# Thom spectrum

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A spectrum of spaces, equivalent to the spectrum associated to a certain structure series (cf. $( B, \phi )$- structure).

Let $( B _ {n} , \phi _ {n} , g _ {n} )$ be a structure series, and let $\xi _ {n}$ be the bundle over $B _ {n}$ induced by the mapping $\phi _ {n} : B _ {n} \rightarrow \mathop{\rm BO} _ {n}$. Let $T _ {n}$ be the Thom space of $\xi _ {n}$. The mapping $g _ {n}$ induces a mapping $S _ {n} : ST _ {n} \rightarrow T _ {n + 1 }$, where $S$ is suspension and $ST \xi _ {n} = T( \xi _ {n} \oplus \theta )$( $\theta$ is the one-dimensional trivial bundle). One obtains a spectrum of spaces $\{ T \xi _ {n} \} = T ( B, \phi , g)$, associated with the structure series $( B _ {n} , \phi _ {n} , g _ {n} )$, and a Thom spectrum is any spectrum that is (homotopy) equivalent to a spectrum of the form $T ( B, \phi , g)$. It represents $( B, \phi )$- cobordism theory. Thus, the series of classical Lie groups $O _ {k}$, $\mathop{\rm SO} _ {k}$, $U _ {k}$, and $\mathop{\rm Sp} _ {k}$ lead to the Thom spectra $\mathop{\rm TBO}$, $\mathop{\rm TBSO}$, $\mathop{\rm TBU}$, and $\mathop{\rm TBSp}$.

Let $\beta _ {n}$ be Artin's braid group on $n$ strings (cf. Braid theory). The homomorphism $\beta _ {n} \rightarrow S _ {n} \subset O _ {n}$, where $S _ {n}$ is the symmetric group, yields a mapping $B \beta _ {n} \rightarrow \mathop{\rm BO} _ {n}$ such that a structure series arises ( $\beta _ {n}$ is canonically imbedded in $\beta _ {n + 1 }$). The corresponding Thom spectrum is equivalent to the Eilenberg–MacLane spectrum $K ( \mathbf Z /2) = \{ K ( \mathbf Z /2, n) \}$, so that $K ( \mathbf Z /2)$ is a Thom spectrum (cf. [1], [2]). Analogously, $K ( \mathbf Z )$ is a Thom spectrum, but using sphere bundles, [3].

#### References

 [1] M. Mahowold, "A new infinite family in " Topology , 16 (1977) pp. 249–256 [2] S. Priddy, " as a Thom spectrum" Proc. Amer. Math. Soc. , 70 : 2 (1978) pp. 207–208 [3] M. Mahowold, "Ring spectra which are Thom complexes" Duke Math. J. , 46 : 3 (1979) pp. 549–559
How to Cite This Entry:
Thom spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Thom_spectrum&oldid=48972
This article was adapted from an original article by Yu.B. Rudyak (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article