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583ea38b4741cafbbd2618cb17cbd0dafad44c44 | subsection | 398 | 1,121 | Equivariant orthogonal spectra | This identificationC Z\cup _{Z\times 1} C Z\ \cong \ Z\wedge ([0,1]\cup _{\lbrace 0,1\rbrace } [0,1] )turns the map p_Z into the mapZ\wedge (t\cup \ast )\ : \ Z\wedge ([0,1]\cup _{\lbrace 0,1\rbrace } [0,1] )\ \longrightarrow \ Z\wedge S^1 \ .So the claim follows from the fact that the map
t\cup \ast :[0,1]\cup _{\lbra... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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c46f8ce814feabe446e25917ac4593533a4c0c6c | subsection | 399 | 1,121 | Equivariant orthogonal spectra | Since the map p_Z\cup * is a G-homotopy equivalence by part (ii), this proves
that (f\wedge S^1)\circ h is G-homotopic to \beta \wedge S^1.Now we are ready to prove the long exact homotopy group sequences for
mapping cones and homotopy fibers.Proposition 1.34
long exact sequence!of equivariant homotopy groups
For ever... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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3a235bb59765c0c0cbda86970d49eda2f1343d1b | subsection | 400 | 1,121 | Equivariant orthogonal spectra | The compositeS^{V\oplus {\mathbb {R}}^{k+1}}\ \xrightarrow{}\ X(V)\wedge S^1 \ \xrightarrow{}\ X(V\oplus {\mathbb {R}})represents an equivariant homotopy class in \pi _k^G(X) and we have\pi _k^G(f)\langle \sigma _{V,{\mathbb {R}}}^{\operatorname{op}}\circ h\rangle \ &= \ \langle f(V\oplus {\mathbb {R}})\circ \sigma _{V... | {
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"Stefan Schwede"
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20109dc3122dec856044ca68798d8e024b295978 | subsection | 401 | 1,121 | Equivariant orthogonal spectra | Indeed, a homotopy inverser \ : \ X\wedge S^1 \ \longrightarrow \ C Y\cup _f C Xis defined by the formular(x\wedge s) \ = \ {\left\lbrace \begin{array}{ll}
\qquad (x,2s) \quad \in C X & \text{\ for $0\le s\le 1/2$, and}\\
(f(x),2-2s) \in C Y & \text{\ for $1/2\le s\le 1$,}
\end{array}\right.}which is to be interpreted ... | {
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91ec4c6851625cc8b74503d18e29f52e84158c2c | subsection | 402 | 1,121 | Equivariant orthogonal spectra | So for every based G-CW-complex A, the long sequence of based sets\cdots \ \longrightarrow \ [A,\Omega Y(V)]^G \ &\xrightarrow{} [A,F(f(V))]^G \\
&\xrightarrow{} \ [A,X(V)]^G \ \xrightarrow{} \ [A,Y(V)]^Gis exact. We take A=S^{V\oplus {\mathbb {R}}^k} and form the colimit over the poset s({\mathcal {U}}_G).
Since filte... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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d75f8e8a8c78b101caecc53888c33b19eab6df8a | subsection | 403 | 1,121 | Equivariant orthogonal spectra | The case of a finite index set I now follows by induction.In the general case we consider the composite{\bigoplus }_{i\in I}\, \pi ^G_k ( X_i )\ \longrightarrow \ \pi ^G_k\left( {\bigvee }_{i\in I}\, X_i \right) \ \longrightarrow \ {\prod }_{i\in I}\, \pi ^G_k ( X_i ) \ ,where the second map is induced by the projectio... | {
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"Stefan Schwede"
] | [
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c3fb2a10ff27c983e0708f2c301588d97c2f66e8 | subsection | 404 | 1,121 | Equivariant orthogonal spectra | We refer to Remark REF
below for more details.We recall that a morphism f:A\longrightarrow B of orthogonal G-spectra
is an h-cofibrationh-cofibration!of orthogonal spectra
if it has the homotopy extension property,
i.e., given a morphism of orthogonal G-spectra \varphi :B\longrightarrow X and a homotopy
H:A\wedge [0,1... | {
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6e15b4bbf85d8636e4318e8c0fc9dc52093b8d5e | subsection | 405 | 1,121 | Equivariant orthogonal spectra | We can thus define a modified connecting homomorphism
\partial :\pi ^G_{k+1}(Y)\longrightarrow \pi ^G_k (F) as the composite\pi ^G_{k+1}(Y) \ \xrightarrow{} \ \pi ^G_k (F(f))\ \ \xrightarrow{} \ \pi ^G_k (F)\ .So we deduce:Corollary 1.36
Let G be a compact Lie group.Let f:A\longrightarrow B be an h-cofibration of orth... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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eceb390f60c1c467fea301ccf6c1ebb7312c2426 | subsection | 406 | 1,121 | Equivariant orthogonal spectra | As a cobase change of the
h-cofibration f, the morphism k is again an h-cofibration,
so its long exact homotopy group sequence shows that \pi _*^G(k)
is an isomorphism.If g is an h-cofibration, then so is its cobase change h.
Moreover, any cokernel C/A of g maps by an isomorphism to
any cokernel D/B of h, since the squ... | {
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"Stefan Schwede"
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41563c986cad89b146cb5ae04dbab8eff0fa286e | subsection | 407 | 1,121 | Equivariant orthogonal spectra | Then the restriction map \operatorname{map}_*(A,X)\longrightarrow \operatorname{map}_*(B,X)
is a strong level fibration of orthogonal G-spectra whose fiber is
isomorphic to\operatorname{map}_*(A/B,X)\ \cong \ \operatorname{map}_*(G/H_+\wedge S^n,X)\ \cong \ \operatorname{map}_*(G/H_+,\Omega ^n X)\ .The G-equivariant st... | {
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"Stefan Schwede"
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1711cfa71f886a87e31a7fcadff672a4f70b9400 | subsection | 408 | 1,121 | Equivariant orthogonal spectra | Then the canonical morphism
f_\infty :Y_0\longrightarrow Y_\infty to a colimit of the sequence \lbrace f_n\rbrace _{n\ge 0}
is a {\underline{\pi }}_*-isomorphism.(i) We let V be a G-representation, and m\ge 0 such that m+k\ge 0.
