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8cedf24d3187e5a673adcb9f7f6e3e742d076f2d | subsection | 698 | 1,121 | Power operations | Finally, the differential of the compositeD(L^m)\ \xrightarrow{}\ \Sigma _m\wr G \ \xrightarrow{}
(\Sigma _m\wr G) / (\Sigma _m\wr H) \ \xrightarrow{} \ (G/H)^mis the identity, so we have indeed defined a slice.
We let\lambda _{\Sigma _m\wr H}^{\Sigma _m\wr G}\ : \ (\Sigma _m\wr G) / (\Sigma _m\wr H)_+\ \longrightarrow... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1e10015f700a1083b774124f2c892eea166c49e9 | subsection | 699 | 1,121 | Power operations | A scaling homotopy thus witnesses that the following
diagram commutes up to (\Sigma _m\wr H)-equivariant based homotopy:@C=15mm{
(G / H )^m_+ [r]^-{(\lambda _H^G)^{(m)}}[d]^\cong _{\gamma _+} &
(S^L )^{(m)} [d]^\cong \\
(\Sigma _m\wr G) / (\Sigma _m\wr H)_+
[r]_-{\lambda _{\Sigma _m\wr H}^{\Sigma _m\wr G} } & S^{L^m} }... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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800e0b35ff6b52c5c1a7c8b71a0accbe8fcf18f5 | subsection | 700 | 1,121 | Power operations | Then the following two diagrams commute:@C=10mm@R=6mm{
R_0^H(S^L) [d]_{P^m} [r]^-{\operatorname{Tr}_H^G} &\pi _0^G(R) [dd]^{P^m}&
\pi _0^H(R) [dd]_{P^m} [r]^-{\operatorname{tr}_H^G} &\pi _0^G(R) [dd]^{P^m} \\
R_0^{\Sigma _m\wr H}( (S^L)^{(m)}) [d]_\cong & \\
R_0^{\Sigma _m\wr H}(S^{L^m}) [r]_-{\operatorname{Tr}_{\Sigma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1f2f5d94aacf32598ec935b9bc4f9bdbc39dac8d | subsection | 701 | 1,121 | Power operations | The identification (S^L)^{(m)}\cong S^{L^m} takes (S^{-\operatorname{Id}_L})^{(m)}
to S^{-\operatorname{Id}_{L^m}}, so the following diagram commutes by naturality of power operations:@C=12mm@R=7mm{
R_0^H(S^L) [r]^-{\varepsilon _L}_-\cong [d]_{P^m} &
R_0^H( S^L) [d]^{P^m} \\
R_0^{\Sigma _m\wr H}( ( S^L)^{(m)}) [d]_\con... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
f0d51e0d3c2f66b7e4612495ada76af2fb3c7ad4 | subsection | 702 | 1,121 | Power operations | The degree zero transfer is obtained from the dimension shifting transfer
by precomposing with the effect of the map S^0\longrightarrow S^L,
the inclusion of the origin into the tangent representation.
If we raise the inclusion of the origin of S^L to the m-th power,
the canonical homeomorphism (S^L)^{(m)}\longrightarr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 691,
"openalex_id": "",
"raw": "J. P. C. Greenlees, J. P. May, Localization and completion theorems for M{\\rm U}-module spectra. Ann. of Math. (2) 146 (1997), 509–544.",
"source_ref_id": "e5b776bc28c1d76efd5f4dda378412379cc... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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05d37fa6eb452f462312ca1c53cd80bc1aad4bb4 | subsection | 703 | 1,121 | Power operations | We will show the relationP^m(x\oplus y)\ = \ \sum _{k=0}^m \, (\psi ^k_+)_*(
(\Sigma _m\wr G)\ltimes _{\Sigma _{k,m-k}\wr G}
(P^k(x)\times P^{m-k}(y)))in the group \pi _0^{\Sigma _m\wr G}(R\wedge \lbrace 1,2\rbrace ^m_+).
Since \pi _0^{\Sigma _m\wr G}(R\wedge -) is additive on wedges, it suffices to show the relation
a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bd71a11708036dffa9a7959671bacd7fd51389c0 | subsection | 704 | 1,121 | Power operations | We obtain(l_k)_*(\operatorname{res}^{\Sigma _m\wr G}_{\Sigma _{k,m-k}\wr G} &(\bar{\psi }^k_*( P^m(x\oplus y)))) \ = \ (l_k\circ \bar{\psi }^k)_*(\operatorname{res}^{\Sigma _m\wr G}_{\Sigma _{k,m-k}\wr G} ( P^m(x\oplus y))) \\
&= \ ( (p^1)^{(k)}\wedge (p^2)^{(m-k)})_*( P^k(x\oplus y)\times P^{m-k}(x\oplus y))\\
&= \ P^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f0808e8ebd59852da3028f944148115c235b60a1 | subsection | 705 | 1,121 | Power operations | We recall from
that an H_\infty -structure is an algebra structure over the monadL{\mathbb {P}}\ : \ {\mathcal {SH}}\ \longrightarrow \ {\mathcal {SH}}on the stable homotopy category that can be obtained
by deriving the `symmetric algebra' monad{\mathbb {P}}\ : \ {\mathcal {S}}p\ \longrightarrow \ {\mathcal {S}}pon th... | {
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"raw": "R. Bruner, J. P. May, J. McClure, M. Steinberger, H_\\infty ring spectra and their applications. Lecture Notes in Mathematics, Vol. 1176. Springer-Verlag, Berlin, 1986. viii+388 pp.",
"source_... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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58da439a81452e2562438f2d408a14f7a110496f | subsection | 706 | 1,121 | Power operations | Then the underlying H_\infty -structure
is given by the composite morphismD_m R \ = \ (E\Sigma _m)_+\wedge _{\Sigma _m} R^{\wedge m} \ \longrightarrow \Sigma _m\backslash R^{\wedge m} \ \xrightarrow{} \ Rwhere the first morphism collapses E\Sigma _m to a point and the second
map is induced by the iterated multiplicatio... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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15f02d1c8c84b37fb389f26bad30b03376b31ef5 | subsection | 707 | 1,121 | Power operations | Then the following diagram commutes:@C=12mm@R=8mm{
R^0(A) [d]_{P^m} [r]^-{D_m} &
[D_m( \Sigma ^\infty _+ A), D_m R] [d]^{(\mu _m)_*}\\
R^0_{\Sigma _m}(A^m) [r]^-U [d]_{R^0_{\Sigma _m}(\Delta )} &
R^0(E\Sigma _m\times _{\Sigma _m} A) [d]^{R^0(E\Sigma _m\times _{\Sigma _m}\Delta )}\\
R^0_{\Sigma _m}(A) [r]_-U & R^0(B\Sig... | {
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"source_... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ca8ac4083c2a583aa45ba6d34b97454698879074 | subsection | 708 | 1,121 | Power operations | The composite{\mathbb {G}}\ = \ \operatorname{Ho}(U)\circ L_{\operatorname{gl}}{\mathbb {P}}\ : \ {\mathcal {GH}}\ \longrightarrow \ {\mathcal {GH}}is then canonically a monad on the global stable homotopy category,
whose algebras we call G_\infty -ring spectra.The underlying non-equivariant homotopy type of a G_\infty... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2b8fd8dea586b33d36e28f1d19b18ee38cd485d7 | subsection | 709 | 1,121 | Power operations | Moreover, the power operations in {\underline{\pi }}_0(R) correspond to the power operations
in {\underline{\pi }}_0(G L_1(R)).In the non-equivariant context,
G L_1(R) is an infinite loop space, i.e., weakly equivalent
to the 0-th space of an \Omega -spectrum of units.
