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1316199ee120da998cb6c0795d05652b3b4cc018 | subsection | 198 | 1,121 | Global power monoids | For every n-tuple G_1,\dots , G_n
of compact Lie groups the functor\operatorname{Ho}(umon)\ \longrightarrow \ \text{(sets)}\ ,\quad X \ \longmapsto \ \pi _0^{G_1}(X)\times \dots \times \pi _0^{G_n}(X)is represented by the free ultra-commutative monoid
{\mathbb {P}}(B_{\operatorname{gl}} G_1 \amalg \ldots \amalg B_{\ope... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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cd9c24443ebbe1fa2ca543bf04eb4fe53342634a | subsection | 199 | 1,121 | Global power monoids | Because{\mathbb {P}}(B_{\operatorname{gl}} G_1 \amalg \ldots \amalg B_{\operatorname{gl}} G_n)\ &\cong \ {\mathbb {P}}(B_{\operatorname{gl}} G_1)\boxtimes \dots \boxtimes {\mathbb {P}}( B_{\operatorname{gl}} G_n)\\
&\cong \ {\coprod }_{j_1,\dots ,j_n\ge 0}\,
B_{\operatorname{gl}} (\Sigma _{j_1}\wr G_1)\boxtimes \dots \... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6342ecdaef3cda8a1449914c7e71937db90a11a9 | subsection | 200 | 1,121 | Global power monoids | We define an isomorphic algebraic category \mathbb {A}^+,
the effective Burnside category.effective Burnside category
Both \operatorname{Nat}^{umon}and \mathbb {A}^+ are
`pre-preadditive' in the sense that all morphism sets are abelian monoids
and composition is biadditive. In \operatorname{Nat}^{umon}, the monoid stru... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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eb57652ce897a09b3810145d8a5d64e1cbffbfb3 | subsection | 201 | 1,121 | Global power monoids | The map(\lbrace 1,\dots ,k\rbrace \times G) \ \amalg \ (\lbrace 1,\dots ,m\rbrace \times G) \ \longrightarrow \ \Phi _{k,m}^*(\lbrace 1,\dots ,k+m\rbrace \times G)that is the inclusion on the first summand and given by
(j,g)\mapsto (k+j,g) on the second summand
is an isomorphism of ((\Sigma _k\wr G)\times (\Sigma _m\wr... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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64d1375f483a42454af117140ac0eb7312d5a370 | subsection | 202 | 1,121 | Global power monoids | Since the isomorphism class of _G G_G is the
identity of G in \mathbb {A}^+, the construction B preserves identities.For the compatibility of B with composition we consider another
operation \beta ^*\circ [k]\in \operatorname{Nat}^{umon}(K,M) and observe that(\beta ^*\circ [k])\circ (\alpha ^*\circ [m])\ = \ (\Psi _{k,... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8c4a7bca4dfb26ed877a283b5988818dd6a33261 | subsection | 203 | 1,121 | Global power monoids | Moreover, a K-G-isomorphism\alpha ^*(\lbrace 1,\dots ,m\rbrace \times G)_G\ \cong \ \beta ^*(\lbrace 1,\dots ,m\rbrace \times G)_Gis given by the action of a unique element \omega \in \Sigma _m\wr G,
and then the homomorphisms \alpha ,\beta :K\longrightarrow \Sigma _m\wr G
are conjugate by \omega . So the functor B is ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4f693d5552407a270d24a330c192e1457628fc33 | subsection | 204 | 1,121 | Global power monoids | Since inner automorphisms
induce the identity in any Rep-functor, we conclude that\Psi _{\bar{g}}^* = \Psi _{\bar{g}\omega }^* \ : \ M(\Sigma _m\wr H) \ \longrightarrow \ M(G)\ .So the transfer \operatorname{tr}_H^G does not depend on the choice of basis \bar{g}.The various properties of the power operations
imply cert... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5aecb8ea2e51466a5207c38ee3598534c2d5cc6d | subsection | 205 | 1,121 | Global power monoids | Then\bar{f}\bar{g}\ = \ (f_1 g_1,\dots ,f_1 g_m,\ f_2 g_1,\dots , f_2 g_m,
\ \dots ,\ f_k g_1,\dots ,f_k g_m)is an H-basis of F. With respect to this basis, the homomorphism
\Psi _{\bar{f}\bar{g}}:F\longrightarrow \Sigma _{k m}\wr H equals the compositeF\ \xrightarrow{} \Sigma _k\wr G
\ \xrightarrow{} \ \Sigma _k\wr (\... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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2b77edfed90c8576271f86548e990fda2369b381 | subsection | 206 | 1,121 | Global power monoids | Then s_1+\dots +s_r=m=[G:H] is the index of H in G,
and this data provides an H-basis of G, namely\bar{g} \ = \ (k^1_1 g_1,\,\dots ,k^1_{s_i}g_1,\,
k^2_1 g_2,\,\dots ,k^2_{s_2}g_2,\,\dots ,
k^r_1 g_r,\,\dots ,k^r_{s_r}g_r)\ .The following diagram of group homomorphisms then commutes:@C=10mm{
K [r]^-{\text{incl}} [d]_{(... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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37818c7c13ba4c2592f8b8ba6dce7068a0a3b116 | subsection | 207 | 1,121 | Global power monoids | From here the double coset formula is straightforward:\operatorname{res}^G_K\circ \operatorname{tr}_H^G \ &= \ \operatorname{res}^G_K\circ \Psi _{\bar{g}}^*\circ [m] \\