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"Stefan Schwede"
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e9f94ab7503b48df7b5bca7f25f60c6ae214a64f | subsection | 409 | 1,121 | Equivariant orthogonal spectra | This is the orthogonal G-spectrum with V-term(\Sigma ^\infty A)(V) \ = \ S^V\wedge A\ ,with O(V)-action only on S^V, with G-action only on A,
and with structure map \sigma _{U,V} given by the canonical homeomorphism
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8197fa14b89fd9bcaf9c452d7958563a3d9c42f6 | subsection | 410 | 1,121 | Equivariant orthogonal spectra | An equivalent condition is that the map \pi _0(f):\pi _0(X)\longrightarrow \pi _0(Y) is surjective,
and for all x\in X the map
\pi _k(f):\pi _k(X,x)\longrightarrow \pi _k(Y,f(x)) is bijective for
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0a2f7e20050b6c653840bab68514a842eea68170 | subsection | 411 | 1,121 | Equivariant orthogonal spectra | Indeed, if any G-CW-structure for S^{V\oplus {\mathbb {R}}^{n+k}} contains
an equivariant cell of the form G/H\times D^j, then
(G/H)^H\times \mathring{D}^j embeds into S^{V^H\oplus {\mathbb {R}}^{n+k}}, and hence\dim (W_G H)+ j \ = \ \dim ((G/H)^H)+ j \ \le \ \dim (V^H)+ n+ k \ .The cellular dimension at H is the maxim... | {
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"Stefan Schwede"
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0b4deb74e312b9c327261d327b67d39ed1987e87 | subsection | 412 | 1,121 | Equivariant orthogonal spectra | Then evaluation at the tautological class is a bijection\operatorname{Nat}^{G{\mathcal {S}}p}(\pi _0^H,\Phi )
\ \longrightarrow \ \Phi (\Sigma ^\infty _+ G/H) \ , \quad \tau \ \longmapsto \ \tau (e_H)\ .To show that the evaluation map is injective
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"Stefan Schwede"
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a84816cd800b69b742a720175d1f9d4bd00bd108 | subsection | 413 | 1,121 | Equivariant orthogonal spectra | So\Phi (f^\sharp )(\tau (e_H)) \ = \ \tau (\pi _0^H(f^\sharp )(e_H)) \ = \ \tau (\pi _0^H(\tilde{\lambda }^U_X)(x))
\ = \ \Phi (\tilde{\lambda }^U_X)(\tau (x)) \ .Since \Phi (\tilde{\lambda }^U_X) is bijective, this proves that \tau
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"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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38964d163c0d93c476546ee35426c9e660969158 | subsection | 414 | 1,121 | Equivariant orthogonal spectra | Hence\Phi (\bar{f}^\sharp )\ = \ \Phi (K\circ i_0)\ = \ \Phi (K)\circ \Phi (i_0)\ = \ \Phi (K)\circ \Phi (i_1)\ = \ \Phi (K\circ i_1)\ = \ \Phi (f^\sharp )\ .This shows that \tau (x) does not change if we modify f by an H-homotopy.Now we let V be another G-representation and \varphi :U\longrightarrow V
a G-equivariant ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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ea518ffb5b7732ae6c3caa7990b0e6cc1e1a995a | subsection | 415 | 1,121 | Equivariant orthogonal spectra | Moreover, the following diagram of orthogonal G-spectra commutes:@C=15mm{
\Sigma ^\infty _+ G/H[r]^-{f^\sharp } [dr]_(.4){ (\psi (U)\circ f)^\sharp } &
\Omega ^U\operatorname{sh}^U X [d]^{\Omega ^U\operatorname{sh}^U \psi } &
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&\Omega ^U\operatorname{sh}^U Y & Y [l]^-{\tilde{\la... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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9df8676d445c75b1b9a66b23975111c39272b94c | subsection | 416 | 1,121 | The Wirthmüller isomorphism and transfers | This section establishes the Wirthmüller isomorphism
that relates the equivariant homotopy groups
of a spectrum over a subgroup to the equivariant homotopy groups
of the induced spectrum, see Theorem REF .
Intimately related to the Wirthmüller isomorphism are various transfers
that we discuss in Constructions REF and R... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a4995a19b14321071df4645743712ff3f896d979 | subsection | 417 | 1,121 | The Wirthmüller isomorphism and transfers | The subgroup H is the (\Sigma _2\ltimes H^2)-orbit of 1\in G, whose stabilizer
is the subgroup \Sigma _2\times \Delta , where \Delta = \lbrace (h,h) \ | \ h\in H\rbrace
is the diagonal subgroup of H^2.
The differential of the projection G\longrightarrow G/H identifies\nu \ = \ ( T_1 G) / ( T_1 H) \ ,the normal space a... | {
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9e713c4ac19bc4ef89f82169b3acd94b05a61c5c | subsection | 418 | 1,121 | The Wirthmüller isomorphism and transfers | This embedding is equivariant for the action of H^2 on the source by(h_1,h_2)\cdot (l, h)\ = \ (h_1 l,\, h_1 h h_2^{-1} )and for the action of \Sigma _2 on the source by\tau \cdot (l, h)\ = \ (-h^{-1} l,\, h^{-1} ) \ .The mapl_H^G \ : \ G \ \longrightarrow \ S^L \wedge H_+is then defined as the H^2-equivariant collapse... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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27a74ed55950bd13176b857170641f94e202e9a5 | subsection | 419 | 1,121 | The Wirthmüller isomorphism and transfers | Then the following triangle commutes up to H-equivariant based homotopy:@C=18mm{ B\wedge (G\ltimes _H A) [d]^\cong _{\text{\em shear}} [dr]^-{B\wedge l_A} &\\
G\ltimes _H(i^*B\wedge A) [r]_-{l_{i^* B\wedge A}} & B\wedge A\wedge S^L }We write down an explicit homotopy: we define the mapK \ : \ ( B\wedge (G\ltimes _H A) ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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74f2e55f8e37c43a211be41575d4ec15cb6e7b38 | subsection | 420 | 1,121 | The Wirthmüller isomorphism and transfers | With this diagonal G-action, (G\ltimes _H Y)(V)
is equivariantly homeomorphic to G\ltimes _H Y(i^* V)
(where H acts diagonally on Y(i^* V)), viaG\ltimes _H Y(i^* V)\ \cong \ (G\ltimes _H Y)(V)\ , \quad [g,y] \ \longmapsto \ [g, Y(l_g)(y)]\ .Under this isomorphism the structure map of the spectrum G\ltimes _H Y
becomes ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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54e94647a6973276a16e097a30f65bfa7309e471 | subsection | 421 | 1,121 | The Wirthmüller isomorphism and transfers | Still, Proposition REF shows
that the above diagram does commute up to based H-equivariant homotopy,
and this is good enough to yield a well-defined homomorphism(l_Y)_* \ : \ \pi ^H_k(G\ltimes _H Y) \ \longrightarrow \ \pi _k^H(Y\wedge S^L) \ .As we just explained, this is an abuse of notation, since (l_Y)_*
is in gene... | {
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5ba18ba36da9adce9def2bef3ac4504c6f4736fd | subsection | 422 | 1,121 | The Wirthmüller isomorphism and transfers | The associated collapse mapc \ : \ S^V \ \longrightarrow \ G\ltimes _H S^Wthen becomes the G-map defined byc(v) \ = \ {\left\lbrace \begin{array}{ll}
\left[g, \frac{w}{1-|w|} \right] &\text{if $v= g\cdot (v_0 +w)$ for some $(g,w)\in G\times D(W)$, and}\\
\quad \infty & \text{else.}
\end{array}\right.}With the collapse ... | {
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"Stefan Schwede"
] | [
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90c26ce57859046296a6a635328b6dceb8bd750d | subsection | 423 | 1,121 | The Wirthmüller isomorphism and transfers | If the composites