This fact has a global generalization as follows.
... | {
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{
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"raw": "M. A. Mandell, J. P. May, Equivariant orthogonal spectra and S-modules. Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108 pp.",
"source_ref_id": "1fe5cf0f78b8a5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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194c1262a4e1a0b9fdc8739765139308234addb2 | subsection | 710 | 1,121 | Power operations | We let\operatorname{Pic}(R)(G)\ = \ \operatorname{Pic}(\operatorname{Ho}(R_G\text{-mod}))be the resulting Picard group, i.e., the set of isomorphism
classes, in the homotopy category of R_G-modules,
of objects that are invertible under the derived smash product.
For a continuous group homomorphism \alpha :K\longrightar... | {
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{
"arxiv_id": "",
"doi": "10.4007/annals.2016.184.1.1",
"end": 1506,
"openalex_id": "https://openalex.org/W1846653373",
"raw": "M. Hill, M. Hopkins, D. Ravenel, On the nonexistence of elements of Kervaire invariant one. Ann. of Math. (2) 184 (2016), 1–262.",
"... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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407bb6168e805dde672ff3433356e1e1f436bb7c | subsection | 711 | 1,121 | Power operations | Despite the strong evidence for its existence,
I cannot presently construct \operatorname{pic}(R) as an ultra-commutative monoid
in our formalism.Example 1.17 (Free global power functors) global power functor!free
For a compact Lie group K we construct a
free global power functor C_K generated by K.
The underlying glob... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3b26d77543e71e07cb846ab6c823606498153b62 | subsection | 712 | 1,121 | Power operations | The multiplication \mu : C_K\Box C_K\longrightarrow C_K
that makes this into a global Green functor restricted to the (m,n)-summand
is the morphism{\mathbf {A}}(\Sigma _m\wr K,-)\Box {\mathbf {A}}(\Sigma _n\wr K,-) \ \longrightarrow \ C_Kthat corresponds, via the universal property of the box product,
to the bimorphism... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7e94e05ebdda9d49d173d6881a232c4ab94020ab | subsection | 713 | 1,121 | Power operations | The unit is the inclusion {\mathbf {A}}(e,-)\longrightarrow C_K of the summand indexed by m=0.The global Green functor C_K can be made into a
global power functor in a unique way such that the relationP^m(1_K) \ = \ 1_{\Sigma _m\wr K}holds in the m-th summand of C_K(\Sigma _m\wr K),
where 1_K\in {\mathbf {A}}(K,K) and ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c00a16839f823e516939d09fc0ad9170a5bedce5 | subsection | 714 | 1,121 | Power operations | The stabilization map \sigma :{\underline{\pi }}_0(B_{\operatorname{gl}}K)\longrightarrow {\underline{\pi }}_0(\Sigma ^\infty _+ B_{\operatorname{gl}}K)
commutes with power operations, so this shows thatP^m(e_K) \ = \ P^m(\sigma ^K(u_K))\ = \ \sigma ^{\Sigma _m\wr K}([m](u_K)) \ = \ \sigma ^{\Sigma _m\wr K}(u_{\Sigma _... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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70f1fe7c58328a1383380b15ee4d3866d074dde2 | subsection | 715 | 1,121 | Comonadic description of global power functors | In this section we show that the category of global power functors
is both monadic and comonadic over the category of global Green functors.
We introduce the functor of exponential sequences
and make it into a comonad on the category of global Green functors.
For a global Green functor R and a compact Lie group G,
Cons... | {
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{
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"raw": "J. Singer, Äquivariante \\lambda -Ringe und kommutative Multiplikationen auf Moore-Spektren. Dissertation, Universität Bonn, 2007.",
"source_ref_id": "dc339c2bff7582a7586049bf7616bda50970a4ec... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8872595506e98a40379ce27e6c47c2ece18802f4 | subsection | 716 | 1,121 | Comonadic description of global power functors | We define a multiplication on the set \exp (R;G) by
coordinatewise multiplication in the rings R(\Sigma _m\wr G), i.e.,(x \cdot y )_m \ = \ x_m\cdot y_m\ .We introduce another binary operation \oplus on \exp (R;G) by( x \oplus y )_m \ = \ \sum _{k=0}^m \ \operatorname{tr}_{k,m-k} (x_k\times y_{m-k}) \ ,where x=(x_m), y... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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48b6d1f89c00c164e5023d97b436e956dd574c80 | subsection | 717 | 1,121 | Comonadic description of global power functors | We parametrize these double cosets by pairs (a,b) of natural numbers satisfying0\le a \le i \ , \quad 0 \le b\le m-i \text{\qquad and\qquad } a + b = k \ .For each such pair we define a permutation \chi (a,b)\in \Sigma _m by\chi (a,b)(j) \ = \ {\left\lbrace \begin{array}{ll}
\ j & \text{ for $1\le j\le a$, }\\
\ j-a+i ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
012a69d3db5396d34feaf0a25ae98f967c9eac2b | subsection | 718 | 1,121 | Comonadic description of global power functors | Now we consider exponential sequences x,y\in \exp (R;G) and calculate\Phi _{i,m-i}^*( (x\oplus y)_m) \ &= \ \sum _{k=0}^m \, \Phi _{i,m-i}^*(\operatorname{tr}_{k,m-k} (x_k\times y_{m-k}))
\\
_(\ref {eq:double coset double symmetric}) \ &= \ \sum _{a,b}
\operatorname{tr}_{\Sigma _i\times \Sigma _{i-a}\times \Sigma _b\ti... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e267f29c4b4f55bec61d609ab1ae958282d93e2c | subsection | 719 | 1,121 | Comonadic description of global power functors | By unraveling the definitions, this becomes the associativity
of the operation \oplus .Also, the following square of group monomorphisms commutes:@C=15mm{
(\Sigma _k\wr G)\times (\Sigma _l\wr G)
[r]^-{\Phi _{k,l}}
[d]_{\text{twist}} &
\Sigma _{k+l}\wr G [d]^{c_\chi } \\
(\Sigma _l\wr G)\times (\Sigma _k\wr G)
[r]_-{\Ph... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
086facc5a1a6b422d2bcdf11fffd8572b07a3e80 | subsection | 720 | 1,121 | Comonadic description of global power functors | Now we assume that m\ge 2; then- \Phi _{i,m-i}^*( y_m) \ &= \ {\sum }_{k=1}^m \, \Phi _{i,m-i}^*(\operatorname{tr}_{k,m-k} (x_k\times y_{m-k})) \\
_(\ref {eq:double coset double symmetric}) \ &= \ \sum _{ a+b\ge 1}
\operatorname{tr}_{\Sigma _a\times \Sigma _{i-a}\times \Sigma _b\times \Sigma _{m-i-b}}^{\Sigma _i\times ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
e251cef757aa7255e6f0f01e8679503d202804fd | subsection | 721 | 1,121 | Comonadic description of global power functors | Then( ( x\oplus y)\cdot z)_m \ &= \ \left({\sum }_{k=0}^m\, \operatorname{tr}_{k,m-k}(x_k\times y_{m-k}) \right)\cdot z_m \\
&= \ {\sum }_{k=0}^m\, \operatorname{tr}_{k,m-k} ( (x_k\times y_{m-k}) \cdot \Phi _{k,m-k}^*(z_m) ) \\
&= \ {\sum }_{k=0}^m\, \operatorname{tr}_{k,m-k} ( (x_k\times y_{m-k}) \cdot (z_k\times z_{m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
16c81bfc6f9f2c97a2282824fd524370c6dfc250 | subsection | 722 | 1,121 | Comonadic description of global power functors | For the course of this proof we call a pair (G,x)
consisting of a compact Lie group G and an element x\in R(G) good
if the relation f(P^m(x))= P^m(f(x)) holds for all m\ge 0.