&= \ ((\Sigma _{s_1}\wr c_{g_1})\circ \Psi _{\bar{k}^1},\dots ,(\Sigma _{s_r}\wr c_{g_r})\circ \Psi _{\bar{k}^r})^*\circ \Phi _{s_1,\dots ,s_r}^*\circ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8c16f64363a86bf2cf219557308fb3a6b86befc0 | subsection | 208 | 1,121 | Global power monoids | With respect to these bases we have\Psi _{\alpha (\bar{k})}\circ \alpha \ = \ (\Sigma _m\wr (\alpha |_L)) \circ \Psi _{\bar{k}}
\ : \ K \ \longrightarrow \ \Sigma _m\wr H \ .So\alpha ^*\circ \operatorname{tr}_H^G \ = \ \alpha ^*\circ \Psi ^*_{\alpha (\bar{k})} \circ [m]
\ &= \ \Psi _{\bar{k}}^*\circ (\Sigma _m\wr (\alp... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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54e9b03eaa6198c59fd302a9ac99b5e3e27a02d9 | subsection | 209 | 1,121 | Global power monoids | This means that(\sigma ;\,g_1,\dots ,g_m)\cdot (\tau _1,\dots ,\tau _m) \ = \ (\tau _1,\dots ,\tau _m) \cdot (\sigma ;\,l_1,\dots ,l_m) \ ,and hence\Psi _{\bar{\tau }}(\sigma ;\,g_1,\dots ,g_m) \ = \ (\sigma ;\,l_1,\dots ,l_m) \ .Because(\Sigma _m\wr q)(\Psi _{\bar{\tau }}(\sigma ;\,g_1,\dots ,g_m))\ = \ (\Sigma _m\wr ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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41a4f009f28b1b8d9487a50c49932998be828ebc | subsection | 210 | 1,121 | Global power monoids | Moreover, K\times _{\alpha } G is isomorphic to
K\times _{\alpha ^{\prime }} G if and only if (L,\alpha ) is conjugate to
(L^{\prime },\alpha ^{\prime }) by an element of K\times G.
So \mathbb {A}^+(G,K) is freely generated by the classes of the
K-G-spaces K\times _{\alpha } G, where (L,\alpha )
runs through the (K\tim... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3d8f66a7358c76717d8a2c5634cbd3038bb687cf | subsection | 211 | 1,121 | Examples | In this section we discuss various examples of ultra-commutative monoids,
mostly of a geometric nature, and several geometrically defined morphisms between them.
We start with the ultra-commutative
monoids {\mathbf {O}} and {\mathbf {SO}} (Example REF ),
{\mathbf {U}} and {\mathbf {SU}} (Example REF ),
\mathbf {Sp} (Ex... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1ca1ee81bca69bdfb26e91095160e8298f58636b | subsection | 212 | 1,121 | Examples | In the cases of the orthogonal, special orthogonal,
unitary, special unitary, spin, spin^c and symplectic groups,
these multiplications are symmetric, so those examples yield ultra-commutative monoids.Definition 3.2
A monoid valued orthogonal spaceorthogonal space!monoid valued
is a monoid object in the category of or... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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92a396e8334f495828b598e672621db047a1e1f4 | subsection | 213 | 1,121 | Examples | Then the multiplication (REF )
makes M into an ultra-commutative monoid.Example 3.6 (Orthogonal group ultra-commutative monoid)
orthogonal group ultra-commutative monoid
We denote by {\mathbf {O}}{\mathbf {O}} - ultra-commutative monoid of orthogonal groups
the orthogonal space that sends an inner product space V
to i... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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1d359e649f3134cbefbc365a3c09c6d2fb063b9f | subsection | 214 | 1,121 | Examples | If \lambda is an irreducible orthogonal G-representation, then
the endomorphism ring \operatorname{Hom}^G_{{\mathbb {R}}}(\lambda ,\lambda ) is a finite-dimensional
skew field extension of {\mathbb {R}}, so it is isomorphic to either {\mathbb {R}}, {\mathbb {C}}
or {\mathbb {H}}; the representation \lambda is according... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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367221e542a1e62159608a735fa5191408af03b5 | subsection | 215 | 1,121 | Examples | We now define an orthogonal space {\mathbf {U}} by{\mathbf {U}}(V)\ = \ U(V_{\mathbb {C}}) \ ,the unitary group of the complexification of V.unitary group
The complexification of every {\mathbb {R}}-linear isometric embedding \varphi :V\longrightarrow W
preserves the hermitian inner products,
so we can define a continu... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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f4a11a98d84bcecaa70ddd6a4f93398d9e465660 | subsection | 216 | 1,121 | Examples | So in the unitary context, we get a decomposition{\mathbf {U}}({\mathcal {U}}_G)^G\ = \ U^G({\mathcal {U}}_G^{\mathbb {C}}) \ = \ {\prod }^{\prime }_{[\lambda ]}\ U^G({\mathcal {U}}_\lambda ^{\mathbb {C}})\ \ \cong \ {\prod }^{\prime }_{[\lambda ]}\ U \ .This weak product is indexed by the isomorphism classes of irredu... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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675063cbc8e03bb5e4a0eb77752c0785e4ac10d3 | subsection | 217 | 1,121 | Examples | We can thus define
the realification morphismrealification morphism!from {\mathbf {U}} to {\mathbf {O}}r \ : \ {\mathbf {U}}\ \longrightarrow \ \operatorname{sh}^\otimes _{\mathbb {C}}( {\mathbf {SO}})at V as the inclusion
r(V): U(V_{\mathbb {C}})\longrightarrow O({\mathbb {C}}\otimes V).