l_A\circ f, l_A\circ f^{\prime }:B\longrightarrow A\wedge S^L are H-equivariantly homotopic,
then the maps f\wedge S^V,f^{\prime }\wedge S^V:B\wedge S^V\longrightarrow (G\ltimes _H A)\wedge S^V
are G-equivariantly homotopic.(i)
The composite (l_H^G\wedge _H S^W)\circ c
is the collapse map based on the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d0819d52f4e0f4c0c4385030eb55638f28485392 | subsection | 424 | 1,121 | The Wirthmüller isomorphism and transfers | Moreover, for every t\in [0,1] the map K(-,-,t):D(L)\times D(W)\longrightarrow V
is a smooth equivariant embedding,
so it gives rise to a collapse map c_t:S^V\longrightarrow S^L\wedge S^W defined byc_t(v) \ = \ {\left\lbrace \begin{array}{ll}
\left( \frac{l}{1-|l|},\frac{w}{1-|w|} \right) &\text{if $v= K(l,w,t)$ for so... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bab792f6c019af7bb9b90fb029254b896c40e426 | subsection | 425 | 1,121 | The Wirthmüller isomorphism and transfers | Otherwise, v=j[g,w]=g\cdot (v_0+w) for some (g,w)\in G\times D(W), and thenr(\psi \wedge v) \ &= \ ((G\ltimes _H (\epsilon \wedge S^W))\circ \text{shear})( \psi \wedge [g,w/(1-|w|)]) \\
&= \ [g, \epsilon (g^{-1}\cdot \psi )\wedge w/(1-|w|)] \ = \ \left[g, \psi (g^{-1})\wedge \frac{w}{1-|w|}\right] \ .We denote by l_A^\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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14367fee02ddce373cc0cb7b80b521330dddfdbe | subsection | 426 | 1,121 | The Wirthmüller isomorphism and transfers | If a vector is of the form v=\zeta (l,w)=s(l)\cdot (v_0+w)
for some (l,w)\in D(L)\times D(W) (necessarily unique), then we setK( a\wedge v,t) \ =\ K( a\wedge (s(l)\cdot (v_0 +w)),\ t) \ =\ \left[ s(t\cdot l),\, a\wedge \frac{-l}{1-|l|}\wedge \frac{w}{1-|w|} \right] \ .For |l|=1 or |w|=1 this formula yields the basepoin... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e5a6bd0990dd2f4513b4d5e7c2ee864b12570646 | subsection | 427 | 1,121 | The Wirthmüller isomorphism and transfers | Since the square (REF ) commutes up to H-equivariant homotopy,
the compositeG\ltimes _H(A\wedge S^V)\ \xrightarrow{}\ \operatorname{map}^H(G,A\wedge S^L)\wedge S^V\ \xrightarrow{}\ G\ltimes _H(A\wedge S^L\wedge S^W)is G-equivariantly homotopic to the G-homeomorphism
G\ltimes _H ((S^{-\operatorname{Id}_L}\wedge S^W)\cir... | {
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"source_ref_id": "e93fc00e1caae8b3ca02f4532365a78bd5... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f348bd2a77a12ca67c6b8ff6b40c02c3ba0f75dc | subsection | 428 | 1,121 | The Wirthmüller isomorphism and transfers | We recall that\varepsilon _L\ : \ \pi _0^H(Y\wedge S^L) \ \longrightarrow \ \pi _0^H(Y\wedge S^L)denotes the effect of the involution of Y\wedge S^L
induced by the linear isometry -\operatorname{Id}_L:L\longrightarrow L given by multiplication by -1.Theorem 2.13 (Wirthmüller isomorphism) Wirthmüller isomorphism
Let H b... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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aff28d85b3ec10ff6b421bf7258b131f0a554ede | subsection | 429 | 1,121 | The Wirthmüller isomorphism and transfers | We contemplate the diagram of based H-maps:@C=20mm{
S^U\wedge S^V @/^1pc/[dr]^(.6){S^U\wedge S^\varphi }@<-6ex>@/_3pc/[ddd]
[d]_{S^U\wedge c} \\
S^U\wedge (G\ltimes _H S^W) [r]_-{S^U\wedge ( l_H^G\wedge _H S^W)}[d]_{\text{shear}}^\cong & S^U\wedge S^L\wedge S^W [d]^{S^U\wedge \tau _{L,W}} \\
G\ltimes _H ( S^U\wedge S^W... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4c71ac6b479da25db1fb0f23b9e57954406c09e1 | subsection | 430 | 1,121 | The Wirthmüller isomorphism and transfers | Upon expanding the definition (REF )
of f\diamond W, the diagram shows that
the class (l_Y)_*(\operatorname{res}^G_H(G\ltimes _H\langle f\rangle )) is also represented by the compositeS^{U\oplus V} \ &\xrightarrow{}\ Y(U)\wedge S^L\wedge S^L\wedge S^W \ \xrightarrow{}\ Y(U)\wedge S^L\wedge S^W\wedge S^L \\
& \xrightarr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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40063fc3fe8c32a82d7372dfa5bd9b47e791ffd9 | subsection | 431 | 1,121 | The Wirthmüller isomorphism and transfers | The maps remain G-homotopic if we furthermore postcompose
with the opposite structure
map \sigma ^{\operatorname{op}}_{U,V}:(G\ltimes _H Y)(U)\wedge S^V\longrightarrow (G\ltimes _H Y)(U\oplus V).
This shows that the stabilizations from the right of f and f^{\prime } by V
become G-homotopic.
Since such a stabilization r... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6613b950743d32fbe3456d8c013f3a99ca451d0f | subsection | 432 | 1,121 | The Wirthmüller isomorphism and transfers | This extension is a rather formal consequence of the fact that
the Wirthmüller maps
commute with the looploop isomorphism
and suspension isomorphismssuspension isomorphism\alpha \ : \ \pi ^G_k(\Omega X)\ \longrightarrow \ \pi ^G_{k+1} (X) \text{\quad and\quad }
-\wedge S^1 \ : \ \pi ^G_k (X)\ \longrightarrow \ \pi ^G_{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2afda5bdfce5d2c3515988c831400b13f70274f8 | subsection | 433 | 1,121 | The Wirthmüller isomorphism and transfers | The Wirthmüller map
\operatorname{Wirth}_H^G \ : \ \pi _k^G(G\ltimes _H Y) \ \longrightarrow \ \pi _k^H(Y\wedge S^L)
is an isomorphism for all k\in {\mathbb {Z}}.(i) The loop and suspension isomorphisms commute with restriction from G to H,
by direct inspection. So the proof comes down
to checking that the following ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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dd6d661dee5b827a95f1b97b70bca812864d52fa | subsection | 434 | 1,121 | The Wirthmüller isomorphism and transfers | The induction starts with k=0, where Theorem REF
provides the desired conclusion.
If k is positive, the compatibility of the Wirthmüller map with the loop isomorphism,
established in part (i), provides the inductive step.
If k is negative,
the compatibility of the Wirthmüller map with the suspension isomorphism,
also ... | {
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"Stefan Schwede"
] | [
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0.03210875391960144,
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0.005425219424068928,
0.005512969568371773,
0.039... |
458d6e4a7c7d198abdde22ec862ac15fee3ad776 | subsection | 435 | 1,121 | The Wirthmüller isomorphism and transfers | We argue by contradiction. If the claim were false,
we could find a compact Lie group G of minimal dimension for which it fails.
We let A be a G-CW-complex whose dimension n is minimal among all counterexamples.
Then A can be obtained from an (n-1)-dimensional subcomplex B
by attaching equivariant cells G/H_i\times D^n... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04095564782619476,
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0... |
86b336305570f3e97f9616258f48a414bc6dc1eb | subsection | 436 | 1,121 | The Wirthmüller isomorphism and transfers | An arbitrary cofibrant based G-space is G-homotopy equivalent
to a based G-CW-complex, so
the groups \pi _*^G(X\wedge A) vanish for all cofibrant A.Now we let K be an arbitrary closed subgroup of G.