Then the naturality relation P^m\circ \alpha ^* = (\Sigma _m\wr \alpha )^*\circ P^m
for a continuous homomorphism \alpha :K\longrightarrow G impl... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0... |
161341cbbbcbd12e9ff420005fca624a6a33cd09 | subsection | 723 | 1,121 | Comonadic description of global power functors | Since (G,1) is good, so is
(G,1-1)=(G,0). This completes the proof that the good elements
form a global Green subfunctor of R.The closure property under power operations is then a consequence of
the transitivity relation and closure under restriction: if (G,x) is good, thenf(P^k(P^m(x)))\ &= \ f(\Psi _{k,m}^*(P^{k m}(x... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.... |
f3991453d75f4e62508ef22af37a193f642aeffe | subsection | 724 | 1,121 | Comonadic description of global power functors | For every global power functor R the map
\mathcal {G}l\mathcal {P}ow(C_K,R)\ \longrightarrow \ R(K) \ , \quad f \ \longmapsto \ f(1_K)
is bijective.(i)
The underlying global functor of C_K is the direct sum of the
represented global functors {\mathbf {A}}(\Sigma _m\wr K,-), for m\ge 0.
So the enriched Yoneda lemma (s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3268efd290a8fa19a1e82b903aa33c27c01339f7 | subsection | 725 | 1,121 | Comonadic description of global power functors | The relation P^m(1_K)=1_{\Sigma _m\wr K} holds in the global power functor C_K;
so since f also commutes with power operations, it is already determined
by the element f(1_K). This shows that the map in question
is injective.Now we show that evaluation at 1_K is also surjective. We let y\in R(K)
be any element. Then th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a0967ccf9b200332b303afdac6c1756feb95968d | subsection | 726 | 1,121 | Comonadic description of global power functors | As an auxiliary tool we introduce a functor\Gamma \ : \ {\mathbf {A}}^{\operatorname{op}}\ \longrightarrow \ \mathcal {G}l\mathcal {P}ow\ .The functor is given by \Gamma (K)=C_K on objects;
on morphisms, the freeness property of Proposition REF (ii)
allows us to define\Gamma \ : \ {\mathbf {A}}(K,G)\ \longrightarrow \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
96879d94b67dd8fc617189bc1083358757d8d173 | subsection | 727 | 1,121 | Comonadic description of global power functors | Indeed, for \tau \in {\mathbf {A}}(K,G), \psi \in {\mathbf {A}}(L,K)
and a morphism of global Green functors f:C_G\longrightarrow R, we have\exp (R;\tau \circ \psi )&(\epsilon _G(f))\ = \ \epsilon _L(f\circ \Gamma (\tau \circ \psi )) \ = \ \epsilon _L(f\circ \Gamma (\psi )\circ \Gamma (\tau )) \\
&= \ \exp (R;\tau )(\e... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9c46cc680c146400b002619f6c4ca6ba64092d5f | subsection | 728 | 1,121 | Comonadic description of global power functors | It is not completely obvious, though, that this construction is additive
in both variables, but we will show that in the next proposition.The construction of exponential sequences is in fact a functor in two variables:
for every operation \tau \in {\mathbf {A}}(G,K)
and all morphisms of global Green functors f:C_G\long... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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-0.0018910372164100409... |
e44043812f0b5e0840b11bc68cc00fa919379e6c | subsection | 729 | 1,121 | Comonadic description of global power functors | For all compact Lie groups G and K, the map
\exp (R;-)\ : \ {\mathbf {A}}(G,K)\times \exp (R;G)\ \longrightarrow \ \exp (R;K)
is biadditive, i.e., \exp (R;-) becomes a global functor in the Lie group variable.
As the Lie group G varies, the ring structures and the functoriality
in the global Burnside category make \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0329... |
64faa3f9240ad69dea3b855976bddb393501a91f | subsection | 730 | 1,121 | Comonadic description of global power functors | If \alpha is surjective, then so is \Sigma _m\wr \alpha , and(\Sigma _m\wr \alpha )^{-1}( (\Sigma _k\wr G)\times (\Sigma _{m-k}\wr G)) \ = \ (\Sigma _k\wr K)\times (\Sigma _{m-k}\wr K)\ .So for epimorphisms, the relation (REF )
is a special case of compatibility of transfer with inflation.
If H is a closed subgroup of ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
e32b7e7cc5f4dc3483de8164c4ae344b316b1021 | subsection | 731 | 1,121 | Comonadic description of global power functors | We can then conclude that \exp (R;\alpha ^*) is additive:(\exp (R;\alpha ^*)(x\oplus y) )_m \ &= \ (\Sigma _m\wr \alpha )^*( (x\oplus y)_m) \ = \ \sum _{k=0}^m \ (\Sigma _m\wr \alpha )^*(\operatorname{tr}_{k,m-k} (x_k\times y_{m-k}) )\\
_(\ref {eq:alpha_after_tr})\ &= \ \sum _{k=0}^m \ \operatorname{tr}_{k,m-k} \left( ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.035431284457445145,
0.015381022356450558,
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0... |
5496fc2c1f128b59b7832cbece944608827504e9 | subsection | 732 | 1,121 | Comonadic description of global power functors | Then\left( \exp (R;\operatorname{tr}_H^G)(\epsilon _H(f))\right)_m \ &= \ \left( \epsilon _G(f\circ \Gamma (\operatorname{tr}_H^G))\right)_m\ = \ (f\circ \Gamma (\operatorname{tr}_H^G))(1_{\Sigma _m\wr G}) \\
&= \ f ( \Gamma (\operatorname{tr}_H^G)(P^m(1_G))) \ = \ f ( P^m( \Gamma (\operatorname{tr}_H^G)(1_G))) \\
&= \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
a44d138236617ceb661c62eec5ffa93629477a79 | subsection | 733 | 1,121 | Comonadic description of global power functors | Thus( \exp (R;\operatorname{tr}_H^G)(x \oplus y) )_m \ &= \ \sum _{k=0}^m \ \operatorname{tr}_{\Sigma _m\wr H}^{\Sigma _m\wr G}(\operatorname{tr}_{k,m-k} (x_k\times y_{m-k}) )\\
&= \ \sum _{k=0}^m \ \operatorname{tr}_{k,m-k} \left( \operatorname{tr}_{(\Sigma _k\wr H)\times (\Sigma _{m-k}\wr H)}^{(\Sigma _k\wr G)\times ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
796d5607496e9724b95ef48e543a060541982849 | subsection | 734 | 1,121 | Comonadic description of global power functors | Then the morphism of global power functors \Gamma (\tau +\tau ^{\prime }):C_K \longrightarrow C_G
satisfies\Gamma (\tau +\tau ^{\prime })(1_{\Sigma _m\wr K}) \ &= \ \Gamma (\tau +\tau ^{\prime })(P^m(1_K)) \ = \ P^m( \Gamma (\tau +\tau ^{\prime })(1_K)) \\
&= \ P^m( \tau +\tau ^{\prime }) \ =\ \sum _{k=0}^m \operatorna... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
5fae263053219215c39aaa9e761ba77a126d2c21 | subsection | 735 | 1,121 | Comonadic description of global power functors | Since we have already established additivity in \tau , we can assume
that \tau =\operatorname{tr}_L^K\circ \alpha ^* is one of the generating operations.
