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ae684b4b93749336f2be05ad63aef9f7fc5099e8 | subsection | 798 | 1,121 | Global Thom spectra | Since all the maps are natural in E, it suffices to check this claim
for the identity of the universal example E={\mathbf {mO}}_{(m)}=\operatorname{sh}^m F_m.
Since the square@C=15mm{
E\wedge S^{\nu _m} [r]^-{\eta } [d]_{\tilde{\lambda }^m_E\wedge S^{\nu _m}} &
\Omega ^m(E\wedge S^{\nu _m}\wedge S^m) [d]^{\Omega ^m(\la... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1ecc4cf70e26c680a62aa135bc77bcf8926439e7 | subsection | 799 | 1,121 | Global Thom spectra | Since all the individual maps in the above composite are bijective,
so is the composite, which proves the representability property
of the pair ({\mathbf {mO}}_{(m)},\tau _m).(ii)
The class (j^m\wedge S^{\nu _m})_*(\tau _m) \wedge S^1 is represented by the composite:S^{\nu _m\oplus {\mathbb {R}}^{m+1}}\ &\xrightarrow{}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3a9279f87e09a227f0fb7a670525513c9112823b | subsection | 800 | 1,121 | Global Thom spectra | The two representatives differ by conjugation with the O(m)-equivariant
linear isometry\nu _m\oplus {\mathbb {R}}\oplus {\mathbb {R}}^m\oplus {\mathbb {R}}\ \longrightarrow \ \nu _m\oplus {\mathbb {R}}\oplus {\mathbb {R}}^m\oplus {\mathbb {R}}\ , \quad (v,u,x,s)\ \longmapsto \ (v,-s,x,u)\ ,so they represent the same cl... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4c1ad81c1f51facce73600cf20c795231966aa14 | subsection | 801 | 1,121 | Global Thom spectra | This proves the relation (\psi ^m\wedge S^{\nu _m})_*(\tau _m)=\tau _{O(m),\nu _m} \cdot p_{O(m)}^*(\sigma ^m) in~{\mathbf {mO}}_m^{O(m)}(S^{\nu _m}).The fact that {\mathbf {mO}} is the global homotopy colimit of the sequence of
orthogonal spectra {\mathbf {mO}}_{(m)} (see Proposition REF )
has the following consequenc... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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56babbf7c08aa6f7ddb88346c0b13d548f71465d | subsection | 802 | 1,121 | Global Thom spectra | The answer given by Theorem REF below
takes the form of a distinguished triangle
in the global stable homotopy category, witnessing that
the mapping cone of j^m:{\mathbf {mO}}_{(m-1)}\longrightarrow {\mathbf {mO}}_{(m)}
`is' the m-fold suspension of the suspension spectrum
of the global classifying space B_{\operatorn... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e1615728fbcb3e01d61add8ade8052f7a113f110 | subsection | 803 | 1,121 | Global Thom spectra | So there is a unique non-trivial morphism of orthogonal spectraa\ :\ F_{m+1}\ \longrightarrow \ \Sigma ^\infty _+ {\mathbf {L}}_{O(m+1),\nu _{m+1}}=\Sigma ^\infty _+ B_{\operatorname{gl}}O(m+1) \ .For every orthogonal spectrum E
the morphism \lambda _E^{m+1}:E\wedge S^{m+1}\longrightarrow \operatorname{sh}^{m+1} E
is a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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62ee98d348f29e5e0a2638fb5186e4612082c8a6 | subsection | 804 | 1,121 | Global Thom spectra | We can thus apply Theorem REF
and obtain a distinguished triangle:F_{m+1} \ \xrightarrow{}\ \Sigma ^\infty _+ B_{\operatorname{gl}}O({m+1})\ \xrightarrow{} \ F_m\ \xrightarrow{} \ F_{m+1} \wedge S^1By Example REF
shifting preserves distinguished triangles;
so the following sequence is also distinguished:\operatorname... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9a12b414cff3dee3f0b94c520f80e0dac7ee87cc | subsection | 805 | 1,121 | Global Thom spectra | The fourth equation is the compatibility of transfer and suspension isomorphism,
see Proposition REF .For calculations of equivariant homotopy groups of {\mathbf {mO}}
we also need to understand the composite:\Sigma ^\infty _+ B_{\operatorname{gl}} O(m+1) \wedge S^m
\ \xrightarrow{} \ {\mathbf {mO}}_{(m)} \ \xrightarro... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b1601a1dc775ac22b816659313f33143b9f3628f | subsection | 806 | 1,121 | Global Thom spectra | We denote by \langle \operatorname{tr}_e^{O(1)}\rangle the global subfunctor of the Burnside ring functor
{\mathbb {A}} generated by \operatorname{tr}_e^{O(1)}\in {\mathbb {A}}(O(1)).Theorem 1.39
The orthogonal spectrum {\mathbf {mO}} is globally connective
and the action of the Burnside ring global functor on the uni... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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26ca7ed6d12fd16251089b86edfaba3c0c1c0f1b | subsection | 807 | 1,121 | Global Thom spectra | Since {\mathbf {mO}} is a colimit of the sequence of closed embeddings
j^m:{\mathbf {mO}}_{(m)}\longrightarrow {\mathbf {mO}}_{(m+1)}, the map\operatorname{colim}_{m}\, {\underline{\pi }}_k({\mathbf {mO}}_{(m)})\ \longrightarrow \ {\underline{\pi }}_k({\mathbf {mO}})induced by the morphisms \psi ^m:{\mathbf {mO}}_{(m)}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2720d53ab7b81127f2ec46a5c5225cf21cbafd69 | subsection | 808 | 1,121 | Global Thom spectra | For a closed subgroup H of G we use the familiar notationt_H^G \ = \ \operatorname{tr}_H^G(p_H^*(1)) \ \in \ \pi _0^G({\mathbf {mO}}) \ ,where p_H:H\longrightarrow e is the unique homomorphism.Proposition 1.40
For every compact Lie group G, an {\mathbb {F}}_2-basis of \pi _0^G({\mathbf {mO}})
is given by the classes t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c081b834cdbc3870c32e0b7ad4bff8abeffbadf2 | subsection | 809 | 1,121 | Global Thom spectra | Conversely, if H is a subgroup of G with finite Weyl group of even order,
then we can choose a subgroup C\le W_G H of order 2.
The preimage L of C under the projection N_G H\longrightarrow W_G H
then contains H as an index 2 subgroup. By the above, the class t_H^G
is then one of the generating elements of \langle \oper... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6ba874b2fc8eba9696d1f4650fa274ffaa4f73ec | subsection | 810 | 1,121 | Global Thom spectra | So in the colimit over V\in s({\mathcal {U}}_G) this gives an isomorphism\Phi _*^G({\mathbf {mOP}}) \ \cong \ {\mathbf {mOP}}_*\left(\mathbf {Gr}({\mathcal {U}}_G^\perp )^G_+\right)to the non-equivariant {\mathbf {mOP}}-homology groups
of the G-fixed point space of \mathbf {Gr}({\mathcal {U}}_G^\perp ),
the disjoint un... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f466b7a0c8efc706a79673760db81b5f36bea2f3 | subsection | 811 | 1,121 | Global Thom spectra | The composite of the isomorphism (REF )
and the isomorphism (REF )
thus takes the wedge summand of \Phi ^G_*({\mathbf {mOP}})
corresponding to {\mathbf {mO}}={\mathbf {mOP}}^{[0]} to the sum of the terms with k=j.