The underlying K-space of A is again cofibrant
by Proposition REF (i),
so we can apply the previous reasoning to K inste... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.024717912077903748,
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... |
35b54c5089b01c88525fe5a1220a0065c3ec83f4 | subsection | 437 | 1,121 | The Wirthmüller isomorphism and transfers | Since the G-action on A is free (away from the basepoint),
all the homomorphisms \alpha _i that occur are injective.Since equivariant homotopy groups take wedges to sums, the
suspension isomorphism and the Wirthmüller
isomorphism allow us to rewrite the equivariant homotopy groups
of (A_n/A_{n-1})\wedge _K C as\pi _*^G... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0... |
7bd72020b114cd50185fc1c5bf3b97b3bb0ae760 | subsection | 438 | 1,121 | The Wirthmüller isomorphism and transfers | Then the induction functorG\ltimes _H - \ : \ H{\mathcal {S}}p\ \longrightarrow \ G{\mathcal {S}}ptakes {\underline{\pi }}_*-isomorphisms of orthogonal H-spectra to
{\underline{\pi }}_*-isomorphisms of orthogonal G-spectra.We let G\times H act on G by left and right translation, i.e., via(g,h)\cdot \gamma \ = \ g \gamm... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.09756876528263092,
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0.01... |
04dbc621607fcca037d9833996a1e59c06278dea | subsection | 439 | 1,121 | The Wirthmüller isomorphism and transfers | The external transfer is additive;
since the dimension shifting and degree zero transfers are obtained from
there by applying morphisms of equivariant spectra, they are also additive.The key properties of these transfer maps are:transfers are transitive (Proposition REF );
transfers commute with inflation maps (Propos... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.0... |
bb24123f247d4eabe8e121fb6dce99e80cbf87ea | subsection | 440 | 1,121 | The Wirthmüller isomorphism and transfers | We start with the compatibility with the loop and suspension isomorphisms.Proposition 2.25
Let H be a closed subgroup of a compact Lie group G,For every orthogonal H-spectrum Y and all k\in {\mathbb {Z}},
the following diagrams commute:
@C=15mm@R=7mm{
\pi _k^H( (\Omega Y)\wedge S^L) [r]^-{G\ltimes _H -}_-{\cong }
[d]... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05474364757537842,
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0... |
3805506c9aff6635f907e7aae2d340bfe91e6bdc | subsection | 441 | 1,121 | The Wirthmüller isomorphism and transfers | For every orthogonal G-spectrum X and all k\in {\mathbb {Z}},
the following diagrams commute:
@C=7mm@R=7mm{
\pi _k^H( (\Omega X)\wedge S^L) [r]^-{\operatorname{Tr}_H^G} [d]^\cong _{\text{\em assembly}}
&\pi _k^G(\Omega X) [dd]_\cong ^\alpha &
\pi _k^H(X\wedge S^L) [r]^-{\operatorname{Tr}_H^G}
[d]^\cong _{-\wedge S^1} ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
db1e10a51ca071961055301df5e34c3830826580 | subsection | 442 | 1,121 | The Wirthmüller isomorphism and transfers | We choose a slices\ : \ D( L ) \ \longrightarrow \ Gas in the construction of the map l_H^G:G\longrightarrow S^L\wedge H_+ in
Construction REF ; so s is a wide smooth embedding
of the unit disc of L satisfyings(0)\ =\ 1 \text{\qquad and\qquad } s(h\cdot l)\ = \ h\cdot s(l)\cdot h^{-1}for all (h,l)\in H\times D(L), and
... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0079-8169(08)x6007-6",
"end": 1222,
"openalex_id": "https://openalex.org/W1515066108",
"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"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.03807879239320755,
0.0... |
85e49d4425c9e45f859f168969776af313f7a81f | subsection | 443 | 1,121 | The Wirthmüller isomorphism and transfers | We choose a slice for the inclusion of K into H,
i.e., a wide smooth embedding \bar{s} : D( \bar{L} ) \longrightarrow H satisfying\bar{s}(0)\ =\ 1 \text{\qquad and\qquad }
\bar{s}(k\cdot \bar{l})\ = \ k\cdot \bar{s}(\bar{l})\cdot k^{-1}for all (k,\bar{l})\in K\times D(\bar{L}), and
such that the differential at 0\in D(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.024873152375221252,
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0.024766335263848305,
0.02... |
baab1a39fb03b661acdf71449ef6a9fb2732d34e | subsection | 444 | 1,121 | The Wirthmüller isomorphism and transfers | So we can – and will – define the map l_K^G:G\longrightarrow S^{L(K,G)}\wedge K_+
from the slices^{\prime } \ : \ D(L(K,G)) \ \xrightarrow[(\ref {eq:split_transitive_tangent})^{-1}]{\cong } \ D( L \oplus \bar{L})\ \xrightarrow{} \ G\ .The maps l_H^G:G\longrightarrow S^L\wedge H_+ respectively
l_K^H:H\longrightarrow S^{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.03708738088607788,
... |
54374dc0f2d36097affeaea8441c5efb9f79dd0d | subsection | 445 | 1,121 | The Wirthmüller isomorphism and transfers | A rescaling homotopy connects \Psi to the canonical homeomorphism
S^L\wedge S^{\bar{L}}\cong S^{L\oplus \bar{L}}, so the following square
commutes up to K^2-equivariant based homotopy:@C=12mm{
G_+ [d]_{l_K^G} [r]^-{l_H^G} &
S^L\wedge H_+[d]^{S^L\wedge l_K^H} \\
S^{L(K,G)}\wedge K_+ [r]_-\cong ^-{(\ref {eq:split_transit... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.005834877956658602,
0.012241803109645844,
0... |
820779a2bb1f18b9ea7ba995cb96950d5cfbe70e | subsection | 446 | 1,121 | The Wirthmüller isomorphism and transfers | Under the homeomorphism between S^{L(K,G)} and S^L\wedge S^{\bar{L}},
this becomes the smash product of the antipodal maps of L and \bar{L}.
So reading the diagram backwards gives a commutative diagram of external transfers\begin{aligned}
@C=7mm@R=7mm{
\pi _k^K(X\wedge S^L\wedge S^{\bar{L}}) [r]^-{H\ltimes _K-} [dd]_\c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.01345792505890131,
0.0034903630148619413,
... |
dbf39314ddeb16685c3a53601e680e89e709991f | subsection | 447 | 1,121 | The Wirthmüller isomorphism and transfers | We claim that then the following square commutes:@C=15mm{
\pi _*^K(X\wedge S^L)[r]^-{\operatorname{tr}_K^H}[d]_{G\ltimes _K -}^{\cong } &
\pi _*^H(X\wedge S^L)[d]^{G\ltimes _H -}_{\cong }\\
\pi _*^G(X\wedge G/K_+)[r]_-{ (X\wedge p_+)_*} &
\pi _*^G(X\wedge G/H_+)}To see this we compose the commutative
diagram (REF )
in ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.0702371671795845,
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0.02784162200987339,
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0.010503537952899933,
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0.04... |
214e9fe41fe6f9b4512d6e9b6a1292f0025e8c82 | subsection | 448 | 1,121 | The Wirthmüller isomorphism and transfers | One of them is the natural isomorphism
of K-spacesK\ltimes _J (\alpha |_J)^*(A) \ \cong \ \alpha ^*(G\ltimes _H A) \ , \quad [k, a]\ \longmapsto \ [\alpha (k), a]\ .Proposition 2.29
Let K and G be compact Lie groups and \alpha :K\longrightarrow G
a continuous epimorphism.