Since\exp (R;\operatorname{tr}_L^K\circ \alpha ^*)\ = \ \exp (R;\operatorname{tr}_L^K)\circ \exp (R;\alpha ^*) \ ,this in turn follows from additivity for restriction... | {
"cite_spans": [
{
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"end": 1609,
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"raw": "J. Singer, Äquivariante \\lambda -Ringe und kommutative Multiplikationen auf Moore-Spektren. Dissertation, Universität Bonn, 2007.",
"source_ref_id": "dc339c2bff7582a7586049bf7616bda50970a4ec... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.05288751795887947,
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0.017090022563934326,
-0.031174033880233765,
0.0... |
b3e9511a607f7c96be577e7b8842f9423b268d86 | subsection | 736 | 1,121 | Comonadic description of global power functors | The natural transformations
\eta \ : \ \exp \ \longrightarrow \ \operatorname{Id}\text{\qquad and\qquad }
\kappa \ :\ \exp \ \longrightarrow \ \exp \circ \exp
make the functor \exp into a comonad on the category of global Green functors.(i) Because the square of group homomorphisms@C=13mm@R=7mm{
(\Sigma _j\wr \Sigma ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.036619290709495544,
0.030668655410408974,
-0.04226476326584816,
-0.0271440502256155,
-0.028639337047934532,
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0.020476287230849266,
0.02758653275668621,
-0.002233395352959633,
0.031767234206199646,
-0.042020637542009354,
... |
778e60da0ae69bb8154ca681c6be6c93f65a52bc | subsection | 737 | 1,121 | Comonadic description of global power functors | However, the square (REF )
does commute up to conjugation by an element of \Sigma _{k m}\wr G.
Since inner automorphisms are invisible through the eyes of a Rep-functor,
we conclude that the relation\left( \Phi _{i,m-i}^*( \kappa (x)_m ) \right)_k \ &= \ (\Sigma _k\wr \Phi _{i,m-i})^*( \Psi _{k,m}^*(x_{k m}) ) \\
&= \ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.027325093746185303,
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0.01413551066070795,
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0.... |
6aac52e35b0b1d77a5959d79352ad4ea1256b38d | subsection | 738 | 1,121 | Comonadic description of global power functors | To a large extent, the group G
acts like a dummy, which is why we omit it from the notation for
\Sigma _k\wr ^{\prime }\Sigma _m and \Sigma _i\times ^{\prime }\Sigma _{k m-i}.
We specify a bijection between the set of double cosets\Sigma _k\wr ^{\prime }\Sigma _m \backslash \Sigma _{k m}\wr G / \Sigma _i\times ^{\prime... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.006185532081872225,
-0.02097204327583313,
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-0.03449550271034241,... |
fcf252a6462b8e12c8c96fdab13840ad6513cec0 | subsection | 739 | 1,121 | Comonadic description of global power functors | Moreover,(\Sigma _k\wr ^{\prime }\Sigma _m)^{\sigma (a_0,\dots ,a_m)}&\cap (\Sigma _i\times ^{\prime }\Sigma _{k m-i})
\ = \ \left( {{\prod }^{\prime }}_{j=0}^m \Sigma _{a_j}\wr ^{\prime }\Sigma _j \right)
\times ^{\prime }
\left( {{\prod }^{\prime }}_{j=0}^m\Sigma _{a_j}\wr ^{\prime }\Sigma _{m-j}\right)and(\Sigma _k\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0223840344697237,
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0.014167401008307934,
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0.0... |
1e18881e5173175ff90d8f766e4ae3f26caa2ac5 | subsection | 740 | 1,121 | Comonadic description of global power functors | If x_i and y_{k m-i} are the respective components of two exponential sequences
x, y\in \exp (R;G), then\sigma (a_0,\dots ,a_m)_\star &\left(\operatorname{res}^{(\Sigma _i\wr G)\times (\Sigma _{k m-i}\wr G)}_{ \left( {\prod ^{\prime }} \Sigma _{a_j}\wr ^{\prime }\Sigma _j \right)
\times ^{\prime }
\left( \prod ^{\prime... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03284107521176338,
0.013620194047689438,
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-0.029544757679104805,
... |
7bc68d33812255ab953aef9b29a5cfbb2be8e670 | subsection | 741 | 1,121 | Comonadic description of global power functors | Expanding this further we arrive at the expression( ( \kappa (x) &\oplus \kappa (y) )_m)_k \ = \ \sum _{a_0+\dots +a_m=k} \operatorname{tr}_{a_0,\dots ,a_m}\left(
{\prod }_{j=0}^m \operatorname{tr}_{j,m-j}( \kappa (x)_j \times \kappa (y)_{m-j} )_{a_j} \right) \\
&= \ \sum _{a_0+\dots +a_m=k} \operatorname{tr}_{a_0,\dot... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
0.024996886029839516,
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-0.008088125847280025,
-0.012902849353849888,... |
4c7fbd0660263018aa307c949123f2900ce3ab49 | subsection | 742 | 1,121 | Comonadic description of global power functors | Here \operatorname{tr}_{a_0,\dots ,a_m} is shorthand notation for the transfer along the monomorphism\Phi _{a_0,\dots ,a_m}\ : \ (\Sigma _{a_0}\wr \Sigma _m\wr G)\times \dots \times (\Sigma _{a_j}\wr \Sigma _m\wr G)\times \dots \times (\Sigma _{a_m}\wr \Sigma _m\wr G)
\ \longrightarrow \ \Sigma _k\wr \Sigma _m\wr G \ ,... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06452910602092743,
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0.02875421941280365,
... |
dd56f0ffab3d747516aefb1e75202da3f31d3152 | subsection | 743 | 1,121 | Comonadic description of global power functors | Indeed, for every closed subgroup H of G, the group
\Sigma _{k m}\wr G consists of a single
double coset for the left (\Sigma _k\wr \Sigma _m\wr G)-action
and right (\Sigma _{k m}\wr H)-action, and(\Sigma _k\wr \Sigma _m\wr G) \ \cap \ (\Sigma _{k m}\wr H) \ = \ \Sigma _k\wr \Sigma _m\wr H \ .So for every x\in \exp (R;... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.038183629512786865,
-0.024219591170549393,
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0.0022205107379704714,
0.012567632831633091,
... |
08eb5ef53b56b5051e944e4593ec7256e805af8b | subsection | 744 | 1,121 | Comonadic description of global power functors | The coassociativity relation\exp (\kappa _M)\circ \kappa _M \ = \ \kappa _{\exp (M)}\circ \kappa _Multimately boils down to the observation that the following square of monomorphisms
commutes:\begin{gathered}
@C=15mm@R=7mm{
\Sigma _k\wr \Sigma _m\wr \Sigma _n\wr G[r]^-{\Sigma _k\wr \Psi _{m,n}}
[d]_{\Psi _{k,m}} &
\Sig... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03505629673600197,
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0.009603106416761875,
-0.025521839037537575,
-0.013172808103263378,
0.0005882760742679238,... |
fb2d1128022a26153ee663d76949e75709373f73 | subsection | 745 | 1,121 | Comonadic description of global power functors | In general, the forgetful functor from any category of coalgebras
to the underlying category has a right adjoint `cofree' functor. In particular,
colimits in a category of coalgebras are created in the underlying category.