So the isomorphisms restrict to an isomorphism\Phi ^G_*({\mathbf {mO}})\ \cong \ {\bigoplus }_{j\ge 0}\ {\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e663bf25732e92efa50682e5f83b6c461233b8d8 | subsection | 812 | 1,121 | Global Thom spectra | In much the same way in Construction REF ,
{\mathbf {mSO}}_{(m)}(V) `is' (by passage to orthogonal complements)
the Thom space over the tautological
bundle over the oriented Grassmannian G r^+_{|V|}(V\oplus {\mathbb {R}}^m)
of oriented |V|-planes in V\oplus {\mathbb {R}}^m.We define a morphismi \ : \ F_{S O(m),\nu _m}\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ca848108bc5456e5725e677f252531a807678a7c | subsection | 813 | 1,121 | Global Thom spectra | As before we denote by V_{\mathbb {C}}={\mathbb {C}}\otimes _{\mathbb {R}}V the complexification
of a real inner product space V.
The value of {\mathbf {bU}} at V is{\mathbf {bU}}(V)\ = \ Gr_{|V|}^{\mathbb {C}}(V_{\mathbb {C}}\oplus {\mathbb {C}}^\infty ) \ ,the Grassmannian of {\mathbb {C}}-linear subspaces of V_{\mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d2d6d023c9266811f6c255427453db55b9a17e5f | subsection | 814 | 1,121 | Global Thom spectra | The structure map of the spectrum {\mathbf {mU}} starts from the
(O(V)\times O(W))-equivariant mapS^{V_{\mathbb {C}}}\wedge T h({\mathbf {bU}}( W))\ &\longrightarrow \quad T h({\mathbf {bU}}(V\oplus W)) \\
v \wedge (x, U) \qquad &\longmapsto \ ( (v,x),\ U\oplus (0\oplus W_{\mathbb {C}}\oplus 0))\ .The structure map\sig... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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58434e5b214cb4a8576c58000322726a03d46aa4 | subsection | 815 | 1,121 | Global Thom spectra | The unit morphism is a global equivalence {\mathbb {S}}\simeq {\mathbf {mU}}_{(0)}
and {\mathbf {mU}}_{(m)} is globally equivalent to the 2 m-th
suspension of the semifree orthogonal spectrum generated by
the tautological unitary representation \nu _m^U of U(m) on {\mathbb {C}}^m:{\mathbf {mU}}_{(m)} \ \simeq \ F_{U(m)... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e4a9bfd96d59ae90549841143dd5cbf4f3121e89 | subsection | 816 | 1,121 | Global Thom spectra | Indeed, the action of the Burnside ring global functor on the element
1\in \pi _0^e({\mathbf {mU}}) induces an isomorphism of global functors{\mathbb {A}}\ \cong \ {\underline{\pi }}_0({\mathbf {mU}}) \ .Moreover, there is an exact sequence of global functors{\mathbf {A}}(U(1),-)\ \longrightarrow \ {\underline{\pi }}_1... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/0040-9383(70)90058-3",
"end": 1836,
"openalex_id": "https://openalex.org/W2011056259",
"raw": "T. tom Dieck, Bordism of G-manifolds and integrality theorems. Topology 9 (1970), 345–358.",
"source_ref_id": "39736b9a7358eb5407b5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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36d150a80d3f83e445fbde61e18bbda6a30261d4 | subsection | 817 | 1,121 | Global Thom spectra | Closely related, strictly commutative ring spectrum models
for these homotopy types have been discussed in various places,
for example , ,
or .For an inner product space V we consider the complex Grassmannian{\mathbf {BU}}(V)\ = \ Gr_{|V|}^{\mathbb {C}}(V_{\mathbb {C}}^2) \ .Over the space {\mathbf {BU}}(V) sits a tau... | {
"cite_spans": [
{
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"doi": "",
"end": 144,
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"raw": "J. P. May, E_{\\infty } ring spaces and E_{\\infty } ring spectra. With contributions by F. Quinn, N. Ray, and J. Tornehave. Lecture Notes in Mathematics, Vol. 77. Springer-Verlag, Berlin-New York, ... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ba94eaa7773ecb803f0d4df983520711c63ba408 | subsection | 818 | 1,121 | Global Thom spectra | As in the orthogonal situation in Theorem REF
the Thom class \sigma _{G,V}^U
is inverse to the image of the inverse Thom class \tau _{G,V}^U,
and \sigma _{G,V}^U restricts to a unitary Euler class
in \pi _{-2 n}^G({\mathbf {MU}}).Euler class!in {\mathbf {MU}}
The proof of Corollary REF generalizes to the
unitary situa... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1c3e9b537978071b33eb72ec7383442a887f9508 | subsection | 819 | 1,121 | Equivariant bordism | equivariant bordism|(In this section we recall equivariant bordism groups and their
relationship to the equivariant homology groups defined by the global Thom spectrum {\mathbf {mO}}
introduced in Example REF .
The main result is Theorem REF
which says that when G is isomorphic to a product of
a finite group and a tor... | {
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"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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bcebb7daee96f80f513d110026786d89a804ce55 | subsection | 820 | 1,121 | Equivariant bordism | The groups \mathcal {N}_n^G(X) are covariantly functorial
in continuous G-maps, by postcomposition.Proposition 2.1
Let G be a compact Lie group.Let \varphi ,\varphi ^{\prime }:X\longrightarrow Y be equivariantly homotopic continuous G-maps.
Then \varphi _*=\varphi ^{\prime }_* as homomorphisms
from \mathcal {N}_n^G(X)... | {
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"doi": "10.1007/bf01456063",
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"raw": "S. Illman, The equivariant triangulation theorem for actions of compact Lie groups. Math. Ann. 262 (1983), 487–501.",
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"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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172b09b825d539afe18953e9246fa1f3dcb9326b | subsection | 821 | 1,121 | Equivariant bordism | The triple (B,H^{\prime },\psi ) thus witnesses that [M,h]=0.
Since \varphi _* is a group homomorphism, it is injective.Property (iii) holds because compact manifolds only have finitely
many connected components, so all
continuous reference maps from singular manifolds or bordisms have image
in a finite union.Now we st... | {
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"raw": "G. E. Bredon, Introduction to compact transformation groups. Pure and Applied Mathematics, Vol. 46. Academic Press, New York-London, 197... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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e1db547034b340b4764d208d74740cfdfeeca543 | subsection | 822 | 1,121 | Equivariant bordism | The sets h^{-1}(X-A) and h^{-1}(X-B) are
G-invariant, disjoint closed subsets of M;
we let r:M\longrightarrow {\mathbb {R}} be a G-invariant smooth separating function
as provided by Lemma REF , i.e.,
such that h^{-1}(X-A)\subseteq r^{-1}(0) and
h^{-1}(X-B)\subseteq r^{-1}(1).