Let H be a closed subgroup of G,
set J=\alpha ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.07285533845424652,
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0.02190541662275791,... |
6c5c15058d3a128d1645cacf4fa11ace44a9154a | subsection | 449 | 1,121 | The Wirthmüller isomorphism and transfers | For every orthogonal G-spectrum X, every g\in G
and all closed subgroups K\le H of G the following diagram commutes:
{
\pi _k^{K^g}(X) [d]_{\operatorname{tr}_{K^g}^{H^g}} [r]^-{g_\star } & \pi _k^K(X) [d]^{\operatorname{tr}_K^H} \\
\pi _k^{H^g}(X) [r]_-{g_\star } & \pi _k^H(X) }(i) The restriction maps commute with th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05913328379392624,
0.022045303136110306,
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0.013791663572192192,
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0.0248829685151577,
0.... |
718fcb293a81976b69224f1b9f3a6f77a0098ce7 | subsection | 450 | 1,121 | The Wirthmüller isomorphism and transfers | The action map for the orthogonal K-spectrum K\ltimes _J \alpha ^*(X)
coincides with the compositeK\ltimes _J \alpha ^*(X)\ \xrightarrow{}\ \alpha ^*(G\ltimes _ H X)\ \xrightarrow{} \ \alpha ^*(X)\ .So the following diagram commutes by naturality of the restriction map:@C=15mm{
\pi _k^G(G\ltimes _H X) [d]_{\pi _k^G(\te... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.030406218022108078,
0.025051183998584747,
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0.02782786823809147,
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0.034906886518001556,
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0.... |
c626b873c2c9f6343b19decc85e3120c28a401b6 | subsection | 451 | 1,121 | Geometric fixed points | In this section we study the
geometric fixed point homotopy groups \Phi ^G_*(X)
of an orthogonal G-spectrum X.
We establish the isotropy separation sequence (REF )
that is often useful for inductive arguments,
and we prove that equivariant equivalences can
also be detected by geometric fixed points,
see Proposition REF... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.05031668394804001,
-0.018552180379629135,
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0.010130466893315315,
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0.020733891054987907,
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-0.02222905121743679,
0.0012882398441433907,
0.... |
3b5bb9e72548c71695cb025450b84395b52b6e8a | subsection | 452 | 1,121 | Geometric fixed points | The 0-th geometric fixed point homotopy groupgeometric fixed points fixed points!geometric
is defined as \Phi ^G_k - geometric fixed point homotopy group\Phi _0^G(X) \ = \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)}\, [S^{V^G}, X(V)^G] \ .If k is an arbitrary integer, we define the k-th
geometric fixed point homoto... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.04789867624640465,
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0.010594150051474571,
0.00... |
f5a0450ec575d26d48272db46beabb3b0dfc57f9 | subsection | 453 | 1,121 | Geometric fixed points | We choose a K-equivariant linear isometric embedding \psi :\alpha ^*({\mathcal {U}}_G)\longrightarrow {\mathcal {U}}_K
of the restriction along \alpha of the complete G-universe into
the complete K-universe. We let f:S^{V^G}\longrightarrow X(V)^G be a based map representing
an element of \Phi _0^G(X), for some V\in s({... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
-0.018020303919911385,
0.006828184239566326,
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0.02156027778983116,
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0.04116746783256531,
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0.016402900218963623,
0.01... |
fb826cc3dc12b6c30da8ea685bf9f59f8f7cba5f | subsection | 454 | 1,121 | Geometric fixed points | For every continuous epimorphism \alpha :K\longrightarrow G
of compact Lie groups the following square commutes:
{
\pi _0^G(X) [r]^-{\Phi ^G}[d]_{\alpha ^*} & \Phi _0^G(X) [d]^{\alpha ^*} \\
\pi _0^K(\alpha ^* X) [r]_-{\Phi ^K} & \Phi _0^K (\alpha ^* X)}(i) We choose a K-equivariant linear isometric embedding \psi :\a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.02270149439573288,
0.... |
a230331ea8282ac1fab5b59320b0c72e7780ccc6 | subsection | 455 | 1,121 | Geometric fixed points | Hence the map l_g^{X(V)}:c_g^*(X(V))\longrightarrow X(V)
is the composite of (c_g^* X)(l_g^V):(c_g^*X)(c_g^* V)\longrightarrow (c_g^* X)(V)
and (l_g^X)(V):(c_g^* X)(V)\longrightarrow X(V).
Since the restriction of l_g^{X(V)} to the G-fixed points is the
identity, the composite of ( (c_g^* X)(l_g^V))^G
and ((l_g^X)(V))^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fcc862b29f12c5788012e453a4fe55cd51aeb762 | subsection | 456 | 1,121 | Geometric fixed points | So combining inflation along c_g
with the effect of l_g^X gives a homomorphismconjugation homomorphism!on geometric fixed point homotopy groupsg_\star \ : \ \Phi _0^{H^g}(X)\ \xrightarrow{}\ \Phi _0^H(c_g^*(X))\ \xrightarrow{}\ \Phi _0^H(X)\ .In the special case when g normalizes H, this is a self-map
of the geometric ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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77f791d7d209ffecfa220bf39fcb525375a03239 | subsection | 457 | 1,121 | Geometric fixed points | We claim that the unit sphere S({\mathcal {U}}_G^\perp )
of this complement is a universal space E{\mathcal {P}}_G.
Indeed, the unit sphere S({\mathcal {U}}_G^\perp ) is G-equivariantly homeomorphic
to the space {\mathbf {L}}({\mathbb {R}},{\mathcal {U}}_G^\perp ), so it is cofibrant as a G-space
by Proposition REF (i... | {
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"raw": "T. Bröcker, T. tom Dieck, Representations of compact Lie groups. Graduate Texts in Math., Vol. 98, Springer-Verlag, New York, 1985. x+313 pp.... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fe403804be3ae3c944f1c421f4b4d8d6107c3254 | subsection | 458 | 1,121 | Geometric fixed points | If we compose the inverse with the geometric
fixed point homomorphism (REF ), we arrive at a homomorphism
\Phi :\pi _k^G(X\wedge \tilde{E}{\mathcal {P}}_G)\longrightarrow \Phi ^G_k(X).Proposition 3.7
For every orthogonal G-spectrum X and every integer k,
the geometric fixed point map\Phi \ : \ \pi _k^G (X\wedge \tilde... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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464ad87c1303b8b41aefa9d5362f6f70574c1d04 | subsection | 459 | 1,121 | Geometric fixed points | The argument in the other dimensions is similar, and
we leave it to the reader.A consequence of the previous proposition is the following
isotropy separation sequence.isotropy separation sequence
The mapping cone sequence of based G-space(E{\mathcal {P}}_G)_+\ \longrightarrow \ S^0 \ \longrightarrow \ \tilde{E}{\mathca... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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43b651f8e9f11bccdf0855b05b4fd7a06ef62a52 | subsection | 460 | 1,121 | Geometric fixed points | Since \Phi _*^G(f):\Phi _*^G(X)\longrightarrow \Phi _*^G(Y)
is also an isomorphism, the isotropy separation sequence
and the five lemma let us conclude that
\pi _*^G(f):\pi _*^G(X)\longrightarrow \pi _*^G(Y) is an isomorphism.The next proposition shows that `geometric fixed points vanish on transfers'.