In our situation that means:Corollary 2.12Colimits in the category of global power functors
exist... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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-0.015514295548200607,
... |
9bd5d8888f0130435ebdfe793adfe340c485ccd2 | subsection | 746 | 1,121 | Comonadic description of global power functors | If E and F are ultra-commutative ring spectra, then the ring spectra
morphisms E\longrightarrow E\wedge F and F\longrightarrow E\wedge F induce morphisms
of global power functors {\underline{\pi }}_0(E) \longrightarrow {\underline{\pi }}_0(E\wedge F)
and {\underline{\pi }}_0 (F) \longrightarrow {\underline{\pi }}_0(E\w... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05168553441762924,
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0.007806987036019564,
0.02948882430791855,
0.025... |
01c8fac93424409a45d9effcd992847c9232dd3d | subsection | 747 | 1,121 | Comonadic description of global power functors | Because \operatorname{res}^G_H(p_G^*(S))=p_H^*(S),
the localization R(H)[p_H^*(S)^{-1}]=R[S^{-1}](H)
is also a localization of R(H) at p_G^*(S) as an R(G)-module.
The reciprocity formula means that the transfer map \operatorname{tr}_H^G:R(H)\longrightarrow R(G)
is a homomorphism of R(G)-modules. The composite R(G)-line... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.06477463245391846,
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0.03... |
701cd015aff4dc25d1510263160b72d10b1da8f6 | subsection | 748 | 1,121 | Comonadic description of global power functors | Let f:R\longrightarrow R^{\prime } be a morphism of global power functors
such that all elements of the set f(e)(S) are invertible in the ring R^{\prime }(e).
Then there is a unique homomorphism of global power functors \bar{f}:R[S^{-1}]\longrightarrow R^{\prime }
such that \bar{f} i=f.(i) We use the comonadic descript... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04135500267148018,
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-0.... |
608aa92f48093460fbacc4e5ab7357bb6f104d3c | subsection | 749 | 1,121 | Comonadic description of global power functors | Similarly,\exp (P[S^{-1}])\circ P[S^{-1}]\circ i \ &= \ \exp (P[S^{-1}])\circ \exp (i)\circ P\ = \ \exp (P[S^{-1}]\circ i)\circ P\\
&= \ \exp (\exp (i)\circ P)\circ P\ = \ \exp (\exp (i))\circ \exp (P)\circ P\\
&= \ \exp (\exp (i))\circ \kappa _R\circ P\ = \ \kappa _{R[S^{-1}]}\circ \exp (i)\circ P\\
&= \ \kappa _{R[S^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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eacd7b8691f9b930751a8d8c96db888a7802e589 | subsection | 750 | 1,121 | Comonadic description of global power functors | This is an equality between morphisms of global Green functors, so the
uniqueness of Proposition REF
shows that \exp (\bar{f})\circ P[S^{-1}]=P^{\prime }\circ \bar{f}, i.e.,
\bar{f} is a morphism of \exp -coalgebras.Example 2.17 (Localization at a subring of {\mathbb {Q}})
We use Theorem REF
to show that global powe... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
3a8fc267e0b271c0080ab04ae5fda970f68f215d | subsection | 751 | 1,121 | Comonadic description of global power functors | ThusP^m(n)\ &= \ P^m(\underbrace{1+\dots +1}_n)\ = \ \sum _{i_1+\dots +i_n=m} \operatorname{tr}_{\Sigma _{i_1}\times \dots \times \Sigma _{i_n}}^{\Sigma _m}( P^{i_1}(1)\times \dots \times P^{i_n}(1)) \\
&= \ n\cdot P^m(1)\ + \sum _{i_1+\dots +i_n=m,\ i_j<m} \operatorname{tr}_{\Sigma _{i_1}\times \dots \times \Sigma _{i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.035299964249134064,
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... |
e32adcbfe3eb6ee9a233eb5a4f2e7fc0ed790e4c | subsection | 752 | 1,121 | Comonadic description of global power functors | In fact, both categories are examples of algebras over
multisorted algebraic theories (also called colored theories).
The `sorts' (or `colors') are the compact Lie groups and the
content of this claim is that the structure of
global Green functors respectively global power functors
can be specified by giving the values... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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-0.020727066323161125... |
f54c8420f23740a325fe6039301eddad4235377c | subsection | 753 | 1,121 | Comonadic description of global power functors | Then we form a coequalizer,
in the category of global power functors{ L^{\prime } @<.4ex>[r] @<-.4ex>[r] & L [r] & F }where the two morphisms from L^{\prime } to L restrict, on each box factor,
to the morphism that represents the respective relation.
The resulting global power functor then represents
the functor \mathc... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 734,
"openalex_id": "",
"raw": "S. Mac Lane, Categories for the working mathematician. Second edition. Graduate Texts in Mathematics, 5. Springer-Verlag, New York, 1998. xii+314 pp.",
"source_ref_id": "05a93e42555651e93a155c... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.... |
b883d25fdf9aed3d0f6777d27f38a24cc8322d20 | subsection | 754 | 1,121 | Global Thom and | The final chapter of this book is devoted to an in-depth study
of interesting examples of ultra-commutative ring spectra, in particular
global Thom spectra and ultra-commutative global models for various flavors
of equivariant K-theory spectra.
In Section we discuss global refinements of
the classical Thom spectrum M O... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0040-9383(69)90005-6",
"end": 2111,
"openalex_id": "https://openalex.org/W2146747536",
"raw": "A. G. Wasserman, Equivariant differential topology. Topology 8 (1969), 127–150.",
"source_ref_id": "73f520bac34f501f9bef5030bd951b8... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.011046072468161583,
0.0... |
568874a11fa5f6c96ceaa71a2a8e0ce25512030a | subsection | 755 | 1,121 | Global Thom and | We also introduce the Bott class in the group \pi _2^e({\mathbf {ku}})
(see Construction REF ) and the more general
equivariant Bott classes associated to G-\operatorname{Spin}^c-representations
(see Construction REF ).The final Section reviews {\mathbf {KU}}, the
periodic global K-theory spectrum, see Construction REF... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.jpaa.2003.07.008",
"end": 1243,
"openalex_id": "https://openalex.org/W2021493603",
"raw": "J. P. C. Greenlees, Equivariant connective K-theory for compact Lie groups. J. Pure Appl. Algebra 187 (2004), 129–152.",
"source_ref_... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f284b1db43a7400dba0c7e8c209c408214337b67 | subsection | 756 | 1,121 | Global Thom spectra | global Thom spectrum|(In this section we discuss two different global forms
of the Thom spectrum M O that represents unoriented bordism,
namely the ultra-commutative Thom ring spectrum {\mathbf {MO}},
and a variation {\mathbf {mO}} that is only E_\infty -commutative.
Both Thom spectra are the homogeneous degree 0 summa... | {
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"Stefan Schwede"
] | [
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2f4ac30a791fde4320418fe0b49afb31b355b38f | subsection | 757 | 1,121 | Global Thom spectra | The transformations labeled \Theta ^G are the equivariant Thom-Pontryagin construction
and its `stabilization'. The upper Thom-Pontryagin
map \Theta ^G:\mathcal {N}_*^G\longrightarrow \pi _*^G({\mathbf {mO}}) is an isomorphism whenever G is isomorphic
to a product of a finite group and a torus;
this result seems to hav... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01187353",
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"raw": "T. Bröcker, E. C. Hook, Stable equivariant bordism. Math. Z. 129 (1972), 269–277.",
"source_ref_id": "808853a7d970e057cfa99f57bb70dbc2e523850... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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e98100a00d63b7fb4ef4a77d97fa1cf313fed66f | subsection | 758 | 1,121 | Global Thom spectra | Multiplication maps
are defined by direct sum, i.e.,\mu _{V,W}\ : \ \mathbf {MGr}(V) \wedge \mathbf {MGr}(W) \ &\longrightarrow \quad \mathbf {MGr}(V\oplus W) \\
(x,U)\wedge (x^{\prime },U^{\prime })\quad &\longmapsto \ ( (x,x^{\prime }),\, U\oplus U^{\prime })\ .Unit maps are defined by\eta (V)\ : \ S^V\ \longrightarr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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5657c8d99b784ebf7b782ef16bfe4dba16c085cb | subsection | 759 | 1,121 | Global Thom spectra | So while the theory \mathbf {MGr} is not globally orientable,
and does not have Thom isomorphisms for equivariant bundles,
informally speaking the inverses of the prospective Thom classes
are already present in \mathbf {MGr}.Remark 1.3 (\mathbf {MGr} as a wedge of semifree spectra)
We recall from Construction REF
tha... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6b1a087a43f7237035432dfe23536b0821b8cb53 | subsection | 760 | 1,121 | Global Thom spectra | For all orthogonal G-representations V and W
the relation
\tau _{G,V}\cdot \tau _{G,W} \ = \ \tau _{G,V\oplus W}
holds in \mathbf {MGr}_0^G(S^{V\oplus W}).