We let t\in (0,1) be any regular value of ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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400ad510a0a1cd18f9124e49eb37100a6459296b | subsection | 823 | 1,121 | Equivariant bordism | Then(\Psi ^{-1}(t),\, H|_{\Psi ^{-1}(t)},\, \psi |_{r^{-1}(t)\cup \bar{r}^{-1}(t)})is a bordism from
(r^{-1}(t),h|_{r^{-1}(t)}) to (\bar{r}^{-1}(t),g|_{\bar{r}^{-1}(t)}).
This shows at the same time that the bordism class is independent of
the choice of separating function and of the choice of representing
singular G-m... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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31c190e8097b2ac7a98757f15a719d6a2891f6bb | subsection | 824 | 1,121 | Equivariant bordism | On the other hand, if we add a disjoint G-fixed
basepoint to an unbased G-space Y, then the composite\mathcal {N}_n^G(Y) \ \xrightarrow{} \ \mathcal {N}_n^G(Y_+) \ \xrightarrow{} \ \widetilde{\mathcal {N}}_n^G(Y_+)is an isomorphism.Construction 2.6 We consider a continuous G-map f:X\longrightarrow Y and letC f\ = \ C X... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a3c48110dbe80078a0044d96b03600919b6274bb | subsection | 825 | 1,121 | Equivariant bordism | In the diagram@C=15mm{
X [r]^-f [d]_{x\mapsto (x,1/2)} & Y [r]^-i [d]_-i & C f @{=}[d] \\
A\cap B [r]_-{\text{incl}} & B [r]_-{\text{incl}} & C f}the right square commutes and the left square commutes up to equivariant homotopy.
Moreover, all vertical maps are equivariant homotopy equivalences, so
they induce isomorphi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0ccdf9613aa356650eb8221bd73bfa604a3cd695 | subsection | 826 | 1,121 | Equivariant bordism | So we can substitute \widetilde{\mathcal {N}}_*^G(B/A)
into the long exact mapping cone sequence of Proposition REF
and obtain a long exact sequence of abelian groups:\dots \ \longrightarrow \ \mathcal {N}_n^G(A) \ \xrightarrow{} \ \mathcal {N}_n^G(B) \ \xrightarrow{} \ \widetilde{\mathcal {N}}_n^G(B/A) \ \longrightar... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e660c8aeec567f98c144fed3846b6f91607f9bb6 | subsection | 827 | 1,121 | Equivariant bordism | We define another embeddingi \ : \ M\times S({\mathbb {R}}\oplus V)\ &\longrightarrow \ B \text{\qquad by}\\
i(m,(x,v))\ &= \ {\left\lbrace \begin{array}{ll}
\quad [(j(m,v), 0]^\text{left} & \text{ for $x\le 0$, and}\\
\quad [(j(m,v), 0]^\text{right} & \text{ for $x\ge 0$.}
\end{array}\right.}Here the superscripts `lef... | {
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"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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d52b75d004ef92ab1bc24a0b1e1c1beef05c61ad | subsection | 828 | 1,121 | Equivariant bordism | Indeed, the composite\mathcal {N}_m^G(X) \otimes \mathcal {N}_n^G(Y) \ &\xrightarrow{} \ \mathcal {N}_{m+n}^G(X\times Y) \ \xrightarrow{} \ \mathcal {N}_{m+n}^G(X\wedge Y) \ \xrightarrow{} \ \widetilde{\mathcal {N}}_{m+n}^G(X\wedge Y)annihilates the image of \mathcal {N}_m^G(\ast )\otimes \mathcal {N}_n^G(Y)
and the im... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2a3c40d2c2e258270c91c57c036c88651e0de631 | subsection | 829 | 1,121 | Equivariant bordism | Then the relationd_{G,V}\wedge \ d_{G,W}\ =\ d_{G,V\oplus W}holds in \widetilde{\mathcal {N}}_{m+n}^G(S^{V\oplus W}).We define a `distorted' version
\tau _V : S({\mathbb {R}}\oplus V) \longrightarrow S^V
of the stereographic projection as the compositeS({\mathbb {R}}\oplus V) \ \xrightarrow{} \ S^V \ \xrightarrow{}\ S^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8199411b3bb30e23001d932af6d79a7f37b209e4 | subsection | 830 | 1,121 | Equivariant bordism | Proposition REF thus shows thatq_*(d_{G,V}&\wedge d_{G,W}) \ = \ q_*(\llbracket S({\mathbb {R}}\oplus V),\tau _V\rrbracket \wedge \llbracket S({\mathbb {R}}\oplus W),\tau _W\rrbracket ) \\
&= \ \llbracket S({\mathbb {R}}\oplus V)\times S({\mathbb {R}}\oplus W),\ q\circ (\tau _V\times \tau _W) \rrbracket \ = \ \llbracke... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e5edcee4474d29126e6a302a214aa90393b7d050 | subsection | 831 | 1,121 | Equivariant bordism | On such points the map f is given byf(x,v,w) \ &= \ q \left( \tau _V\left(\frac{|v|^2-1}{|v|^2+1},\frac{2 v}{|v|^2+1}\right),\ \tau _W\left(\frac{|w|^2-1}{|w|^2+1},\frac{2 w}{|w|^2+1}\right)\right)\\
&= \ q(J(v),J(w))\ ,whereas\Pi _{V\oplus W}(-x,-v,-w) \ = \ \left( \frac{-v}{1+x},\quad \frac{-w}{1+x} \right) \ .If the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4b1e4b05433279ec8792dc613190c64dd038f40a | subsection | 832 | 1,121 | Equivariant bordism | The cone of this map isX^\diamond \ = \ X\times [0,1] / \sim \ ,the unreduced suspension of X, where X\times \lbrace 0\rbrace and X\times \lbrace 1\rbrace
are collapsed to one point each.
Since X has a G-fixed point,
the map f_*:\mathcal {N}_*^G(X)\longrightarrow \mathcal {N}_*^G(\ast )
is a split epimorphism. So the ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c032f3785b5ed26333daa0ca706b166e51a1b6da | subsection | 833 | 1,121 | Equivariant bordism | We define a continuous map H:M\times S({\mathbb {R}}\oplus {\mathbb {R}})\longrightarrow X^\diamond byH(m,(x,y)) \ = \ {\left\lbrace \begin{array}{ll}
\ [h(m), (y+1)/2] & \text{ for $x \le 0$, and }\\
\quad [x_0, (y+1)/2] & \text{ for $x \ge 0$.}
\end{array}\right.}Then the following square commutes up to G-equivariant... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f024dc6b0f90e9ab4499b393884fe934e5190652 | subsection | 834 | 1,121 | Equivariant bordism | If G acts trivially on V, then it is equivariantly isomorphic to {\mathbb {R}}^n
for some n.The bordism theories \mathcal {N}_*^G for different compact Lie groups
are related by geometrically defined restriction and induction maps.
Every continuous group homomorphism \alpha :K\longrightarrow G is automatically smooth
(... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5b69b88ae2744f1ad79d29a4bb940d88704ec386 | subsection | 835 | 1,121 | Equivariant bordism | If H has finite index in G, then the induction \operatorname{ind}_H^G preserves
the dimension, and then it satisfies the double coset formula.
