Mather-Thurston's theorem

In the mathematics is set of Mather Thurston a theorem from the geometric topology. It is named after John Mather and William Thurston .

It says that for every manifold there is an isomorphism of cohomology groups ${\displaystyle M}$

${\displaystyle H^{*}({\text{Homeo}}(M);\mathbb {Z} )\cong H^{*}(B{\text{Homeo}};\mathbb {Z} )}$

Has. The left side is the group cohomology of the group of homeomorphisms of and the right side is the cohomology of the classifying space of this homeomorphism group with the compact open topology . The isomorphism is induced by the continuous mapping , with the left side being the homeomorphism group with the discrete topology . ${\displaystyle {\text{Homeo}}(M)}$ ${\displaystyle M}$ ${\displaystyle B{\text{Homeo}}(M)}$ ${\displaystyle {\text{Homeo}}(M)^{\delta }\to {\text{Homeo}}(M)}$

In the case of oriented manifolds, an isomorphism is obtained correspondingly for the group of orientational homeomorphisms . For example, for , the circle, is homotopy-equivalent to , so , and one obtains that is generated by the Euler class . ${\displaystyle {\text{Homeo}}^{+}(M)}$ ${\displaystyle H^{*}({\text{Homeo}}^{+}(M);\mathbb {Z} )\cong H^{*}(B{\text{Homeo}}^{+}(M);\mathbb {Z} )}$ ${\displaystyle M=S^{1}}$ ${\displaystyle {\text{Homeo}}^{+}(S^{1})}$ ${\displaystyle S^{1}=U(1)}$ ${\displaystyle H^{*}(B{\text{Homeo}}^{+}(S^{1});\mathbb {Z} )\cong H^{*}(BU(1);\mathbb {Z} )\cong H^{*}(\mathbb {C} P^{\infty };\mathbb {Z} )\cong \mathbb {Z} \left[e\right]}$ ${\displaystyle H^{*}({\text{Homeo}}^{+}(S^{1});\mathbb {Z} )}$ ${\displaystyle e\in H^{2}({\text{Homeo}}^{+}(S^{1});\mathbb {Z} )}$

Another formulation of Mather-Thurston's theorem states that for and all the mapping is an isomorphism of homology groups (but not a homotopy equivalence ). Here is the group of - diffeomorphisms with compact support , the classifying space of Haefliger structures (with regard to the discrete topology) and its -fold iterated loop space . ${\displaystyle M=\mathbb {R} ^{p}}$ ${\displaystyle r\geq 0}$ ${\displaystyle H_{*}({\text{Diff}}_{K}^{r}(\mathbb {R} ^{p});\mathbb {Z} )\to H_{*}(\Omega ^{p}(B\Gamma _{p}^{r});\mathbb {Z} )}$ ${\displaystyle {\text{Diff}}_{K}^{r}(\mathbb {R} ^{p})}$ ${\displaystyle C^{r}}$ ${\displaystyle B\Gamma _{p}^{r}}$ ${\displaystyle \Omega ^{p}(B\Gamma _{p}^{r})}$ ${\displaystyle p}$

For example, it follows from Mather-Thurston's theorem that there is a contiguous space. The topology of classifying space is an important tool in understanding codimension scrolling on manifolds. ${\displaystyle B\Gamma _{p}^{\infty }}$ ${\displaystyle (p+1)}$ ${\displaystyle B\Gamma _{p}^{\infty }}$ ${\displaystyle p}$

literature

• W. Thurston: Foliations and groups of diffeomorphisms , Bull. Amer. Math. Soc. 80, 304-307
• T. Tsuboi: Homology of diffeomorphism groups, and foliated structures , Sūgaku 36 (4), 320-343
• T. Tsuboi: Classifying spaces for groupoid structures , Contemp. Math. 498, 67-81
• S. Nariman: A local to global argument on low dimensional manifolds , Trans. Amer. Math. Soc. 373 (2), 1307-1342