For all orthogonal G-representations V and all k\ge 1
the relation
P^k(\tau _{G,V})\ = \ \tau _{\Sigma _k\wr G,V^k}
holds in \mathbf {MGr}_0^{\Sigma _k\wr G}(S^{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f3d87f7961a5521e492a286822051f37609ee85d | subsection | 761 | 1,121 | Global Thom spectra | We also defined \varepsilon _V:\pi _k^G(X\wedge S^V)\longrightarrow \pi _k^G(X\wedge S^V)
as the effect of the antipodal map of S^V.
Now we observe that multiplication by the class \tau _{G,V}
factors as the composite\pi _k^G(\mathbf {MGr}\wedge A)\ &\xrightarrow{}
\ \pi _k^G(\operatorname{sh}^V \mathbf {MGr}\wedge A) ... | {
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"source_ref_id": "39736b9a7358eb5407b5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3775fe365de3cbd81038a818e6c590895702740b | subsection | 762 | 1,121 | Global Thom spectra | The structure maps are given by{\mathbf {O}}(V,W) \wedge {\mathbf {MOP}}(V) \ &\longrightarrow \qquad {\mathbf {MOP}}(W)\\
(w,\varphi ) \wedge (x,U) \quad &\longmapsto \ ((w,0)+{\mathbf {BOP}}(\varphi )(x),\,{\mathbf {BOP}}(\varphi )(U))\ .Multiplication maps\mu _{V,W}\ : \ {\mathbf {MOP}}(V) \wedge {\mathbf {MOP}}(W) ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b24ac329a71ce75672f9d0d53b1f52424bd56422 | subsection | 763 | 1,121 | Global Thom spectra | Explicitly, {\mathbf {MO}}(V) is the Thom space of the tautological |V|-plane
bundle over G r_{|V|}(V^2).{\mathbf {MO}} - global Thom spectrumRemark 1.8 Certain variations of the construction of {\mathbf {MO}} and {\mathbf {MOP}} are possible, and
have been used at other places in the literature.
Indeed, if U is any eu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f6d5a1d6b75b38a6816d8191b73833dea842cbc2 | subsection | 764 | 1,121 | Global Thom spectra | We let t\in \pi _{-1}({\mathbf {MOP}}^{[-1]})
be the class represented by the point(0,\lbrace 0\rbrace ) \ \in \ T h(G r_0({\mathbb {R}}^2)) \ = \ {\mathbf {MOP}}^{[-1]}({\mathbb {R}}) \ .We let \sigma \in \pi _1({\mathbf {MOP}}^{[1]})
be the class represented by the mapS^2 \ \longrightarrow \ T h(G r_2({\mathbb {R}}^2... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bfc3c99fa11cc86abe105ea4220c98676b2ee12e | subsection | 765 | 1,121 | Global Thom spectra | Indeed, the suspension of the defining representative (REF )
for t differs from the defining representative
for \tau _{e,{\mathbb {R}}} by the inversion map -\operatorname{Id}:S^1\longrightarrow S^1.
So t\wedge S^1=-\tau _{e,{\mathbb {R}}}; however, since 2=0 in \pi _0^e({\mathbf {MOP}}),
this yields the claim.The next... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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70e13c5347b28ff58a60bed6529576c63da5d895 | subsection | 766 | 1,121 | Global Thom spectra | For every compact Lie group G, every based G-space A
and all k\in {\mathbb {Z}}, the maps
{\mathbf {MOP}}_{k+1}^G(A) \ \xrightarrow{} {\mathbf {MOP}}_k^G(A)
\text{\quad and\quad }
{\mathbf {MOP}}_k^G(A) \ \xrightarrow{} {\mathbf {MOP}}_{k+1}^G(A)
are mutually inverse isomorphisms.
For every representation V of a com... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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88ba93bc7533675245a56ababfd691d54f099f90 | subsection | 767 | 1,121 | Global Thom spectra | Expanding the definition of \mu _{{\mathbb {R}}.{\mathbb {R}}}
identifies this composite as the mapS^2 \ \longrightarrow \ {\mathbf {MOP}}^{[0]}({\mathbb {R}}\oplus {\mathbb {R}}) \ , \quad x\ \longmapsto \ ( ({\mathbb {R}}\oplus \tau _{{\mathbb {R}},{\mathbb {R}}}\oplus {\mathbb {R}})(0,0,x), 0\oplus {\mathbb {R}}\opl... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1e09a7bc7196f24d4ee68a9e63263e15d6b73136 | subsection | 768 | 1,121 | Global Thom spectra | So(j_{\mathbf {MOP}}^V)_*(\Phi [f])\ &= \ [j_{\mathbf {MOP}}^V(U\oplus V\oplus V)\circ \mu _{U\oplus V,V}\circ (f\wedge s_{G,V})] \\
&= \ [\mu _{U\oplus V,V\oplus V}\circ ({\mathbf {MOP}}(U\oplus V)\wedge j_{\mathbf {MOP}}^V(V))\circ (f\wedge s_{G,V})] \\
&= \ [\mu _{U\oplus V,V\oplus V}\circ (f\wedge ( j_{\mathbf {MOP... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e05ddddc9613dbb7e8f0f270856b043cd4e790e1 | subsection | 769 | 1,121 | Global Thom spectra | In particular, it commutes with the action of the element p_G^*(t),
where t\in \pi _{-1}^e({\mathbf {MOP}}) is the periodicity element defined
in (REF ). Since p_G^*(t) is invertible by part (i)
and \pi _0^G(j_{\mathbf {MOP}}^V) is an isomorphism, the map \pi _k^G(j_{\mathbf {MOP}}^V) is
then an isomorphism for every i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2e940862637af3837ae7c29e3d7ef763051fd692 | subsection | 770 | 1,121 | Global Thom spectra | The adjoint S^V\longrightarrow \operatorname{map}_*(S^V,{\mathbf {MOP}}(V)) of s_{G,V} represents the Thom class
\sigma _{G,V} - Thom class in {\mathbf {MOP}}^0_G(S^V)Thom class!in {\mathbf {MOP}}\sigma _{G,V}\ \in \ {\mathbf {MOP}}^0_G(S^V) \ = \ \pi _0^G(\operatorname{map}_*(S^V,{\mathbf {MOP}}))in the G-equivariant ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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997232f9c3f5ecd6ae066be12de5b0effceb5e08 | subsection | 771 | 1,121 | Global Thom spectra | If we let f=s_{G,V}^\sharp :S^V\longrightarrow \Omega ^V {\mathbf {MOP}}(V) be adjoint to
the defining representative for \sigma _{G,V}, then the composite comes out as the mapS^{V\oplus V} \ \longrightarrow \ {\mathbf {MOP}}(V \oplus V) \ , \quad (v,w)\ \longmapsto \ ( (v,0,-w,0),V\oplus 0\oplus V\oplus 0)\ .This comp... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f3ad84888145519d978689c0e24ef1d7439d2575 | subsection | 772 | 1,121 | Global Thom spectra | Since it is also left inverse to multiplication by \sigma _{G,V},
this proves the first claim.Construction 1.16 (Thom classes for equivariant vector bundles)
The Thom spectrum {\mathbf {MOP}} comes with a distinguished orientation,
given by Thom classes for equivariant vector bundles.