So for fixed n\ge 0 the coefficient groups \mathcal {N}_n^G(\ast )
almost form a global functor; the only missing structure are the
transfer maps for closed inclusions that are ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a32e555455eabb59798028f5319b2dd6e8bc52d7 | subsection | 836 | 1,121 | Equivariant bordism | For an H-space Y the H-equivariant continuous mapl_Y \ : \ G\times _H Y \ \longrightarrow \ Y_+\wedge S^Lwas defined in Construction REF .Proposition 2.12
For every closed subgroup H of a compact Lie group G and every
H-space Y, the composite\mathcal {N}_n^H(Y) \ \xrightarrow{} \ \mathcal {N}_{n+d}^G(G\times _H Y) \ \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
60c956247befe65f87a702c64b6d088712f7a990 | subsection | 837 | 1,121 | Equivariant bordism | So explicitly,l_H^G(g) \ = \ {\left\lbrace \begin{array}{ll}
( l / (1-|l|) )\wedge h & \text{ if $g=s(l)\cdot h$ with $(l,h)\in D(L)\times H$, and }\\
\quad \infty & \text{ if $g$ is not in the image of $\bar{s}$.}
\end{array}\right.}We obtain a smooth H-equivariant embeddingj\ :\ M\times D(L) \ \longrightarrow \ G\tim... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2521eca02918b2cf200265259e9ab4075e3a83be | subsection | 838 | 1,121 | Equivariant bordism | We can thus conclude that((l_Y)_*\circ \operatorname{res}^G_H\circ (G\times _H-))&[M,h] \ = \ \llbracket M\times S({\mathbb {R}}\oplus L), f\rrbracket \\
&= \ [M,h]\wedge \llbracket S({\mathbb {R}}\oplus L),\Psi \rrbracket \ = \ [M,h]\wedge d_{H,L}\ .Remark 2.13 (Failure of the Wirthmüller isomorphism
in equivariant bo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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37c549667e340f56c64af8eba7d8c39b8e055d75 | subsection | 839 | 1,121 | Equivariant bordism | We will now identify the `geometric fixed point term'
in the isotropy separation sequence for equivariant bordism.Construction 2.14 As before we let E{\mathcal {P}} be a universal space for the family of proper closed subgroups
of G. So E{\mathcal {P}} is a cofibrant G-space without G-fixed points,
and (E{\mathcal {P}}... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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651aa8bd5c087f396bdc775c0ccb8b5ea49dfe96 | subsection | 840 | 1,121 | Equivariant bordism | Then for every fixed point x\in M^{(j)},(d i)(T_x (M^{(j)})) \ = \ ( (d i)(T_x M) )^G \ ,i.e., the tangent space inside M^{(j)} `is' the G-fixed part
of the tangent space in M.
So we can define a continuous map\nu _j \ : \ M^{(j)}\ \longrightarrow \ ( G r_j(V^\perp ))^G\ \xrightarrow{}\ G r_j^{G,\perp }by sending a fix... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f0c55194f2fb8d4da00e6b67dffb2f44eb9f510b | subsection | 841 | 1,121 | Equivariant bordism | The space (G r_j(V))^G is a smooth manifold,
and by smooth approximation
we can assume without loss of generality that f is a smooth map.
We define a closed smooth n-dimensional manifold byM\ = \ \lbrace (n,x,v)\in N \times S({\mathbb {R}}\oplus V) \ |\ v\in f(n) \rbrace \ .Another way to say this is that M is a double... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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59a0c9e252562d2dea26d7913b71f99f8a35cfbb | subsection | 842 | 1,121 | Equivariant bordism | We use this data to `cut out' the fixed points M_0^G from M by replacing a
tubular neighborhood by the sphere bundles of the maps F_j; this produces
a new singular G-manifold over \tilde{E}{\mathcal {P}}, bordant to (M,h),
that has no more fixed points over 0.The construction is done separately
and disjointly over each... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a1b4ddbb21dd28d86630ec251b8015bc30999bfe | subsection | 843 | 1,121 | Equivariant bordism | The space W is a topological (n+1)-manifold whose boundary
is the union of two disjoint parts that we now parametrize.
An obvious embedding is given by\psi \ : \ M \ \longrightarrow \ W \ ,\quad \psi (m)=[m,0] \ .A second embeddingi \ : \ \bar{M}\ = \ ( M-\psi (\mathring{D}(\nu )))\cup _{S(\nu )} S(F) \ \longrightarrow... | {
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{
"arxiv_id": "",
"doi": "",
"end": 770,
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"raw": "T. tom Dieck, Algebraic topology. EMS Textbooks in Mathematics. European Mathematical Society (EMS), Zürich, 2008. xii+567 pp.",
"source_ref_id": "acf30c1e28a49... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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01c00c1d90876159fdc025ba739cc129a7f5beda | subsection | 844 | 1,121 | Equivariant bordism | The triple (W,H,\psi +i) is then a bordism that witnesses the relation
\llbracket M,H \psi \rrbracket =\llbracket \bar{M},H i\rrbracket in the group
\widetilde{\mathcal {N}}_n^G(\tilde{E}{\mathcal {P}}).
The map H i:\bar{M}\longrightarrow \tilde{E}{\mathcal {P}} sends all of \bar{M}^G to the fixed point \infty ,
so H i... | {
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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92472f6fb4d09da5e5be08ee36b30a390ce49afc | subsection | 845 | 1,121 | Equivariant bordism | The pair (D M,h) is then a singular G-manifold over \tilde{E}{\mathcal {P}},
and it represents a bordism class[D M,h] \quad \in \ \mathcal {N}_n^G(\tilde{E}{\mathcal {P}})\ .The fact that the map (REF ) is an isomorphism
is Satz 3 in .For finite groups, Stong shows in
that the geometric fixed point map\mathcal {N}_*^G... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 235,
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"raw": "T. tom Dieck, Orbittypen und äquivariante Homologie. I. Arch. Math. (Basel) 23 (1972), no. 1, 307–317.",
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"start": 174
},
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b26ec47dfe44f7ff49df487d9e7da80dac7d3b9a | subsection | 846 | 1,121 | Equivariant bordism | Since there is only one non-trivial irreducible
C-representation, the 1-dimensional sign representation,
every linear subspace of {\mathcal {U}}_C^\perp is C-invariant.
Hence G r_j^{C,\perp } is just a Grassmannian of j-planes
in an infinite dimensional {\mathbb {R}}-vector space,
hence a classifying space of the ortho... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/memo/0103",
"end": 1719,
"openalex_id": "https://openalex.org/W2075517255",
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"source_ref_id": "5d2099d8d92285b911f... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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34540a7a03a678f3175ceea670af1bbab544fde3 | subsection | 847 | 1,121 | Equivariant bordism | An \mathcal {N}_*-linear map\Gamma \ : \ \mathcal {N}_*^C \ \longrightarrow \ \mathcal {N}_{*+1}^Cof degree 1 is given by sending the class of
a manifold M with involution \tau :M\longrightarrow M to the manifoldS({\mathbb {C}})\times _C M \ = \ (S({\mathbb {C}})\times M ) / (z,m)\sim (-z,\tau m) \ .So S({\mathbb {C}})... | {
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"doi": "10.1090/s0002-9939-1972-0290379-9",
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"raw": "J. C. Alexander, The bordism ring of manifolds with involution. Proc. Amer. Math. Soc. 31 (1972), 536–542.",
"source_ref_id": ... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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68bf1eb6f219febd687e54a63876e9b5e32a3feb | subsection | 848 | 1,121 | Equivariant bordism | We can assume without loss of generality that V is a subrepresentation
of the chosen complete G-universe {\mathcal {U}}_G.
