Blakers–Massey theorem

In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey,[1][2][3] gave vanishing conditions for certain triad homotopy groups of spaces.

Description of the result

This connectivity result may be expressed more precisely, as follows. Suppose X is a topological space which is the pushout of the diagram

${\displaystyle A{\xleftarrow {\ f\ }}C{\xrightarrow {\ g\ }}B}$,

where f is an m-connected map and g is n-connected. Then the map of pairs

${\displaystyle (A,C)\rightarrow (X,B)}$

induces an isomorphism in relative homotopy groups in degrees ${\displaystyle k\leq (m+n-1)}$ and a surjection in the next degree.

However the third paper of Blakers and Massey in this area[4] determines the critical, i.e., first non-zero, triad homotopy group as a tensor product, under a number of assumptions, including some simple connectivity. This condition and some dimension conditions were relaxed in work of Ronald Brown and Jean-Louis Loday.[5] The algebraic result implies the connectivity result, since a tensor product is zero if one of the factors is zero. In the non simply connected case, one has to use the nonabelian tensor product of Brown and Loday.[5]

The triad connectivity result can be expressed in a number of other ways, for example, it says that the pushout square above behaves like a homotopy pullback up to dimension ${\displaystyle m+n}$.

Generalization to higher toposes

The generalization of the connectivity part of the theorem from traditional homotopy theory to any other infinity-topos with an infinity-site of definition was given by Charles Rezk in 2010.[6]

Fully formal proof

In 2013 a fairly short, fully formal proof using homotopy type theory as a mathematical foundation and an Agda variant as a proof assistant was announced by Peter LeFanu Lumsdaine;[7] this became Theorem 8.10.2 of Homotopy Type Theory – Univalent Foundations of Mathematics.[8] This induces an internal proof for any infinity-topos (i.e. without reference to a site of definition); in particular, it gives a new proof of the original result.

References

1. Blakers, Albert L.; Massey, William S. (1949). "The homotopy groups of a triad". Proceedings of the National Academy of Sciences of the United States of America. 35 (6): 322–328. Bibcode:1949PNAS...35..322B. doi:10.1073/pnas.35.6.322. MR 0030757. PMC 1063027. PMID 16588898.
2. Blakers, Albert L.; Massey, William S. (1951), "The homotopy groups of a triad. I", Annals of Mathematics, (2), 53 (1): 161–204, doi:10.2307/1969346, JSTOR 1969346, MR 0038654
3. Hatcher, Allen, Algebraic Topology, Theorem 4.23
4. Blakers, Albert L.; Massey, William S. (1953). "The homotopy groups of a triad. III". Annals of Mathematics. (2). 58 (3): 409–417. doi:10.2307/1969744. JSTOR 1969744. MR 0058971.
5. Brown, Ronald; Loday, Jean-Louis (1987). "Homotopical excision, and Hurewicz theorems, for n-cubes of spaces". Proceedings of the London Mathematical Society. (3). 54 (1): 176–192. doi:10.1112/plms/s3-54.1.176. MR 0872255.
6. Rezk, Charles (2010). "Toposes and homotopy toposes" (PDF). Prop. 8.16.
7. "The Blakers-Massey theorem in homotopy type theory (talk at Conference on Type Theory, Homotopy Theory and Univalent Foundations)". 2013.
8. The Univalent Foundations Program (2013). Homotopy type theory: Univalent foundations of mathematics. Institute for Advanced Study.

External links

• Blakers–Massey theorem at the nLab
• tom Dieck, Tammo (2008). Algebraic Topology. EMS Textbooks in Mathematics. European Mathematical Society. Theorem 6.4.1