Here \operatorname{sh}^\otimes... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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a98be4a05fd9c8ebff5142813bef1887809d8eb2 | subsection | 218 | 1,121 | Examples | Quaternionic representations decompose canonically into isotypical
summands, and this results in a product decomposition for the
G-fixed subgroup\mathbf {Sp}({\mathcal {U}}_G)^G\ = \ ( Sp({\mathcal {U}}_G^{\mathbb {H}}))^G \ = \ {\prod }^{\prime }_{[\lambda ]}\, (Sp({\mathcal {U}}_\lambda ^{\mathbb {H}}))^G\ ,where the... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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3d08e91073b681edd3140519e6773f98c8c1099c | subsection | 219 | 1,121 | Examples | The compositeV \ \xrightarrow{}\ T V \ \xrightarrow{} \ \operatorname{Cl}(V)is {\mathbb {R}}-linear and injective, and we denote it by v\mapsto [v].We recall that orthogonal vectors of V anti-commute in the Clifford algebra:
given v, \bar{v}\in V with \langle v,\bar{v}\rangle =0, then[v][\bar{v}] + [\bar{v}][ v] \ = \ ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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c339a089e64b586e36057882aebbe96d11e55e80 | subsection | 220 | 1,121 | Examples | The map \operatorname{Pin}(\varphi ) induced by a linear isometric
embedding \varphi :V\longrightarrow W is homogeneous, so it restricts to a homomorphism\operatorname{Spin}(\varphi )\ = \ \operatorname{Pin}(\varphi )|_{\operatorname{Spin}(V)} \ : \ \operatorname{Spin}(V)\ \longrightarrow \ \operatorname{Spin}(W)betwee... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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"math.AT"
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298a1ea6fad2c00044ee6e3e4c976f3807e11825 | subsection | 221 | 1,121 | Examples | The pin^c grouppin^c group of V is the subgroup\operatorname{Pin}^c(V) \ \subset \ ({\mathbb {C}}\otimes _{\mathbb {R}}\operatorname{Cl}(V))^\timesgenerated inside the multiplicative group
by the unit scalars \lambda \otimes 1 for all \lambda \in U(1)
and the elements 1\otimes [v] for all unit vectors v\in S(V).
The pi... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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175a7b0317d4c46419dba6784e52edd2165fe8e2 | subsection | 222 | 1,121 | Examples | As V varies, the spin^c groups from a group valued orthogonal subspace {\mathbf {Spin}}^c
of {\mathbf {Pin}}^c.spin^c group ultra-commutative monoid{\mathbf {Spin}}^c - ultra-commutative monoid of spin^c groups
As for {\mathbf {Spin}}, the images of the homomorphisms
\operatorname{Spin}^c(i_V) and \operatorname{Spin}^c... | {
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"Stefan Schwede"
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3027a90a77aa9702fa24c6a4500a120c2056a59b | subsection | 223 | 1,121 | Examples | For every element x\in \operatorname{Pin}^c(V) the twisted conjugation mapc_x \ : \ {\mathbb {C}}\otimes _{\mathbb {R}}\operatorname{Cl}(V)\ \longrightarrow \ {\mathbb {C}}\otimes _{\mathbb {R}}\operatorname{Cl}(V) \ , \quad c_x(y)\ = \ \alpha (x)y x^{-1}is an automorphism of {\mathbb {Z}}/2-graded {\mathbb {C}}-algebr... | {
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739b534e19819226826a0f03e6e12f4f1bdea0d3 | subsection | 224 | 1,121 | Examples | Clifford algebra|)There is yet another interesting morphism of group valued orthogonal spacesl \ : \ {\mathbf {U}}\ \longrightarrow \ \operatorname{sh}^\otimes _{\mathbb {C}}({\mathbf {Spin}}^c)that lifts the forgetful realification morphism (REF )
through\operatorname{sh}^\otimes _{\mathbb {C}}(\operatorname{ad})\ :\ ... | {
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7bf90c0a65ac444119a97d574659a29a7649efe6 | subsection | 225 | 1,121 | Examples | The structure map induced by a linear isometric embedding \varphi :V\longrightarrow W
is given by \mathbf {Gr}(\varphi )(L) = \varphi (L).
A commutative multiplication on \mathbf {Gr} is given by direct sum:\mu _{V,W}\ : \ \mathbf {Gr}( V ) \times \mathbf {Gr}( W ) \ \longrightarrow \ \mathbf {Gr}( V\oplus W) \ ,\quad ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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4c7ac2b4e7ceca9266e8cab85eba311686f5b095 | subsection | 226 | 1,121 | Examples | We define a map\mathbf {Gr}(V)^G \ = \ {\coprod }_{m\ge 0} \, \left( Gr_m(V) \right)^G
\ \longrightarrow \ \mathbf {RO}^+(G)from this fixed point space to the monoid of isomorphism classes
of G-representations
by sending L \in \mathbf {Gr}(V)^G to its isomorphism class.
The isomorphism class of L only depends on the pa... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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29465e931f68c03f1c0e1821f1835321825d7e2c | subsection | 227 | 1,121 | Examples | If \lambda is any irreducible orthogonal G-representation,
then \pi _0^G(\tau ) sends its class to the automorphism -\operatorname{Id}_\lambda .