We use the inner product on V to define the normal bundle \nu
of the embedding at x\in M by\nu _x \ = \ V - (d i)(T_x M)\ ,the orthogonal complement of the image of the tangent space T_x M in V.
B... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ef8ac28a44a98166da0c897ca3b017bc1b20d2aa | subsection | 849 | 1,121 | Equivariant bordism | This isotopy induces a
homotopy between the two collapse maps and shows that altogether
the normal class \langle M\rangle is independent of the wide embedding.Part (ii) of the following proposition refers to an external
multiplication morphism\mu _{A,B}\ :\ (\mathbf {MGr}\wedge A_+)\wedge (\mathbf {MGr}\wedge B_+)\ \co... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8a3d73d237672da6b07e7d0e6e5a4159a79ca2de | subsection | 850 | 1,121 | Equivariant bordism | Then the class \langle \partial B\rangle is in the kernel of the homomorphism
(j\wedge \iota _+)_* \ : \ \mathbf {MGr}_0^G(\partial B_+) \ \longrightarrow \ (\operatorname{sh}\mathbf {MGr})_0^G(B_+) \ ,
where \iota :\partial B\longrightarrow B is the inclusion.(i)
We let p^1:(M\cup N)_+\longrightarrow M_+ and p^2:(M\c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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601e5650399ad9eb33bf7ddb0edddf4ac050a9fe | subsection | 851 | 1,121 | Equivariant bordism | The collapse mapS^{V\oplus W}\ \xrightarrow{} \ \mathbf {MGr}(V\oplus W)\wedge (M\times N)_+is equivariantly homotopic to the compositeS^V\wedge S^W\ \xrightarrow{} \ &( \mathbf {MGr}(V) \wedge M_+)\wedge (\mathbf {MGr}(W)\wedge N_+) \\
&\xrightarrow{} \ \mathbf {MGr}(V\oplus W)\wedge (M\times N)_+\ .This shows the des... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9047d39055d6de3f83489520831fed3429700b29 | subsection | 852 | 1,121 | Equivariant bordism | We choose an equivariant collar, i.e., a smooth G-equivariant embeddingc\ : \ \partial B\times [0,1) \ \longrightarrow \ Bsuch that c(-,0):\partial B\longrightarrow B is the inclusion and the image of c is
an open neighborhood of the boundary inside B.
Then we choose a smooth function\kappa \ : \ [0,1]\ \longrightarrow... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7f3fd7f24605df68504df0a10701d5166f6628d4 | subsection | 853 | 1,121 | Equivariant bordism | We consider (b,v,t)\in D(\nu ) where b\in B and (v,t)\in V\oplus {\mathbb {R}} is
normal to i(B) at i(b).
If b\in U, then the normal vector must lie in V\oplus 0, i.e., t=0.
The map \kappa then takes (b,v,0) to\left[ \left(\frac{v}{1-|v|}, \nu _b \right) , i_2(b) \right] \wedge bin the cone of \mathbf {MGr}(V)\wedge B_... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e8e6a28299d76c01df67a50977b0878f800de5a5 | subsection | 854 | 1,121 | Equivariant bordism | Thusp_*(\langle B\rangle ^{\operatorname{rel}}) \ = \ \iota _* \langle \partial B\rangle \wedge S^1in the group \pi _1^G(\mathbf {MGr}\wedge B_+\wedge S^1).
The relation (REF ) thus follows from
the definition of the connecting homomorphism (REF )
as the composite of p_* and the inverse suspension isomorphism.The inver... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.02... |
b2c09a14c0e8b1689a2cc3d035963ff4a15f2c97 | subsection | 855 | 1,121 | Equivariant bordism | Then the map\psi \ : \ G\times _H M\ \longrightarrow \ V\oplus W \ , \quad [g,m]\ \longmapsto \ (i(g H),\ g\cdot j(m))is a G-equivariant wide smooth embedding. We base the collapse map for
the G-manifold G\times _H M on the embedding \psi .The differential at the coset H of the embedding i is a linear embeddingL \ = \ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
311a853d90b9fe4f3b598a868f2b66595653785c | subsection | 856 | 1,121 | Equivariant bordism | The compositeD(L)\times M \ \xrightarrow{} \ G \ \xrightarrow{} \ V\oplus Wis given by the formula(l,m)\ \longmapsto \ ( i(s(l)\cdot H),\, s(l)\cdot j(m))\ .We define a homotopy of smooth wide H-equivariant embeddings[0,1]\times D(L)\times M \ \longrightarrow \ V\oplus W
\text{\quad by\quad } (t,l,m)\ \longmapsto \ ( i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
ee2f59703bb2fb07865b5699f3e35fe30d1456b3 | subsection | 857 | 1,121 | Equivariant bordism | The compositeO(m+1)/O(m)\ \xrightarrow{}\ S({\mathbb {R}}\oplus \nu _m) \ \xrightarrow{}\ S^{\nu _m}is O(m)-equivariantly homotopic to the map l_{O(m)}^{O(m+1)}:O(m+1)/O(m)\longrightarrow S^{\nu _m}
that appears in the Wirthmüller isomorphism,
where \psi (A\cdot O(m))=A\cdot (0,\dots ,0,1).