In fact, it is o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d32f7fc4ec9859617cade2dc49809dc89216cd37 | subsection | 461 | 1,121 | Geometric fixed points | The dimension shifting transfer is defined as the composite\pi _0^H (X\wedge S^L) \ \xrightarrow{}\ \pi _0^G (G\ltimes _H X) \ \xrightarrow{}\ \pi _0^G (X)\ .The geometric fixed point map is natural for G-maps,
so the
composite \Phi ^K\circ \operatorname{res}^G_K\circ \text{act}_*:\pi _0^G (G\ltimes _H X) \longrightarr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9e07341ffb41d412aac30748ddc927ef7d432d70 | subsection | 462 | 1,121 | Geometric fixed points | Inspection of Construction REF
reveals that the transfer \operatorname{tr}_K^G(1) in \pi _0^G({\mathbb {S}})
is represented by the G-mapS^V \ \xrightarrow{} \ G\ltimes _K S^W\ \xrightarrow{} \ S^Vwhere c is the collapse map based on any wide embedding
of i:G/K\longrightarrow V into a G-representation,
W is the orthogo... | {
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"s... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8212c42027ccf5d7d729802bc49f4412ff76f9a8 | subsection | 463 | 1,121 | Geometric fixed points | On the other hand, the class 1 is invariant under the action of the Weyl group,
and hence\Phi ^K(\operatorname{res}^G_K(\operatorname{tr}^G_K(1)))\ = \ \lambda \cdot \sum _{ g K\in W_G K }\,
g_\star (\Phi ^K(1)) \ = \ \lambda \cdot |W_G K|\cdot \Phi ^K(1)\ .Since the abelian group \Phi _0^K({\mathbb {S}}) is freely gen... | {
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{
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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17492c442b83c84b84d3cafa1851fde92666290c | subsection | 464 | 1,121 | Geometric fixed points | Passing to the colimit over n proves the claim.If H is a closed subgroup of a compact Lie group G,
and Y is the underlying H-space of a G-space,
then the normalizer N_G H leaves Y^H invariant,
and the action of N_G H factors over an
action of the Weyl group W_G H=N_G H/H on Y^H.
This, in turn, induces an action of the ... | {
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"raw": "L. G. Lewis, Jr., J. P. May, M. Steinberger, Equivariant stable homotopy theory. Lecture Notes in Mathematics, Vol. 1213, Springer-Verlag, 1986. x+... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a6795cec5c09c1f7e5a9dba0d1a809face2b46b5 | subsection | 465 | 1,121 | Geometric fixed points | Let z\in \pi _0^G(\Sigma ^\infty _+ Y) be a class
such that for every closed subgroup K of G with finite Weyl group
the geometric fixed point class
\Phi ^K( \operatorname{res}^G_K(z))\ \in \ \Phi _0^K(\Sigma ^\infty _+ Y)
is trivial. Then z=0.(i)
In (REF ) we recalled property (i) when G is a trivial group.
For the t... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c4a5620e73bc66fa6dd9e98a2b41fdc4c4b735aa | subsection | 466 | 1,121 | Geometric fixed points | For any representation M of a finite group W
the norm mapN \ : \ M \ \longrightarrow \ M \ , \quad x\ \longmapsto \ \sum _{w\in W} w\cdot xfactors over the group of coinvariantsM_W\ = \ M / \langle x- w x\, |\, x\in M, w\in W\rangle \ .For the integral permutation representation M={\mathbb {Z}}[S] of a W-set S,
a speci... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f2fef2250f6e26d9ded2b92ffcc7b5b792549cc9 | subsection | 467 | 1,121 | Geometric fixed points | ThenT(z) \ = \ \operatorname{tr}_K^G(\text{incl}_*(p_K^*(y)))\ +\ {\sum }_{i=1}^m\, \operatorname{tr}_{H_i}^G(y_i)with y an element of \pi _0^e(\Sigma ^\infty _+ Y^K)
with non-zero image in the W_G K-coinvariants,
and with certain closed subgroups H_i of G
that are not conjugate to K and `no larger' in the sense that
e... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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33fc99a59c579f34edb8328f50b7ec14989658d2 | subsection | 468 | 1,121 | Geometric fixed points | Now we let G be a non-trivial compact Lie group.
We start with the special case Y=G/K_+
for a proper closed subgroup K of G.
The compositeG\ltimes _K{\mathbb {S}}\ \xrightarrow{}\ G\ltimes _K (\Sigma ^\infty _+ G/K) \ \xrightarrow{}\ \Sigma ^\infty _+ G/Kis an isomorphism of orthogonal G-spectra.
Hence the composite\pi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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77edf74951c3ac90cca35f0938dab51c3959eb17 | subsection | 469 | 1,121 | Geometric fixed points | Indeed, the following diagram commutes by the various naturality properties:@C=11mm@R=8mm{
\pi _0(\lbrace 0\rbrace )@<-4ex>@/_1pc/[ddd]_(.3){\sigma ^L} [rr]^-{\pi _0((e K)_*)}
[d]^{p_L^*\circ \sigma ^e} &&
\pi _0( (G/K)^L) [d]_-{p_L^*\circ \sigma ^e} @<0ex>@/^2pc/[ddr]^(.7){\sigma ^L} \\
\pi _0^L(\Sigma ^\infty S^0)[dd... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2e7cf334035b0b09ee95debd6e75b05b9366c3e2 | subsection | 470 | 1,121 | Geometric fixed points | If the Weyl group of L in the ambient group G happens to be infinite,
then \operatorname{tr}_L^G=0 and the generator is redundant.
Otherwise e K is an L-fixed point of G/K, so the generator is one
of the classes mentioned in the statement of (i).
This shows the generating property for the G-space G/K.Next we observe th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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029f981359c658c61d71c1bc82a9663c7b8022e6 | subsection | 471 | 1,121 | Geometric fixed points | The long exact isotropy separation sequence (REF )
thus decomposes into short exact sequences and the map(q_*,\,\text{incl}_*\circ p_G^*)\ : \ \pi _*^G(\Sigma ^\infty _+ (Y\times E{\mathcal {P}}_G) ) \oplus \pi _*^e(\Sigma ^\infty _+ Y^G)
\ \longrightarrow \ \pi _*^G(\Sigma ^\infty _+ Y)is an isomorphism, where q:Y\tim... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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de755ae48b034de1e0d814af0340d8b81d454539 | subsection | 472 | 1,121 | Geometric fixed points | So Theorem REF (i)
says that the map\psi _G \ : \ A(G)\ \longrightarrow \ \pi _0^G({\mathbb {S}})\ , \quad [G/H]\ \longmapsto \operatorname{tr}_H^G(1)is an isomorphism, a result that is originally due to Segal .