The group {\mathbf {O}}(\lambda )^G is isomorphic to O(1), U(1)
or Sp(1) depending on whether \lambda is of real, complex or
quaternionic type. In the real case, the map -\ope... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d55625f7b24c2ccd9f5a75bd4e0e2ffb7ff73da1 | subsection | 228 | 1,121 | Examples | A multiplication on \mathbf {Gr}^{\operatorname{or}} is given by direct sum:\mu _{V,W}\ : \ \mathbf {Gr}^{\operatorname{or}}( V ) \times \mathbf {Gr}^{\operatorname{or}} ( W ) \ &\longrightarrow \ \mathbf {Gr}^{\operatorname{or}}( V\oplus W) \\
((L,[b_1,\dots ,b_m]),\, (L^{\prime },[b_1^{\prime },\dots ,b_n^{\prime }])... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ebfc90e8b4737e1b7ada40ca5d16752cb129aa0c | subsection | 229 | 1,121 | Examples | Moreover, the forgetful map \mathbf {Gr}^{\operatorname{or},\operatorname{ev}}\longrightarrow \mathbf {Gr}
to the additive Grassmannian is a homomorphism
of ultra-commutative monoids.Example 3.16 (Complex and quaternionic Grassmannians)
The complex additive Grassmannian \mathbf {Gr}^{\mathbb {C}}additive Grassmannian!... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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da25237f02f5dc6e105fe03cc290b920494de43b | subsection | 230 | 1,121 | Examples | This orthogonal space is isomorphic to \mathbf {Gr}^{{\mathbb {C}},[m]} via{\mathbf {L}}^{\mathbb {C}}({\mathbb {C}}^m, V_{\mathbb {C}}) / U(m) \ \cong \ \mathbf {Gr}^{{\mathbb {C}},[m]}(V)\ ,\quad \varphi \cdot U(m)\ \longmapsto \ \varphi ({\mathbb {C}}^m)\ .Proposition REF (i) then exhibits a global equivalenceB_{\o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ac610375eb4de71318e69aa7047dc0853b78ba17 | subsection | 231 | 1,121 | Examples | A complex subspace of V_{\mathbb {C}} is invariant under \psi _V if and
only if it is the complexification of an {\mathbb {R}}-subspace of V
(namely the \psi _V-fixed subspace of V).
So the morphism c is an isomorphism of \mathbf {Gr} onto
the \psi -invariant ultra-commutative submonoid (\mathbf {Gr}^{\mathbb {C}})^\ps... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0ad3a9b45bcf4583fc0563525e15dfdbc83ba6b0 | subsection | 232 | 1,121 | Examples | The isomorphisms are also compatible
with complexification and realification, in the sense of the commutative diagram:@C6mm{
{\underline{\pi }}_0(\mathbf {Gr}) [r]^-{{\underline{\pi }}_0(c)} [d]_{(\ref {eq:pi^G Gr to RO^+ G})}^\cong &
{\underline{\pi }}_0(\mathbf {Gr}^{\mathbb {C}}) [r]^-{{\underline{\pi }}_0(r)} [d]^{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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853ce1db3c369dfb1c1533b7d313c326fcff93cf | subsection | 233 | 1,121 | Examples | If W is another inner product space, then the two direct summand inclusions
induce algebra homomorphisms\operatorname{Sym}(V)\ \xrightarrow{}\ \operatorname{Sym}(V\oplus W) \ \xleftarrow{} \ \operatorname{Sym}(W)\ .We use the commutative multiplication on \operatorname{Sym}(V\oplus W)
to combine these into an {\mathbb ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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76253f2a7a3a124abdfaacc0a0878332ad70c0f4 | subsection | 234 | 1,121 | Examples | For an inner product space V we let
i:V\longrightarrow \operatorname{Sym}(V) be the embedding as the linear summand of the
symmetric algebra. Then as V varies, the maps\mathbf {Gr}(V)\ = \ {\coprod }_{n\ge 0} Gr_n(V) \ \longrightarrow \ {\coprod }_{n\ge 0} Gr_n(\operatorname{Sym}(V)) \ = \ \mathbf {Gr}_\otimes (V)sendi... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3018d7919610b72621e38f155a6ce3e6a783e912 | subsection | 235 | 1,121 | Examples | Because{\mathbf {P}}(V)\ = \ \mathbf {Gr}_\otimes ^{[1]}(V) \ = \ P(\operatorname{Sym}(V))is the projective space of the symmetric algebra of V,
we use the symbol {\mathbf {P}} and refer to it as the
global projective space.global projective space{\mathbf {P}} - global projective space
The multiplication is given by te... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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53fce8ba0173b0c2f77f36fb5a58057a651d50fa | subsection | 236 | 1,121 | Examples | For n\ge 1 and 0\le i \le n, the face map d_i:M^n\longrightarrow M^{n-1}
is given byd_i(x_1,\dots x_n) \ = \left\lbrace \begin{array}{ll}
(x_2,\dots ,x_n) & \mbox{for $i=0$,} \\
(x_1,\dots ,x_{i-1},x_i\cdot x_{i+1},x_{i+2},\dots ,x_n) & \mbox{for $0<i< n$,} \\
(x_1,\dots ,x_{n-1}) & \mbox{for $i=n$.}
\end{array} \right... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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00eabe040d630c60bba72410a46013d1b14bb8da | subsection | 237 | 1,121 | Examples | The cofree functor takes a space A to the orthogonal space R A with
values(R A)(V)\ = \ \operatorname{map}({\mathbf {L}}(V,{\mathbb {R}}^\infty ),A) \ .We endow the cofree functor with a lax symmetric monoidal transformation\mu _{A,B}\ : \ R A \boxtimes R B \ \longrightarrow \ R(A\times B)\ .To construct \mu _{A,B} we ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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57d5141cbb39a8b98cd5b3a5ce00470357ca3b28 | subsection | 238 | 1,121 | Examples | We saw in Proposition REF
that there is then a unique structure of global power monoid on
{\underline{\pi }}_0(R(B A)), and the power operations are characterized by the relation[m](u_A)\ = \ p_m^*(u_A)where u_A\in \pi _0^A(R(B A)) is a tautological class
and p_m:\Sigma _m\wr A\longrightarrow A is the homomorphism def... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3279069450d23a0a492587ca51e792aeeec6645b | subsection | 239 | 1,121 | Examples | This \Sigma _m-action is faithful, so the semifree orthogonal space {\mathbf {L}}_{\Sigma _m,{\mathbb {R}}^m}
is a global classifying space for the symmetric group.