So we can argue:(\mathbf {MG... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0... |
1bfd021ce7dce730907e903c855e879d52bb55d5 | subsection | 858 | 1,121 | Equivariant bordism | We define\Theta ^G[M,h]\ = \ (b\wedge h)_* \langle M\rangle \cdot p_G^*(\sigma ^m) \ \in \ {\mathbf {mO}}_m^G(X)\ ,i.e., we take the image of the normal class of M under the homomorphism(b\wedge h)_*\ : \ \mathbf {MGr}_0^G(M_+)\ \longrightarrow \ {\mathbf {mOP}}_0^G(X)and multiply by the unit p_G^*(\sigma ^m) in \pi _m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2854e2700b7a54c0b5f5c550880d9633ae7ddbb9 | subsection | 859 | 1,121 | Equivariant bordism | We choose a null-bordism (B,H:B\longrightarrow X,\psi :M\cong \partial B),
so that H\circ \iota \circ \psi =h. Then(b\wedge h)_*\langle M\rangle \ &= \ (b\wedge (H\circ \iota \circ \psi ))_*\langle M\rangle \ = \ ({\mathbf {mOP}}\wedge H)_*( (b\wedge \iota _+)_*\langle \partial B\rangle )\ = \ 0\ .Multiplying by p_G^*(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0ba056abee35884a73350c591c4a4bafde8b85b0 | subsection | 860 | 1,121 | Equivariant bordism | In the special case where H has finite index in G, then the
tangent representation is zero, so in this special case \bar{\tau }_{H,L}=1 and part (v)
of the following theorem specializes to the simpler relation\Theta ^G( G\times _H y) \ = \ G\ltimes _H \Theta ^H(y)\ .Theorem 2.31The Thom-Pontryagin map \Theta ^G is addi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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892abc44bfe85110133abdfa9710f77848b3ddb2 | subsection | 861 | 1,121 | Equivariant bordism | Then(b\wedge (q\circ (h\times g)))_* \langle M\times N\rangle \ &= \ (b\wedge (q\circ (h\times g)))_* ( (\mu _{M,N})_*(\langle M\rangle \times \langle N\rangle ) )\\
&= \ (\mu _{M,N})_* ((b\wedge h)_* \langle M\rangle \times (b\wedge g)_* \langle N\rangle ) \\
&= \ (b\wedge h)_* \langle M\rangle \wedge (b\wedge g)_* \l... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f4ea7e9ff20a6b31f50bf0681b487cdcc15d81ac | subsection | 862 | 1,121 | Equivariant bordism | Compatibility of the Thom-Pontryagin construction with the boundary homomorphism
amounts to the commutativity of the following square:@C=18mm{
\widetilde{\mathcal {N}}_{m+1}^G(C f)[r]^-{p_*} [d]_{\Theta ^G} &
\widetilde{\mathcal {N}}_{m+1}^G(X_+\wedge S^1)[r]^-{(-\wedge d_{G,{\mathbb {R}}})^{-1}}_-{\cong }
[d]_{\Theta ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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531d9493692aba21772c6e778a29f816594d3ef0 | subsection | 863 | 1,121 | Equivariant bordism | Then\operatorname{Wirth}_H^G ( (b\wedge (G\ltimes _H h))_*\langle G\times _H M\rangle ) \ &= \ (b\wedge h\wedge S^L)_*( \operatorname{Wirth}_H^G \langle G\times _H M\rangle ) \\
&= \ (b\wedge h\wedge S^L)_*( \langle M\rangle \wedge \tau _{H,L}) \\
&= \ (b\wedge h)_*\langle M\rangle \wedge (b\wedge S^L)_*(\tau _{H,L}) \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7ff2943a9659455956502ca4cafc3558436ad311 | subsection | 864 | 1,121 | Equivariant bordism | Then (\tilde{E}{\mathcal {P}})^G=S^0, consisting of the two cone points,
and (\tilde{E}{\mathcal {P}})^H is contractible for every proper subgroup of G.Proposition 2.32
The Thom-Pontryagin map\Theta ^G \ : \ \widetilde{\mathcal {N}}_*^G(\tilde{E}{\mathcal {P}})\ \longrightarrow \ {\mathbf {mO}}_*^G(\tilde{E}{\mathcal ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3e93b58839316d1393f2bbc27d252327cd31e129 | subsection | 865 | 1,121 | Equivariant bordism | The compositeS^{V^G}\ \xrightarrow{} ( T h(G r_{|V|-n}(V)) )^Gwith the projection to the j-th summand is then on the nose the mapS^{V^G}\ \xrightarrow{} \ T h(G r_{\dim (V^G)+j}(V^G))\wedge ( G r_j(V^\perp ))^G_+ \ ,the smash product of the collapse map for the non-equivariant manifold M^{(j)},
based on the embedding i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02566923",
"end": 1038,
"openalex_id": "https://openalex.org/W1989427081",
"raw": "R. Thom, Quelques propriétés globales des variétés différentiables. Comment. Math. Helv. 28 (1954), 17–86.",
"source_ref_id": "a8a88e3f6c8d61... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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982ec685a60dd881abacb79392cb9d781e459661 | subsection | 866 | 1,121 | Equivariant bordism | To show that \Theta ^G is an isomorphism,
we exploit that \mathcal {N}_*^G and {\mathbf {mO}}_*^G
are both equivariant homology theories and \Theta ^G
is a morphism of homology theories. This reduces the claim to
the special case X=G/H of an orbit for a closed subgroup H of G.
The argument for an orbit falls into two c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8272c0b9d37217290e1e2723115892e56c308e18 | subsection | 867 | 1,121 | Equivariant bordism | We then get compatible long exact isotropy separation sequences:@C=5mm{
\cdots [r] & \mathcal {N}_*^G(E{\mathcal {P}}) [r]^-{p_*} [d]_{\Theta ^G} &
\mathcal {N}_*^G(\ast ) [r]^-{i_*}[d]_{\Theta ^G} &
\widetilde{\mathcal {N}}_*^G(\tilde{E}{\mathcal {P}}) [r]^-{\partial } [d]^{\Theta ^G} &
\mathcal {N}_{*-1}^G(E{\mathcal... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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62dd5531e53dc0f0eba9718b188b32a21f6c5ac9 | subsection | 868 | 1,121 | Equivariant bordism | This shows that\operatorname{ind}_e^{C_2}(1) \ = \ 0 \text{\quad in \quad } \mathcal {N}_0^{C_2}\ ,where 1\in \mathcal {N}_0^e is the bordism class of a point.
The action on the class 1 thus factors over a morphism of restricted global functors{\mathbb {A}}^{\operatorname{res}} /\langle \operatorname{ind}_e^{C_2}\rangl... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1090/memo/0103",
"end": 1769,
"openalex_id": "https://openalex.org/W2075517255",
"raw": "R. E. Stong, Unoriented bordism and actions of finite groups. Mem. Amer. Math. Soc. 103 (1970), 80 pp.",
"source_ref_id": "5d2099d8d92285b911f... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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282b45dadee45a4399acf9a443c98a521d67cfe3 | subsection | 869 | 1,121 | Equivariant bordism | In finite and abelian compact Lie groups,
every subgroup inclusion with finite Weyl group is necessarily of finite index,
so for finite and abelian compact Lie groups, there is no difference in
the two kinds of quotients.
This is an independent verification of Theorem REF
in dimension 0. Moreover, we conclude that the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01187353",
"end": 1703,
"openalex_id": "https://openalex.org/W1987656475",
"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"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d9ffc01793ef9aa23e5dd488d70ddfdea334b398 | subsection | 870 | 1,121 | Equivariant bordism | More precisely, their definition comes down to\tilde{\mathfrak {N}}^{G:S}_m(X)\ = \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)}\, \widetilde{\mathcal {N}}_{m+|V|}^G(X\wedge S^V) \ ;for V\subset W, the structure map in the colimit system is the multiplication\widetilde{\mathcal {N}}_{m+|V|}^G(X\wedge S^V) \ \xrighta... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ff71421d34fdb2819be78bb318f5f1342648a51f | subsection | 871 | 1,121 | Equivariant bordism | Morally, the reason for this is that formally inverting the classes d_{G,V}
forces the Wirthmüller isomorphism to hold,
so in stable equivariant bordism this potential obstruction
to representability by a global homotopy type vanishes.Theorem 2.37
For every compact Lie group G and every cofibrant based G-space X,
the ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
9bf7c01e50072b797a494030e4886a872dac2a5a | subsection | 872 | 1,121 | Equivariant bordism | We let V be any G-representation,
and observe that the fixed point inclusion i:V^G\longrightarrow V
induces a G-homotopy equivalence\tilde{E}{\mathcal {P}}\wedge i \ : \ \tilde{E}{\mathcal {P}}\wedge S^{V^G} \ \longrightarrow \ \tilde{E}{\mathcal {P}}\wedge S^V \ .In the commutative diagram@C=13mm@R=7mm{
\widetilde{\ma... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.01... |
3b3b17dce59f4964b166c340f6877069227ef897 | subsection | 873 | 1,121 | Equivariant bordism | Hence the composite is an isomorphism, which provides an alternative proof
that stable equivariant bordism
agrees with equivariant {\mathbf {MO}}-homology, which is the main result
of the paper by Bröcker and Hook.