The Burnside rings for different groups are related by restriction
homomorphisms \alpha ^*:A(G)\longrightar... | {
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{
"arxiv_id": "",
"doi": "10.1007/bfb0075778",
"end": 212,
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"raw": "G. Segal, Equivariant stable homotopy theory. Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 59–63, 1971.",
"sour... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3b79cb56af047a9cb585eb051a9da84052c4f7eb | subsection | 473 | 1,121 | Geometric fixed points | Moreover, if z\in \pi _0^K({\mathbb {S}}) is represented by the K-map
f:S^V\longrightarrow S^V for some K-representation V, then\Phi ^K(z)\ = \ [f^K\ : \ S^{V^K}\longrightarrow S^{V^K}]\ = \ \deg (f^K)\cdot \Phi ^K(1)\ .So in terms of the basis \Phi ^K(1), the geometric fixed point
homomorphism \Phi ^K:\pi _0^K({\mathb... | {
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"raw": "T. tom Dieck, Transformation groups and representation theory. Lecture Notes in Mathematics, Vol. 766. Springer-Verlag, Berlin, 1979. viii+309 pp.",... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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01a18e4b9e69f289586955c5d10e5f2199d28367 | subsection | 474 | 1,121 | Geometric fixed points | The induced morphism of suspension spectra
\Sigma ^\infty _+ i:\Sigma ^\infty _+\lbrace y_0\rbrace \longrightarrow \Sigma ^\infty _+ Y
is then an h-cofibration of orthogonal G-spectra,
so it gives rise to a long exact sequence
of homotopy groups as in Corollary REF (i).
The cokernel of i_+:\lbrace y_0\rbrace _+\longri... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d40a80065022b7cab5b5535137fc9460c37fb276 | subsection | 475 | 1,121 | The double coset formula | The main aim of this section is to establish
the double coset formula for the composite of a transfer
followed by a restriction to a closed subgroup,
see Theorem REF below.
We also discuss various examples and special cases in
Examples REF through REF .
For finite groups (or more generally for transfers along finite in... | {
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b5b35a277fd18d7920eb73858f33ac97cb27a923 | subsection | 476 | 1,121 | The double coset formula | The next result determines the image of the class [c_B]
under the geometric fixed point map.In (REF )
we defined the map \sigma ^K:\pi _0(B^K)\longrightarrow \pi _0^K(\Sigma ^\infty _+ B) that
produces equivariant stable homotopy classes from fixed point information.Proposition 4.2
Let K be a compact Lie group.For eve... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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aa49a8ad212579fae5903362c97440bede2ac227 | subsection | 477 | 1,121 | The double coset formula | Then the compositeS^{V^K} \ \xrightarrow{} \ S^{V^K}\wedge B^K_+ \ \xrightarrow{}\ S^{V^K}\wedge M_+coincides withS^{V^K} \ \xrightarrow{} \ S^{V^K}\wedge M_+ \ ,the collapse map (REF )
based on the non-equivariant wide smooth embedding (i^K)|_M:M\longrightarrow V^K.
Since M is path connected, the group of based homoto... | {
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{
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"doi": "10.1016/0040-9383(75)90029-4",
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"raw": "J. C. Becker, D. H. Gottlieb, The transfer map and fiber bundles. Topology 14 (1975), 1–12.",
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"Stefan Schwede"
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37c53337e5f6fc6da23279b0a67f968bd23e972e | subsection | 478 | 1,121 | The double coset formula | On the other hand, if g\in G is such that g H\in M\subset (G/H)^K, then
K^g\le H and\sigma ^K [M] \ &= \ g_\star (\sigma ^{K^g}\langle e H\rangle )\ = \ g_\star ( \operatorname{res}^H_{K^g}(e_H))\ ;this proves the desired relation for the universal class e_H.Now we can proceed towards the double coset formula
for a tra... | {
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"Stefan Schwede"
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] | 2,018 | en | Mathematics | [
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323520f37d7fb1c7b25bda8d0d97624ab078a2f4 | subsection | 479 | 1,121 | The double coset formula | In particular, if the action happens to have only one orbit type
(i.e., all stabilizer groups are conjugate), then the quotient space K\backslash B
is a manifold and inherits a smooth structure from B.The terms in the double coset formula will be indexed by path components
of the orbit type orbit manifolds K\backslash ... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01301261",
"end": 615,
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"Stefan Schwede"
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e337c955de02a6b321fe738ee6089e0e9127386e | subsection | 480 | 1,121 | The double coset formula | Indeed since the integral homology groups of \bar{M} and \delta M
are finitely generated and vanish for almost all degrees,
the same is true for the relative singular homology groups H_*(\bar{M},\delta M;{\mathbb {Z}}),
and the internal Euler characteristic satisfies\chi ^\sharp (M)\ =\ \sum _{n\ge 0} (-1)^n\cdot \text... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1be1cc2e16c90da273c3403f2e336acbee2cb4b5 | subsection | 481 | 1,121 | The double coset formula | The double coset formula expresses the composite \operatorname{res}^G_K\circ \operatorname{tr}_H^G
as a sum of terms, indexed by all connected components M
of orbit type orbit manifolds K\backslash (G/H)_{(L)}.
The coefficient of the contribution of M is the
internal Euler characteristic \chi ^\sharp (M).Our next resul... | {
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"Stefan Schwede"
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d391eff29ccff564534d9e0fbc476c2074091872 | subsection | 482 | 1,121 | The double coset formula | The internal Euler characteristic is multiplicative on smooth fiber bundles
with closed fiber, so\chi ^\sharp (p^{-1}(M)\cap N)\ =\ \chi ^\sharp (M)\cdot \chi (W[M,N])\ .The internal Euler characteristic is additive on disjoint unions, so\chi ^\sharp ( N\cap B_{(J)} ) \ = \sum _{M\in \pi _0 ( K\backslash B_{(J)})}
\chi... | {
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"Stefan Schwede"
] | [
"math.AT"
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2df307c1ca67525edf682ba4d047c915f09bcfbe | subsection | 483 | 1,121 | The double coset formula | Then the relation[c_B] \ = \ \sum _{(J)\le K}\sum _{M\in \pi _0(K\backslash B_{(J)})}\ \ \chi ^\sharp (M)\cdot \operatorname{tr}_J^K (\sigma ^J \langle b_M\rangle )holds in the group \pi _0^K(\Sigma ^\infty _+ B).
Here the sum runs over all connected components M
of all orbit type orbit manifolds K\backslash B_{(J)},
a... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f84d0961938ae9181a3de6c710be97421c5bae95 | subsection | 484 | 1,121 | The double coset formula | The element k\in K is chosen so that k J\in W\subset (K/J)^L;
in particular this forces the relation L\le {^k J}\le {^k\text{stab}(b_M)},
and thus k b_M\in B^L.