The homeomorphisms{\mathbf {L}}({\mathbb {R}}^m,V)/\Sigma _m \ \cong \ \mathbf {F}^{[m]}(V) \ , \quad \varphi \cdot \Sigma _m\ \longmapsto \ \lbrace \varph... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
3293,
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f94ef1f821f6ed2c7f062d72a4b71feae07b8cd7 | subsection | 240 | 1,121 | Examples | So they assemble into a map\pi _0^G ( \mathbf {F})\ = \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)} \, \pi _0( \mathbf {F}(V)^G ) \ \longrightarrow \ {\mathbb {A}}^+(G)\ ,and this map is a monoid isomorphism with respect to the disjoint union
of G-sets on the target.
Moreover, the isomorphisms are compatible with r... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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bbc170463c30baeaa22becf973635f89b6991d58 | subsection | 241 | 1,121 | Examples | The induced morphism of global power monoids
is linearization: the square of monoid homomorphisms@C=18mm{
\pi _0^G(\mathbf {F})[d]_{(\ref {eq:pi^G F to A^+ G})}^\cong [r]^-{\pi _0^G(\operatorname{span})}&
\pi _0^G(\mathbf {Gr})[d]^{(\ref {eq:pi^G Gr to RO^+ G})}_\cong \\
{\mathbb {A}}^+(G)[r]_-{[S]\,\longmapsto \, [{\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9c4020455f2145ab1dfbe5d634bc0df000562c53 | subsection | 242 | 1,121 | Examples | The unit is the identity of S^V.The equivariant homotopy set
\pi _0^G(\Omega ^\bullet {\mathbb {S}}) is equal
to the stable G-equivariant 0-stem \pi _0^G({\mathbb {S}}),
compare Construction REF below.
The monoid structure on \pi _0^G(\Omega ^\bullet {\mathbb {S}})
arising from the multiplication on \Omega ^\bullet {\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c5a617517a44af2c031aaeb3e482299e62945a46 | subsection | 243 | 1,121 | Examples | If \lambda is such an irreducible G-representation,
then the image of the \lambda -indexed copy of {\mathbb {Z}}/2
is represented by the antipodal map of S^\lambda .In (REF ) we defined
a morphism of ultra-commutative monoids \tau :\mathbf {Gr}\longrightarrow {\mathbf {O}}.
The composite morphism\mathbf {Gr}\ \xrightar... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bfb0085965",
"end": 1093,
"openalex_id": "https://openalex.org/W205144100",
"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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7b48d0125f9791a38d9f7b300f93be2823968aed | subsection | 244 | 1,121 | Examples | The morphism of ultra-commutative monoids (REF )
realizes the exponential morphism in the sense that the following diagram
of monoid homomorphisms commutes:@C=6mm{
\pi _0^G(\mathbf {Gr})[d]_{(\ref {eq:pi^G Gr to RO^+ G})}^\cong [rr]^-{\pi _0^G(J\circ \tau )} &&
\pi _0^G(\Omega ^\bullet {\mathbb {S}}) @{=}[r] & (\pi _0^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fb2a3cf7d95ae85db7f6d484b34a137e874b1479 | subsection | 245 | 1,121 | Examples | We close this section with a discussion of
the complex representation ring global functor (Example REF ),
and a global view on `explicit Brauer induction' (Remark REF ).Example 3.1 (Burnside ring global functor) Burnside ring global functor
The Burnside ring global functor {\mathbb {A}}={\mathbf {A}}(e,-)
is the unit o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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697d9ec90c054484b08f0c7c861a12fc56c7957f | subsection | 246 | 1,121 | Examples | Indeed, for the additive generator [G/H]=t_H
of {\mathbb {A}}(G) this is the relation (REF ),
and for general finite G-sets it follows from the additivity formula
for power operations and the fact that for two finite G-sets S and T
the power (S\amalg T)^m is (\Sigma _m\wr G)-equivariantly isomorphic to
the coproduct{\c... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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e3890fc18914f967b4c3bf39313139d6837258e2 | subsection | 247 | 1,121 | Examples | The power operations satisfy
P^m(1_A)\ = \ p_m^* \text{\quad in\quad } {\mathbf {A}}(A,\Sigma _m\wr A) \ ,
the inflation operation of the continuous homomorphism
p_m\ :\ \Sigma _m\wr A\ \longrightarrow \ A \ , \quad (\sigma ;\,a_1,\dots ,a_m)\ \longmapsto \ a_1\cdot \ldots \cdot a_m \ .Moreover, for every global pow... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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4a0dd25ea9c817e1d49f82ac9b0a7bdebebe80cf | subsection | 248 | 1,121 | Examples | Since the global functor {\mathbf {A}}_A\Box {\mathbf {A}}_A
is representable by A\times A, the Yoneda lemma reduces
the multiplicativity property to the relation\exp ({\mathbf {A}}_A,\mu ^*)((p^*_m)_m) \ = \ \exp ({\mathbf {A}}_A,q_1^*)((p_m^*)_m)\ \cdot \ \exp ({\mathbf {A}}_A,q_2^*)((p_m^*)_m)in the ring \exp ({\mat... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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677c0623b2e712db2f0ba6d3eb0029fd3ce9e9db | subsection | 249 | 1,121 | Examples | Theorem REF
shows that {\underline{\pi }}_0(\Sigma ^\infty _+ R(B A)) is
freely generated, as a global functor,
by the stable tautological class e_A\in \pi _0^A(\Sigma ^\infty _+ R(B A)),
the stabilization of the unstable tautological class u_A.