Strictly speaking there is a bit more work involved
in the translation, because our group {\mathbf {MO}}_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf01187353",
"end": 666,
"openalex_id": "https://openalex.org/W1987656475",
"raw": "T. Bröcker, E. C. Hook, Stable equivariant bordism. Math. Z. 129 (1972), 269–277.",
"source_ref_id": "808853a7d970e057cfa99f57bb70dbc2e523850a... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3da973e93e4c3bb1807a089aa6c928c6748123db | subsection | 874 | 1,121 | Connective global | In this section we define and discuss the ultra-commutative ring spectrum {\mathbf {ku}},
the connective global K-theory spectrum,
see Construction REF .
Our construction is an elaboration of a model of non-equivariant connective K-theory
by Segal ,
constructed from certain {\mathbf {\Gamma }}-spaces of
`orthogonal sub... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
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"raw": "G. Segal, K-homology theory and algebraic K-theory. K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), pp. 113–127. Lecture Notes in Mathematics, Vol. 575, Springer-Verl... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5603180f219fa46f665b1f5abbe2adeb3e16a1f1 | subsection | 875 | 1,121 | Connective global | For a based map \alpha :A\longrightarrow B the induced map
{{C}}({\mathcal {U}},\alpha ):{{C}}({\mathcal {U}},A)\longrightarrow {{C}}({\mathcal {U}},B) sends (E_a) to (F_b) whereF_b \ = \ {\bigoplus }_{\alpha (a)=b}\, E_a \ .Then {{C}}({\mathcal {U}}) is a {\mathbf {\Gamma }}-space whose underlying space is{{C}}({\math... | {
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"raw": "G. Segal, K-homology theory and algebraic K-theory. K-theory and operator algebras (Proc. Conf., Univ. Georgia, Athens, Ga., 1975), pp. 113–127. Lecture Notes in Mathematics, Vol. 575, Springer-Ver... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bd0fd1e2de76c7c0c8ec62b868c4eee6723ae481 | subsection | 876 | 1,121 | Connective global | For U\subset V the map in the colimit system is induced by
the \ast -homomorphism \operatorname{End}_{\mathbb {C}}(U)\longrightarrow \operatorname{End}_{\mathbb {C}}(V)
that extends an endomorphism by 0 on the orthogonal complement.
A homeomorphism{{C}}({\mathcal {U}},K)\ \longrightarrow \ \operatorname{colim}_{V\in s(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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47a4d594165066aa5bd9bad608fc01e8bb19adf8 | subsection | 877 | 1,121 | Connective global | As n varies, these maps are compatible
with the equivalence relation and so they assemble into a continuous map\index {subject}{eigenspace decomposition}
{{C}}({\mathcal {U}},S^1) \ = \ \int ^{n_+\in {\mathbf {\Gamma }}} {{C}}({\mathcal {U}},n_+) \times (S^1)^n\ &\longrightarrow \quad U({\mathcal {U}})\\
[E_1,\dots ,E_... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1ae422ae1659d4ac06de0c3c5f3db1e15c590e26 | subsection | 878 | 1,121 | Connective global | We explain the complex version; the real version
works in much the same way.
For a {\mathbb {C}}-vector space V we denote by\operatorname{Sym}^n(V) \ = \ V^{\otimes n}/\Sigma _nthe n-th symmetric power of V and by \operatorname{Sym}(V)=\bigoplus _{n\ge 0}\operatorname{Sym}^n(V)
the symmetric algebra of V.
If W is anoth... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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09896ff07fb90c164de065581caaa4dbf283dd90 | subsection | 879 | 1,121 | Connective global | We omit the straightforward verification that this inner product is indeed
given by the formula in the statement of the proposition,
and that it is natural for linear isometric embeddings.The algebra isomorphism (REF ) is the sum of the embeddings\operatorname{Sym}^m(V)\otimes \operatorname{Sym}^n(W)\quad &\longrightar... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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723498a0a1b5524c69885a31ed0c03a41ee7527d | subsection | 880 | 1,121 | Connective global | The last equation uses that (v,0) and (0,w) are orthogonal
in V\oplus W for all v\in V and w\in W.The case n=2 gives an idea of the induced inner product on \operatorname{Sym}^n(V):
if \lbrace e_1,\dots ,e_k\rbrace is an orthonormal basis of V, then the vectors1/\sqrt{2}\cdot e_i^2 \quad (1\le i\le k)\text{\qquad and\q... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ccdfe69224159dbb14d940a8f0eb43f43adc8c5d | subsection | 881 | 1,121 | Connective global | An O(V)-equivariant unit map is given by\iota _V \ : \ S^V\ \longrightarrow \ {{C}}(\operatorname{Sym}(V_{\mathbb {C}}),S^V) = {\mathbf {ku}}(V) \ , \quad v \ \longmapsto \ [{\mathbb {C}}\!\cdot \! 1;\, v]\ ,where {\mathbb {C}}\!\cdot \! 1 is the homogeneous summand of degree 0
in the symmetric algebra, i.e., the line ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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efb7a08c96a105813c6f11df8de943b3a58cdc0f | subsection | 882 | 1,121 | Connective global | As V varies, the maps \psi (V) form an automorphism
\psi :{\mathbf {ku}}\longrightarrow {\mathbf {ku}} of the ultra-commutative ring spectrum {\mathbf {ku}}.Remark 3.10 (Connective real global K-theory)
There is a straightforward real analog {\mathbf {ko}} of the
complex connective global K-theory spectrum {\mathbf {k... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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41645d26e121f6eade4c2892578b25ac07df3360 | subsection | 883 | 1,121 | Connective global | This means that {\mathbf {ko}} `is'
the \psi -fixed orthogonal ring subspectrum of {\mathbf {ku}}; more formally,
the complexification morphism (REF ) is an isomorphism
from {\mathbf {ko}} onto the \psi -fixed orthogonal ring subspectrum of {\mathbf {ku}}.The next proposition justifies the adjective `connective'
that w... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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aecba7326f4b3a6e026bd39473c8ba3ae3600671 | subsection | 884 | 1,121 | Connective global | We argue that the {\mathbf {\Gamma }}-G-space {{C}}({\mathcal {U}},-) is G-cofibrant.
The actions of G respectively \Sigma _n on an n-tuple of
orthogonal subspaces are componentwise respectively by permuting the entries, i.e.,(\sigma ,g)\cdot (E_1,\dots ,E_n) \ = \ (g\cdot E_{\sigma ^{-1}(1)},\dots ,g\cdot E_{\sigma ^{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cf5e9687d851da97597517ece5255e24857e6b5b | subsection | 885 | 1,121 | Connective global | The full configuration space {{C}}({\mathcal {U}},n_+)
decomposes as the disjoint union of (\Sigma _n\times G)-invariant subspaces
indexed by the \Sigma _n-orbits on {\mathbb {N}}^n,{{C}}({\mathcal {U}},n_+) \ = \ \coprod _{i \Sigma _n \in {\mathbb {N}}^n/\Sigma _n}\, \Sigma _n\times _{\Gamma _i} {{C}}({\mathcal {U}}; ... | {
"cite_spans": [
{
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"raw": "T. tom Dieck, Transformation Groups. De Gruyter Studies in Mathematics, 8. Walter de Gruyter & Co., Berlin, 1987. x+312 pp.",
"sou... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b8e23c444c049ba89e2f21e579dba21cc15f9551 | subsection | 886 | 1,121 | Connective global | So every based continuous G-map S^V\longrightarrow {\mathbf {ku}}({\mathbb {R}}^k\oplus V) is equivariantly
null-homotopic by ,
and the set [S^V,{\mathbf {ku}}({\mathbb {R}}^k\oplus V)]^G has only one element.