The third equation uses that for every M,(K/J)^L \ = \ \coprod _{N\in \pi _0(B^L)}\, W[M,N] \ ,that the Euler characteristic is additive on disjoint unions,
an... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5e8f9d65501ba3232b737217be09a5d889912beb | subsection | 485 | 1,121 | The double coset formula | The internal Euler characteristic is
additive for such stratifications, i.e.,\chi ( N ) \ = \ \sum _{(J)\le K}\, \chi ^\sharp ( N\cap B_{(J)} ) \ .An application of Proposition REF (i) gives\Phi ^L(\operatorname{res}^K_L [c_B] ) \ &= \ \sum _{N\in \pi _0(B^L)} \chi (N)\cdot \Phi ^L(\sigma ^L[N]) \\
&= \ \sum _{(J)\le ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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96252c241cea85d475fd061467b3e978662fbd09 | subsection | 486 | 1,121 | The double coset formula | Then for every orthogonal G-spectrum X the relation\operatorname{res}^G_K\circ \operatorname{tr}_H^G \ = \ \sum _{M}\ \chi ^\sharp (M)\cdot \operatorname{tr}_{K\cap {^g H}}^K \circ g_\star \circ \operatorname{res}^H_{K^g\cap H}holds as homomorphisms \pi _0^H(X)\longrightarrow \pi _0^K(X).
Here the sum runs over all con... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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baaa931204ec2c4397d88e104166ef41960999a5 | subsection | 487 | 1,121 | The double coset formula | On the other hand,\sigma ^J \langle g_M H\rangle \ &= \ (g_M)_\star (\sigma ^{J^{g_M}}\langle e H\rangle )\\
&= \ (g_M)_\star ( \operatorname{res}^H_{J^{g_M}}(\sigma ^H\langle e H\rangle ))\ = \ (g_M)_\star ( \operatorname{res}^H_{K^{g_M}\cap H}(e_H))\ ,so this proves the double coset formula for the universal class e_... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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f9e418afbca7d9511160d9f17c728ed6452a21a2 | subsection | 488 | 1,121 | The double coset formula | Since the intersection K\cap {^g H} also has finite index in K,
only finite index transfers show up in the double coset formula,
which specializes to the relation\operatorname{res}^G_K\circ \operatorname{tr}_H^G \ = \ \sum _{[g]\in K\backslash G/H}\ \operatorname{tr}_{K\cap {^g H}}^K \circ g_\star \circ \operatorname{r... | {
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"source_ref_id": "44beee2b5c8d... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ef40fce3138b26184d925dd1fa5e9a76d4c4aca1 | subsection | 489 | 1,121 | The double coset formula | Altogether this specifies a homeomorphism
from the double coset space to the interval [0,1/2]
that sends a double coset K\cdot A\cdot H
to the minimum of |a_{11}|^2 and |a_{21}|^2.
The inverse homeomorphism isg \ : \ [0, 1/2]\ \cong \ K \backslash U(2) / H\ , \quad g(t)\ = \ K\cdot \begin{pmatrix} \sqrt{1-t} & \sqrt{t}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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eaa027bf8d71300004cb428095623b2054fdb998 | subsection | 490 | 1,121 | The double coset formula | We write U(k,n-k) for the subgroup of those elements of U(n)
that leave the subspaces {\mathbb {C}}^k\oplus 0
and 0\oplus {\mathbb {C}}^{n-k} invariant. We also use the analogous notation for more than two
factors. As subgroups we takeH\ =\ U(1,n-1) \text{\qquad and\qquad }
K \ = \ U(k,n-k)\ ,where 1\le k\le n-1.unitar... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d79901681b0944acd1def1566e1bd2fa0acdcd39 | subsection | 491 | 1,121 | The double coset formula | As representatives of the orbit types we can chooseg(0)= \begin{pmatrix}
0 & 0 & \cdots & 0 & 1\\
0 & 1 & \cdots & 0 & 0\\
\vdots & \vdots & \ddots & \vdots & \vdots \\
0 & 0 & \cdots & 1 & 0\\
-1 & 0 & \cdots & 0 & 0\end{pmatrix}
\ , \quad g(1/2)= \frac{1}{\sqrt{2}}\begin{pmatrix}
1 & 0 & \cdots & 0 & 1\\
0 & 1 & \cdo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1e2c4a98566742fde7b80ddee44254b7e50a908b | subsection | 492 | 1,121 | The double coset formula | A G-Mackey functorG-Mackey functor@G-Mackey functor|seeMackey functor
M consists of the following data:an abelian group M(H) for every subgroup H of G,
conjugation homomorphisms g_\star :M(H^g)\longrightarrow M(H) for all H\le G
and g\in G,conjugation homomorphism!in a Mackey functor
restriction homomorphisms \operat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0.01669112779200077,
0.03... |
e2d0392c268f3b94e8735cd7fb2d3a4bf02b32fc | subsection | 493 | 1,121 | The double coset formula | (Double coset formula)
for every pair of subgroups K, L of H the relation
\operatorname{res}^H_L\circ \operatorname{tr}_K^H \ = \ \sum _{[h]\in L\backslash H/K}
\ \operatorname{tr}^L_{L\cap ^h K}\circ h_\star \circ \operatorname{res}^K_{L^h\cap K}
holds as maps M(K)\longrightarrow M(L);
here [h] runs over a set of re... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1020,
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"raw": "A. Dress, Notes on the theory of representations of finite groups. Duplicated notes, Bielefeld, 1971.",
"source_ref_id": "bc83b2fca4f6da54156b60d16e5f60a79afd0032",
"start": 873
},
... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bd07387b0c86e34607c04ebdd5adab48df5b35f5 | subsection | 494 | 1,121 | The double coset formula | This should serve as motivation for the
following algebraic interlude about Mackey functors for finite groups,
where we study the process of `dividing out transfers' systematically.Construction 4.15 We let G be a finite group and M a G-Mackey functor.
For a subgroup H of G we let t_H M be the subgroup of M(H)
generated... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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14de41cc03d27d867b68a29778b72b92e799096f | subsection | 495 | 1,121 | The double coset formula | Moreover, if we choose representatives of the conjugacy
classes of subgroups, then projection from the full product
(over all subgroups of G) to the product indexed by the representatives restricts
to an isomorphism\left( {\prod }_{H\le G}\, \tau _H M\right)^G \ \xrightarrow{}\ {\prod }_{(H)}\, \left(\tau _H M\right)^{... | {
"cite_spans": [
{
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"doi": "10.1515/crll.1988.384.24",
"end": 774,
"openalex_id": "https://openalex.org/W1917296184",
"raw": "J. Thévenaz, Some remarks on G-functors and the Brauer morphism. J. Reine Angew. Math. 384 (1988), 24–56.",
"source_ref_id": "9be4ad9590... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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... |
49d42a3d0b41137842ed7385aaefcfcace0fd81a | subsection | 496 | 1,121 | The double coset formula | So the kernel of \psi ^M_G is precisely the
subgroup of those elements of M(G) that restrict to degenerate
elements on all subgroups.We write any given element x\in K_j asx \ = \ \operatorname{tr}_{H_j}^G(y) \ + \ \bar{x}for suitable y\in M(H_j) and with \bar{x} a sum of transfers from the groups
H_1,\dots ,H_{j-1}.
Fo... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cb887137920bf010091efca9ee8c22651d2575dd | subsection | 497 | 1,121 | The double coset formula | We let I_j denote the subgroup of \prod _{i=1}^n (\tau _{H_i} M)^{W_G H_i}
consisting of those tuplesx \ = \ (x_i)_{1\le i\le n}such that x_{j+1}=x_{j+2}=\dots =x_n=0.
This defines a nested sequence0 \ = \ I_0\ \ \subseteq \ I_1\ \ \subseteq \ I_2\ \ \subseteq \ \dots \ I_{n-1}\ \ \subseteq \ I_n \ = \ \prod _{i=1,\dot... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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... |
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