The characterization of the multiplication and power operations on {\math... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a1a1cc901ec6943938fe880c12df8a6485dd88bf | subsection | 250 | 1,121 | Examples | We already know thatthe class u_G freely generates {\underline{\pi }}_0(B_{\operatorname{gl}}G) as a Rep-functor
(Proposition REF (ii)),
the class u_G^{umon}=\pi _0^G(\eta )(u_G) freely generates
{\underline{\pi }}_0({\mathbb {P}}(B_{\operatorname{gl}}G)) as a global power monoid (Theorem REF (ii)), and
the class e... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d06bd32b2e818dbc7b29ccfd69324f93c11fdb9d | subsection | 251 | 1,121 | Examples | For every global power functor R and every element x\in R(G)
there is a unique morphism of global power functors
f:{\underline{\pi }}_0(\Sigma ^\infty _+ {\mathbb {P}}(B_{\operatorname{gl}} G))\longrightarrow R such that f(u_G^{ucom}) = x.(i)
The ultra-commutative monoid {\mathbb {P}}(B_{\operatorname{gl}}G) is the dis... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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44081eab3b4ceafa6daaae03cdb2cf49b3738781 | subsection | 252 | 1,121 | Examples | Hence the morphism\psi \ = \ {\bigoplus }_{m\ge 0}\psi _m \ : \ C_G \ = \ &{\bigoplus }_{m\ge 0}\,{\mathbf {A}}(\Sigma _m\wr G,-)\\
\longrightarrow \ &{\bigoplus }_{m\ge 0 }\,{\underline{\pi }}_0(\Sigma ^\infty _+{\mathbb {P}}^m(B_{\operatorname{gl}}G))\ = \ {\underline{\pi }}_0(\Sigma ^\infty _+{\mathbb {P}}(B_{\opera... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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9ca0a413d40feeddfac967a9af80d01c95ee8f7e | subsection | 253 | 1,121 | Examples | In this upgraded setting, the global power functor R[M]
has a similar characterization as in the previous paragraph:
for a global power functor S, we let\operatorname{Mon}(S) \ = \ \lbrace x\in S(e)\ | \ \text{$P^m(x)=(p_{\Sigma _m})^*(x)$ for all $m\ge 1$}\rbracebe the set of monoid-like elements of S;
here p_{\Sigma ... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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aa55c89a50d52de7a4be8f0fe3a80500ea66b384 | subsection | 254 | 1,121 | Examples | Hence the morphism\eta _{\underline{B}}\ : \ \exp (\underline{B}) \ \longrightarrow \ \underline{B}is an isomorphism of global Green functors. So when restricted
to constant global Green functors, the exp comonad is isomorphic to the
identity. Thus \underline{B} has a unique structure of coalgebra
over the comonad \exp... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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dfb9db14f63ff27cf14b9fadb23e984ca8f45577 | subsection | 255 | 1,121 | Examples | The Eilenberg-Mac Lane spectrum
{\mathcal {H}}A{\mathcal {H}}A - Eilenberg-Mac Lane spectrum of an abelian group
is defined at an inner product space V by({\mathcal {H}}A)(V) \ = \ A[S^V] \ ,the reduced A-linearizationlinearization!of a space
of the V-sphere.
The orthogonal group O(V) acts through the action on S^V and... | {
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"Stefan Schwede"
] | [
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3b468ba3481dc618a45979689cf3eb95d27beab3 | subsection | 256 | 1,121 | Examples | Since {\mathcal {H}}A is globally connective (by the next proposition),
there is a unique morphism\rho \ : \ {\mathcal {H}}A \ \longrightarrow \ H\underline{A}in the global stable homotopy category that realizes the morphism on \pi _0^e.Proposition 3.8
For every abelian group A the Eilenberg-Mac Lane spectrum {\mathc... | {
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5bc0ccf579c012ca7672181daebfaae8f7718221 | subsection | 257 | 1,121 | Examples | This shows that {\mathcal {H}}A is a {{\mathcal {F}}in}-\Omega -spectrum
for the constant global functor \underline{A}.We offer an independent proof of the {{\mathcal {F}}in}-\Omega -property
via the {\mathbf {\Gamma }}-G-space techniques of Segal and Shimakawa , ,Gamma-space@{\mathbf {\Gamma }}-space!equivariant
in ou... | {
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3f6b44d97d5d6185d2d1eaed589a94d8f278fcd3 | subsection | 258 | 1,121 | Examples | Before we do so, we compare {\mathcal {H}}{\mathbb {Z}} to
the `infinite symmetric product of the sphere spectrum'.Example 3.9 (Infinite symmetric product) infinite symmetric product spectrum
There is no essential difference if we consider
the infinite symmetric product S\! p^\infty (i.e., the reduced free abelian mono... | {
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ecfd229112424b6ed51d04b9e53d56ab40072f76 | subsection | 259 | 1,121 | Examples | Proposition REF provides homeomorphisms(S\! p^\infty (S^V))^G \ \cong \ (S\! p^\infty (S^{V^{G^\circ }}))^{\bar{G}}
\text{\qquad and\qquad }
({\mathbb {Z}}[S^V])^G \cong \ ({\mathbb {Z}}[S^{V^{G^\circ }}])^{\bar{G}} \ .Since V^{G^\circ } is an orthogonal representation of
the finite group \bar{G} the map(S\! p^\infty (... | {
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966908538671f8d2f564e20413b72d233fdf66fc | subsection | 260 | 1,121 | Examples | Theorem 3.12 of shows that
the global functor {\underline{\pi }}_0(S\! p^n) is the quotient