These Thom classes generalize the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c97a825711c868e2afc351d684fc8a6ed6024cde | subsection | 773 | 1,121 | Global Thom spectra | It is straightforward to see that the Thom classes just defined
are natural for pullback of bundles, compatible with
restriction along continuous homomorphisms, and
the Thom class of an exterior product of bundles is
the exterior product of the Thom classes.The Thom diagonal of the G-vector bundle \xi :E\longrightarrow... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5d7f54e61d7a8a6f1525c2a4ebc5f31282b352cf | subsection | 774 | 1,121 | Global Thom spectra | If V has nonzero G-fixed points, then the inclusion i:S^0\longrightarrow S^V
is G-equivariantly null-homotopic, so e(V)=0 whenever V^G\ne 0.Remark 1.18 (Shifted Thom and Euler classes in {\mathbf {MO}}) shifted inverse Thom class!in {\mathbf {MO}}
The author thinks that the periodic theory {\mathbf {MOP}} is the most
n... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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97f94108aef03437eb94bca4dfd9c8f49be6c1fa | subsection | 775 | 1,121 | Global Thom spectra | More precisely, the maps\bigoplus _{n\in {\mathbb {Z}}} \, \pi _n^G({\mathbf {MO}})\ \longrightarrow \ \pi _0^G({\mathbf {MOP}})
\text{\quad and\quad }
\bigoplus _{n\in {\mathbb {Z}}} \, {\mathbf {MO}}^{-n}_G(S^V)\ \longrightarrow \ {\mathbf {MOP}}^0_G(S^V) \ ,given on the n-th summand by multiplication by p_G^*(t^n),
... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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665c6d5349e38448a7eb5952efada82a506ce8c6 | subsection | 776 | 1,121 | Global Thom spectra | The morphism (REF )
induces natural transformations of equivariant homology theories(a\wedge A)_* \ : \ \mathbf {MGr}_*^G(A)\ \longrightarrow \ {\mathbf {MOP}}_*^G(A)for all compact Lie groups G and all based G-spaces A.
We observe that(a\wedge S^V)_*(\tau _{G,V})\ = \ \tau _{G,V} \ ,i.e., the morphism a takes the \mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fc158033a21f3cae4da8282c3b469181552fcf98 | subsection | 777 | 1,121 | Global Thom spectra | We show two separate statements that amount to the injectivity
respectively surjectivity of the map a^\sharp .(a)
We show that for every class x in the kernel
of the map (a\wedge A)_*:\mathbf {MGr}_0^G(A) \longrightarrow {\mathbf {MOP}}^G_0(A),
there is a G-representation V such that x\cdot \tau _{G,V}=0.
Indeed, we ca... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ee6e8ead65642a166726037f129f7e2d049aa642 | subsection | 778 | 1,121 | Global Thom spectra | Since \lambda ^V_{\mathbf {MGr}\wedge A}:\mathbf {MGr}\wedge A\wedge S^V\longrightarrow \operatorname{sh}^V\mathbf {MGr}\wedge A is
a {\underline{\pi }}_*-isomorphism by Proposition REF (ii),
there is a unique class x\in \mathbf {MGr}_0^G(A\wedge S^V) such that(\lambda ^V_{\mathbf {MGr}\wedge A})_*(x)\ = \ [f]\ .On th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1b17375cb6d856a35813ae92b746fe30344cacd7 | subsection | 779 | 1,121 | Global Thom spectra | The equivariant homology theory represented by {\mathbf {mO}} is the natural target
of the equivariant Thom-Pontryagin map from equivariant bordism,
and that map is trying hard to be an isomorphism, see Theorem REF below.We recall that the value of {\mathbf {bOP}} at an inner product space V is{\mathbf {bOP}}(V)\ = \ {... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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07a41c461c20d49c26cab20e4a437fa41f27f942 | subsection | 780 | 1,121 | Global Thom spectra | Multiplication maps\mu _{V,W}\ : \ {\mathbf {L}}(({\mathbb {R}}^\infty )^2,{\mathbb {R}}^\infty )_+ \wedge {\mathbf {mOP}}(V) \wedge {\mathbf {mOP}}(W) \ \longrightarrow \ {\mathbf {mOP}}(V\oplus W)are defined by sending \psi \wedge (x,U)\wedge (x^{\prime },U^{\prime })
to (\psi _\sharp (x,x^{\prime }),\psi _\sharp (U\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8df24f7d6004fd4d8bef60b78aa3409aad739965 | subsection | 781 | 1,121 | Global Thom spectra | We define continuous maps\psi _{V,W}\ : \ {\mathbf {mOP}}(V) \wedge {\mathbf {mOP}}(W) \ &\longrightarrow \quad {\mathbf {mOP}}(V\oplus W) \text{\qquad by} \\
\psi _{V,W}((x,U),(x^{\prime },U^{\prime }))\ &\ = \ (\psi _\sharp (x,x^{\prime }),\psi _\sharp (U\oplus U^{\prime })) \ ,where \psi _\sharp was defined in (REF ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d4ce82f70213b8ec43dd27b57a4d7de212ddd7c4 | subsection | 782 | 1,121 | Global Thom spectra | However, an E_\infty -multiplication does not entitle us to power operations.Given a compact Lie group G and based G-spaces A and B,
we define a multiplication\cdot \ : \ {\mathbf {mOP}}^G_k(A)\times {\mathbf {mOP}}_l^G(B) \ \longrightarrow \ {\mathbf {mOP}}_{k+l}^G(A\wedge B)as the composite\pi _k^G({\mathbf {mOP}}\we... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f2bc1470abe1aa1bded48b4bc8440259e1ba25b1 | subsection | 783 | 1,121 | Global Thom spectra | A shift morphism of orthogonal G-spectra j_{\mathbf {mOP}}^V:{\mathbf {mOP}}\longrightarrow \operatorname{sh}^V{\mathbf {mOP}}
is defined as for \mathbf {MGr} and {\mathbf {MOP}}: the value at an inner product space U is the mapj_{\mathbf {mOP}}^V(U)\ : \ {\mathbf {mOP}}(U)\ &\longrightarrow \ {\mathbf {mOP}}(U\oplus V... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b515ba2231ac6c9e1aef6c769993869738575b65 | subsection | 784 | 1,121 | Global Thom spectra | In particular, exterior multiplication by the inverse Thom class \tau _{G,{\mathbb {R}}}
of the trivial 1-dimensional G-representation
is invertible in equivariant {\mathbf {mOP}}-homology.
For every compact Lie group G, every based G-space A
and every integer k the multiplication map
{\mathbf {mOP}}_{k+1}^G(A) \ \xr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3bb18b436d6c53258e99fcc01b89a6e7a3862f4a | subsection | 785 | 1,121 | Global Thom spectra | Now we add {\mathbf {mOP}} to this picture, which turns out to be
an intermediate localization.
