopy groups. For example are

${\displaystyle \mathbb {R} P^{m}\times S^{n}}$ and ${\displaystyle S^{m}\times \mathbb {R} P^{n}}$

related CW complexes with isomorphic homotopy groups. For example, if is odd and even, is but ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle H_{m+n}(\mathbb {R} P^{m}\times S^{n})=\mathbb {Z} }$and , ${\displaystyle H_{m+n}(S^{m}\times \mathbb {R} P^{n})=0}$

which is why the two spaces cannot be (weakly) homotopy equivalent.

For topological spaces that are not CW complexes, Whitehead's theorem i applies. A. not. The space that can be seen as a union of

${\displaystyle \left\{\left(x,\sin {\frac {1}{x}}\right):x\in (0,1]\right\}\cup \{(0,0)\}}$

with an and connecting circular arc is not a CW complex, all of its homotopy groups are trivial, so the constant mapping to a point is a weak homotopy equivalence. But it is not a homotopy equivalence, the space is not contractible. ${\displaystyle (0,-1)}$ ${\displaystyle (1,\sin(1))}$

There is another theorem about weak homotopy equivalences, known as Whitehead's Theorem:

A continuous mapping between simply connected spaces is a weak homotopy equivalence if and only if it induces an isomorphism of the singular homology groups .

Chain homotopy equivalence

Two chain complexes and are called chain homotopy equivalent if there are chain homomorphisms ${\displaystyle (A_{\bullet },d_{A,\bullet })}$ ${\displaystyle (B_{\bullet },d_{B,\bullet })}$

${\displaystyle f_{\bullet }:(A_{\bullet },d_{A,\bullet })\to (B_{\bullet },d_{B,\bullet }),g_{\bullet }:(B_{\bullet },d_{B,\bullet })\to (A_{\bullet },d_{A,\bullet })}$

there, so that and are chain homotop to the identity mappings. ${\displaystyle f\circ g}$ ${\displaystyle g\circ f}$

A chain homotopy equivalence between two chain complexes induces an isomorphism of the homology groups.

A homotopy equivalence between topological spaces induces a chain homotopy equivalence of their singular chain complexes .

Homology theories

For every homology theory in the sense of Eilenberg-Steenrod , according to the homotopy axiom :

Let there be two continuous mappings that are homotopic . Then the two induced group homomorphisms are identical. ${\displaystyle f,g\colon (X,A)\rightarrow (Y,B)}$ ${\displaystyle f_{*},g_{*}\colon H_{n}(X,A)\rightarrow H_{n}(Y,B)}$

From this it follows in particular that a homotopy equivalence induces an isomorphism for every (generalized) homology theory. (Analogous for cohomology theories.)

From Hurewicz's theorem it follows that even any weak homotopy equivalence induces an isomorphism of the singular homology groups (and singular cohomology groups).

literature

• A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. xii + 544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0
• JHC Whitehead, Combinatorial homotopy. I., Bull. Amer. Math. Soc., 55, 213-245 (1949)
• JHC Whitehead, Combinatorial homotopy. II., Bull. Amer. Math. Soc., 55: 453-496 (1949)

Web links

• Counterexamples in Algebraic Topology

Individual evidence

1. ↑ Hatcher (op.cit.), Proposition 4.21