of the Burnside ring global functor by the global subfunctor generated by
the element n\cdot 1- t_{\Sigma _{n-1}}^{\Sigma _n} in {\mathbb {A}}(\Sigma _n),{\underline{\pi }}_0(S\! p^n) \ \cong \ {\mathbb {A}}/\langle n\cdot 1 \ -... | {
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99d6d6549f637c58a2619fa0a1425a400f1a9bc4 | subsection | 261 | 1,121 | Examples | An explicit example for which \pi _0^G( {\mathcal {H}}{\mathbb {Z}}) has rank bigger than one
is G=S U(2).special unitary group!S U(2)
Then the classes 1 and \operatorname{tr}_N^{S U(2)}(1) are a {\mathbb {Z}}-basis for
\pi _0^{S U(2)}( {\mathcal {H}}{\mathbb {Z}}) modulo torsion,
see ,
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eda6e5cd75733c311600712b4c36abeccb236152 | subsection | 262 | 1,121 | Examples | Since the restriction of {\underline{\pi }}_0({\mathcal {H}}A) to finite groups is a
constant global functor, the relation\operatorname{Tr}_{C^{\prime }}^{U(1)}\circ (-\wedge S^1)&\circ p_{C^{\prime }}^* \ = \ \operatorname{Tr}_C^{U(1)}\circ \operatorname{tr}_{C^{\prime }}^C\circ (-\wedge S^1)\circ p_{C^{\prime }}^* \\... | {
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fe9834177ee048eb9fc41853feae65016b771718 | subsection | 263 | 1,121 | Examples | The isotropy separation sequence (REF )
thus shows that the map E{\mathcal {P}}\longrightarrow \ast induces an isomorphism\pi _1^{U(1)}({\mathcal {H}}A\wedge E{\mathcal {P}}_+)\ \cong \ \pi _1^{U(1)}({\mathcal {H}}A) \ ,where E{\mathcal {P}} is a universal U(1)-space for the family of proper closed (i.e., finite)
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5c8aa6d8f4024bd5c7fd38a19b62e59af407651d | subsection | 264 | 1,121 | Examples | Since {\mathcal {H}}A\wedge X_+ is a mapping telescope, its equivariant homotopy
groups can be calculated as the colimit of the sequence:\pi _1^{U(1)}({\mathcal {H}}A\wedge U(1)_+)
\ \longrightarrow \ \dots \ \longrightarrow \ \pi _1^{U(1)}({\mathcal {H}}A\wedge (U(1)/C_{n !})_+) \ \longrightarrow \ \dotsWe rewrite thi... | {
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7c650806366cc5bb39f51dda5011075cc7f6534d | subsection | 265 | 1,121 | Examples | The global homotopy type of b E is
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45cd4cdcee4d9a99485f391b85b62e6e8527302e | subsection | 266 | 1,121 | Examples | If W is another G-representation, then(V\oplus W)^{\otimes m} \text{\qquad and\qquad }
{\bigoplus }_{k=0}^m \, \operatorname{tr}_{ (\Sigma _k\wr G)\times (\Sigma _{m-k}\wr G)}^{\Sigma _m\wr G}
(V^{\otimes k}\otimes W^{\otimes (m-k)})are isomorphic as (\Sigma _m\wr G)-representations,
because tensor product distributes ... | {
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{
"arxiv_id": "",
"doi": "",
"end": 1136,
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"raw": "R. Brauer, On Artin's L-series with general group characters. Ann. of Math. (2) 48 (1947), 502–514.",
"source_ref_id": "221ee824045151374c8f1ca810e2aaac826f4c90",
"start": 881
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... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d164dff093c8a696d9dc55ca1be788f0ef785c5e | subsection | 267 | 1,121 | Examples | Any unitary representation of a compact Lie group G of dimension n
is isomorphic to \alpha ^*(\tau _n) for a continuous homomorphism
\alpha :G\longrightarrow U(n); so the class of such a representation equals\alpha ^*(i_!(q^*(x)))\ \in \ \mathbf {RU}(G) \ .So the global functor \mathbf {RU} is generated by the single c... | {
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{
"arxiv_id": "",
"doi": "10.1007/bf01394272",
"end": 461,
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"source_ref_id": "979e93cae9b68dba5d84e67fa6e77495efb7b0fa... | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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acd870fb8d90a23788e359a56e7cd20455d21c0f | subsection | 268 | 1,121 | Examples | The value of b_G at the 1-dimensional representation with
character \chi :G\longrightarrow U(1) is given by
b_G[\chi ^*(\tau _1)]\ = \ \chi ^* \ \in \ {\mathbf {A}}(U(1),G)\ .Every class in \mathbf {RU}(G) is a formal difference of classes of
actual representations, and every n-dimensional representation
is the restri... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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3a731b5d98e543aeacdd7bd909588aa22f8f3fa9 | subsection | 269 | 1,121 | Examples | To write the class b_G[V]
as a {\mathbb {Z}}-linear combination of transfers of 1-dimensional representations
of subgroups of G, one would now have to write
the classifying homomorphism \alpha :G\longrightarrow U(n) for V
as the composite of an epimorphism and a subgroup inclusion and then
expand the term \alpha ^*\cir... | {
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{
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"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8f5272aa368c8755bf5a05d10e9353661fddc6b0 | subsection | 270 | 1,121 | Global forms of | In this section we discuss different orthogonal monoid spaces whose
underlying non-equivariant homotopy type is B O,
a classifying space for the infinite orthogonal group.