As we will now explain, {\mathbf {mOP}}-theory is obtained from
\mathbf {MGr}-theory by inverting the inverse Thom classes of all trivial representation.
Then {\mathbf {MOP}}-theory is obtained from
{\mathbf {mOP}}-theory by... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ce2faa3c2f71845e7c1eb8435e84b15f14276d1e | subsection | 786 | 1,121 | Global Thom spectra | This relation is again immediate from the explicit
representatives of the inverse Thom classes in (REF )
respectively (REF ).We define a localized version of equivariant \mathbf {MGr}-homology by\mathbf {MGr}^G_k(A)[\tau _{G,{\mathbb {R}}}^{-1}]\ = \ \operatorname{colim}_{n\ge 0}\, \mathbf {MGr}_k^G(A\wedge S^n) \ ,the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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577c700548a8b0e312704ce910cef86b3f62253f | subsection | 787 | 1,121 | Global Thom spectra | Since \psi ^n= \psi ^{n+1}\circ (\operatorname{sh}^n j_{\mathbf {MGr}}),
the morphisms \psi ^n are compatible with the sequence of morphisms\mathbf {MGr}\ \xrightarrow{}\ \operatorname{sh}\mathbf {MGr}\ \xrightarrow{}\ \operatorname{sh}^2 \mathbf {MGr}\ \xrightarrow{}\ \cdots \ \ .Moreover, the morphisms \psi ^n expres... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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140fd93b3aaed45bd52668bd247fda6cab37733d | subsection | 788 | 1,121 | Global Thom spectra | The diagram@C=12mm{
\pi _k^G(\mathbf {MGr}\wedge A\wedge S^{n-1})[r]^-{\cdot \tau _{G,{\mathbb {R}}}}[d]_{\lambda ^{n-1}_{\mathbf {MGr}\wedge A}}^\cong &
\pi _k^G(\mathbf {MGr}\wedge A\wedge S^n)[d]^{\lambda ^n_{\mathbf {MGr}\wedge A}}_\cong [r]^-{\pi _k^G(b\wedge A\wedge S^n)}& \pi _k^G({\mathbf {mOP}}\wedge A\wedge S... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a1aad2291478d701b62e09187f0d2b19fe7df1a2 | subsection | 789 | 1,121 | Global Thom spectra | We define a localized version of equivariant {\mathbf {mOP}}-homology by{\mathbf {mOP}}^G_0(A)[1/\tau ]\ = \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)}\, {\mathbf {mOP}}_0^G(A\wedge S^V) \ ;for V\subset W, the structure map in the colimit system
is the multiplication{\mathbf {mOP}}_0^G(A\wedge S^V) \ \xrightarrow{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ce922b599324a93b99e209a4be04950ea80e95f7 | subsection | 790 | 1,121 | Global Thom spectra | So combining these two theorems yields:Corollary 1.29
For every compact Lie group G, every based G-space A
and every integer k the mapa^\sharp \circ ((b\wedge A)_*[1/\tau ])^{-1}\ : \ {\mathbf {mOP}}_k^G(A)[1/\tau ] \ \longrightarrow \ {\mathbf {MOP}}_k^G(A)is an isomorphism.While the authors thinks that the periodic ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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aa0e91096128b82ec7d5cfc724265e91610bad9c | subsection | 791 | 1,121 | Global Thom spectra | Both theories {\mathbf {mOP}} and {\mathbf {MOP}} are periodic (in the {\mathbb {Z}}-graded sense),
i.e., the maps\bigoplus _{m\in {\mathbb {Z}}} \, {\mathbf {MO}}_m^G(A)\ \longrightarrow \ {\mathbf {MOP}}_0^G(A) \text{\quad and\quad }
\bigoplus _{m\in {\mathbb {Z}}} \, {\mathbf {mO}}_m^G(A)\ \longrightarrow \ {\mathbf... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5914710b80a683d66fe90e6e7bf841d27d91ffba | subsection | 792 | 1,121 | Global Thom spectra | So we conclude:Corollary 1.31
For every compact Lie group G, every based G-space A
and every integer m the mapa^\sharp \circ ( (b\wedge A)_*[1/\bar{\tau }])^{-1}\ : \ {\mathbf {mO}}_m^G(A)[1/\bar{\tau }] \ \longrightarrow \ {\mathbf {MO}}_m^G(A)is an isomorphism.Now we investigate the global homotopy type of the Thom ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3a445511c4403164d852f758981ee1b3900ddcd3 | subsection | 793 | 1,121 | Global Thom spectra | The value of {\mathbf {bO}}_{(m)} at V is{\mathbf {bO}}_{(m)}(V)\ = \ G r_{|V|}(V\oplus {\mathbb {R}}^m) \ ,the Grassmannian of |V|-planes in V\oplus {\mathbb {R}}^m.
Over the space {\mathbf {bO}}_{(m)}(V) sits a tautological euclidean |V|-plane bundle,
with total space consisting of pairs
(x,U)\in (V\oplus {\mathbb {R... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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951de81fa4769b1d38cc7c5e166c1725e7ae7a76 | subsection | 794 | 1,121 | Global Thom spectra | Since shift and suspension are globally equivalent
(by Proposition REF (i)),
{\mathbf {mO}}_{(m)} is globally equivalent to the
m-fold suspension of the orthogonal spectrum M_{\operatorname{gl}} T (m)=F_{O(m),\nu _m}{\mathbf {mO}}_{(m)}\ = \ \operatorname{sh}^m F_m\ \simeq _{\operatorname{gl}} \ F_m \wedge S^m\ = \ M_... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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61f677b0a93088b00f78cc0aab790296bdb0a22e | subsection | 795 | 1,121 | Global Thom spectra | We definej^m\ = \ \operatorname{sh}^m i \ :\ {\mathbf {mO}}_{(m)} = \operatorname{sh}^m F_m \ \longrightarrow \ \operatorname{sh}^m(\operatorname{sh}F_{m+1})\ = \ \operatorname{sh}^{m+1}F_{m+1}\ = \ {\mathbf {mO}}_{(m+1)}\ .We define a morphism\psi ^m \ : \ {\mathbf {mO}}_{(m)}\ = \ \operatorname{sh}^m F_m \ \longright... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
d2529545a52fae45e1d80541dc574aef70eced17 | subsection | 796 | 1,121 | Global Thom spectra | The underlying non-equivariant statement, i.e., that M O is
a homotopy colimit of the spectra \Sigma ^m M T(m),
can for example be found in .
The identification of {\mathbf {mO}} as a homotopy colimit of semifree orthogonal spectra
now allows an algebraic description of \llbracket {\mathbf {mO}},E\rrbracket ,
the group... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/s11511-009-0036-9",
"end": 141,
"openalex_id": "https://openalex.org/W2019538266",
"raw": "S. Galatius, U. Tillmann, I. Madsen, M. Weiss, The homotopy type of the cobordism category. Acta Math. 202 (2009), no. 2, 195–239.",
"s... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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740afb6bad8dfb15d1bd4c4d402ee8b4708f8235 | subsection | 797 | 1,121 | Global Thom spectra | The morphism \psi ^m:{\mathbf {mO}}_{(m)}\longrightarrow {\mathbf {mO}} sends \tau _m
to the shifted inverse Thom class \bar{\tau }_{O(m),\nu _m}
defined in (REF ).shifted inverse Thom class!in {\mathbf {mO}}(i)
In (REF ) we defined a distinguished equivariant homotopy classa_m \ = \ a_{O(m),\nu _m} \ \in \ \pi _0^{O(m... | {
"cite_spans": []
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"Stefan Schwede"
] | [
"math.AT"
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