Passage to the colimit over V\in s({\mathcal {U}}_G) proves the claim.Every {\mathbb {C}}-linear isometric emb... | {
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"sour... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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274f7bf93bf5cb5ee5854ae2b5209f4d0456db41 | subsection | 887 | 1,121 | Connective global | When we evaluate F (or rather its prolongation)
at a G-space K, it comes with a (G\times G)-action;
one action comes from the `external' action on F, the other one from the
G-action on K via the continuous functoriality of F. In such a situation,
we always consider F(K) as a G-space
via the diagonal action of this (G\t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bc57dcdabb779ce09d3ab1f7cfda0db5507a1179 | subsection | 888 | 1,121 | Connective global | We define a morphism of (G\times \Sigma _n)-spaces\lambda _n\ : \ {{C}}({\mathcal {U}},1_+)^n \ \longrightarrow \ {{C}}({\mathbb {C}}^n\otimes {\mathcal {U}},n_+) \ ;here the \Sigma _n-action on the target is diagonally, from
the permutation action on n_+ and on the tensor factor {\mathbb {C}}^n.
The map \lambda _n sen... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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b8b0a6bfff6aeff5a4a3757603a6695d761635bd | subsection | 889 | 1,121 | Connective global | For 1\le j\le n, we define a 1-parameter family of unit vectorsu_j \ : \ [0,1] \ \longrightarrow \ {\mathbb {C}}^n
\text{\qquad by\qquad }
u_j(t)\ = \ t\cdot e_j + \sqrt{\frac{1-t^2}{n-1}}\cdot \sum _{k\ne j} e_k\ .This provides a homotopyH \ : \ {{C}}({\mathcal {U}},n_+)\times [0,1]\ \longrightarrow \ {{C}}({\mathbb {... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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05096eaca291e57375058891bca1acad5650f302 | subsection | 890 | 1,121 | Connective global | We let H be a closed subgroup of G,
\alpha :H\longrightarrow \Sigma _n a continuous homomorphism,
and \Gamma =\lbrace (h,\alpha (h)) \ | \ h\in H\rbrace \le G\times \Sigma _n
the graph of \alpha .
We let a_1,\dots , a_k\in \lbrace 1,\dots ,n\rbrace be a set of representatives
of the orbits of the H-action on \lbrace 1,... | {
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... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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04be2dc5b21eacadef93004af7ea7732d6680d8e | subsection | 891 | 1,121 | Connective global | We provide the additional arguments
in Theorem REF ;
to apply this, we need that the {\mathbf {\Gamma }}-G-space {{C}}({\mathcal {U}},-)
is special by part (i) and G-cofibrant by Example REF .The orthogonal spectrum {\mathbf {ku}} is trying to be a {{\mathcal {F}}in}-global \Omega -spectrum.
However, the global \Omega ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e21d35062b30c687b91ccadfb2fb19fa161694d8 | subsection | 892 | 1,121 | Connective global | This gives a G-equivariant isometric embedding
\operatorname{Sym}(W^\ast _{\mathbb {C}})\longrightarrow L^2(W;{\mathbb {C}}) with dense image.
So if there was any complex G-representation that did not embed
equivariantly into \operatorname{Sym}(W^\ast _{\mathbb {C}}), then it would also
not embed into L^2(W;{\mathbb {C... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ec4cd4896e8a73866f3530a2add48a0e0d5fc7df | subsection | 893 | 1,121 | Connective global | The second map is a G-weak equivalence by Theorem REF (ii).Now we will justify that for every finite group G
the underlying orthogonal G-spectrum of {\mathbf {ku}} represents
connective G-equivariant topological K-theory.Construction 3.22 We define a morphism of orthogonal spacesc \ : \ \mathbf {Gr}^{\mathbb {C}}\ \lo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.02... |
aba31cd57352311522a16457ce78edfdc4322459 | subsection | 894 | 1,121 | Connective global | So given an inner product space V we define\operatorname{eig}(V) \ : \ {\mathbf {U}}(V)\ = \ U(V_{\mathbb {C}}) \ \longrightarrow \ \operatorname{map}_*(S^V, {\mathbf {ku}}(V\oplus {\mathbb {R}}) )
= \left( \Omega ^\bullet (\operatorname{sh}{\mathbf {ku}}) \right)(V)by\operatorname{eig}(V)(A)(v) \ = \ [E(\lambda _1),\d... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.004998131189495325,
0.024647273123264313,
-0.022510666400194168,
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0.0012543019838631153,... |
8f1d2a90c54b7394bde0a0a98057bf1d36c3e88a | subsection | 895 | 1,121 | Connective global | We may thus show that the morphism\overline{\operatorname{eig}}\ : \ \overline{{\mathbf {U}}}\ \longrightarrow \ \Omega ^\bullet (\operatorname{sh}{\mathbf {ku}})is a {{\mathcal {F}}in}-global equivalence.
We show the stronger statement that for every finite group G and
every ample G-representation V the map
\overline{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.035835959017276764,
0.04047571122646332,
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-0.005288398824632168,
0... |
3500033fe3f6ff28c8abd0eac915b33594399d0f | subsection | 896 | 1,121 | Connective global | The morphism of orthogonal spaces c: \mathbf {Gr}^{\mathbb {C}}\longrightarrow \Omega ^\bullet {\mathbf {ku}}
defined in (REF ) is not a homomorphism of ultra-commutative monoids,
nor is it a loop map; so it is not a priori clear whether
the induced map on equivariant homotopy sets is a monoid homomorphism.Theorem 3.26... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0472089983522892,
-0.002452762331813574,
-0.0303486417979002,
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0.01546422764658928,
0.0... |
a145cfb1d6ebd8a1e2d87155bc3e8c99506d8e56 | subsection | 897 | 1,121 | Connective global | Under this isomorphism, the morphism\Omega ^\bullet \tilde{\lambda }_{{\mathbf {ku}}} \ : \ \Omega ^\bullet {\mathbf {ku}}\ \longrightarrow \Omega ^\bullet (\Omega (\operatorname{sh}{\mathbf {ku}}))becomes the morphism(\Omega ^\bullet {\mathbf {ku}})\circ i \ : \ \Omega ^\bullet {\mathbf {ku}}\ \longrightarrow \ \opera... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.03582978621125221,
0.018830476328730583,
-0.0181590486317873,
0.0035421589855104685,
-0.0034086366649717093,
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-0.0005297978641465306,
-0.023286309093236923,
-0.012062796391546726,
-0.005150150507688522,
... |
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