Each example is interesting in its own right,
and as a whole, the global forms of B O are a great illustration of how
non-equivariant homotopy types... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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5fd1cb1512e9ec5f7b75d9238d04d3899950639d | subsection | 271 | 1,121 | Global forms of | All weak morphisms above can be arranged to preserve the E_\infty -multiplications,
so they induce additive maps of abelian monoids on \pi _0^G
for every compact Lie group G.As we explain in Example REF ,
the bar construction model makes sense more generally for
monoid valued orthogonal spaces; in particular, applying ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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d4196445961812038520d59c0f2a60c4103b7a30 | subsection | 272 | 1,121 | Global forms of | In particular, the orthogonal group O(V) acts on {\mathbf {BOP}}(V)
through its diagonal action on V^2.We make the orthogonal space {\mathbf {BOP}} into an ultra-commutative monoid
by endowing it with multiplication maps\mu _{V,W}\ : \ {\mathbf {BOP}}(V) \times {\mathbf {BOP}}(W) \ \longrightarrow \ {\mathbf {BOP}}(V\o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fa20c325bdc0e2b6bbb8fa8510f03fc9d2173fb6 | subsection | 273 | 1,121 | Global forms of | This orthogonal space
is a `multiplicative shift' of \mathbf {Gr}
in the sense of Example REF ,
it admits a commutative multiplication in much the same way as \mathbf {Gr}, and the maps\mathbf {Gr}(V)\ \longrightarrow \ \mathbf {Gr}^{\prime }(V) \ , \quad L \ \longmapsto \ L\oplus 0form a global equivalence of ultra-co... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6eec4be4093f0b62f127297410da6a463d95d753 | subsection | 274 | 1,121 | Global forms of | We let
\alpha :A\longrightarrow R(V) and \beta :A\longrightarrow R(W) be two G-maps
that represent classes in [A,R]^G. Then their sum is defined as[\alpha ] +[\beta ] \ = \ [\mu _{V,W}(\alpha ,\beta )] \ ,where \mu _{V,W}:R(V)\times R(W)\longrightarrow R(V\oplus W) is the
(V,W)-component of the multiplication of R.
The... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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40866b4ed6091aab964798285b3663cae94a420e | subsection | 275 | 1,121 | Global forms of | We can thus define a G-equivariant homotopyK \ : \ [0,1]\times G r(V^2) \ \longrightarrow \ G r(V^2\oplus V^2) \text{\quad by\quad }
K(t,L) \ =\ (L\oplus 0\oplus 0) + H_L(t,L^\perp )\ .ThenK(0,L)\ =\ (L\oplus 0\oplus 0) + H_L(0,L^\perp ) \ = \ (L\oplus 0) +(0\oplus L^\perp ) \ = \ L\oplus L^\perpandK(1,L)\ &=\ (L\oplus... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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ed39e28285368baa2e68d7ddb25b5e638a6e54c3 | subsection | 276 | 1,121 | Global forms of | The composite of \alpha and the orthogonal complement map
(-)^\perp :{\mathbf {BOP}}(V)\longrightarrow {\mathbf {BOP}}(V) represents another class in [A,{\mathbf {BOP}}]^G,
and[\alpha ]+ [(-)^\perp \circ \alpha ]\ &= \ [\mu ^{{\mathbf {BOP}}}_{V,V}\circ (\operatorname{Id}, (-)^\perp )\circ \alpha ]\ = \ [c_{V\oplus V\o... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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008e27133775fb68c9b087c093ffe0193ae42d06 | subsection | 277 | 1,121 | Global forms of | We let c_V:A\longrightarrow \mathbf {Gr}(V) denote the constant map with value V
and \chi :V^4\longrightarrow V^4 the linear isometry defined by\chi (v_1,v_2,v_3,v_4)\ = \ (v_2,v_3,v_1,v_4)\ .We observe that the following diagram commutes:@C=10mm{
{\mathbf {BOP}}(V)@{=}[d] [rr]^-{(i(V)\circ c_V,\operatorname{Id})} &&
{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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c3a0adef3ff73009345a788ccb9ee689106bf1f1 | subsection | 278 | 1,121 | Global forms of | The map i(V):\mathbf {Gr}(V)\longrightarrow {\mathbf {BOP}}(V)=\mathbf {Gr}(V\oplus V)
factors as the composite\mathbf {Gr}(V)\ \xrightarrow{}\ \mathbf {Gr}(V)\times \mathbf {Gr}(V)\ \xrightarrow{}\ \mathbf {Gr}(V\oplus V) \ ,so[c_V]\ + a \ &= \ [c_V]+[\alpha ]\ = \ [\mu _{V,V}^{\mathbf {Gr}}\circ (c_V,\alpha )] \\
&= ... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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37861d8955831f66bee92ae5094138980a08f1e3 | subsection | 279 | 1,121 | Global forms of | We pull back the tautological G-vector bundle \gamma _V over \mathbf {Gr}(V) and obtain
a G-vector bundle f^\star (\gamma _V):E\longrightarrow A over A with total spaceE\ = \ \lbrace (v,a)\in V\times A\ | \ v\in f(a)\rbrace \ .The G-action and bundle structure are as
a G-subbundle of the trivial bundle V\times A.
Since... | {
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0655cc931f837631a4a7053c5675404fbf1eac95 | subsection | 280 | 1,121 | Global forms of | So there is a unique homomorphism of abelian groups\langle -\rangle \ : \ [A,{\mathbf {BOP}}]^G \ = \ \operatorname{colim}_{V\in s({\mathcal {U}}_G)}\, [A,\, {\mathbf {BOP}}(V) ]^G \ \longrightarrow \ {\mathbf {KO}}_G(A)such that the following square commutes:\begin{aligned}
@C=20mm{
[A,\mathbf {Gr}]^G [d]_{[A,i]^G} [r... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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] | 2,018 | en | Mathematics | [
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d91058a81fe3a26b1f07d634af894450a04cc11c | subsection | 281 | 1,121 | Global forms of | As G varies the isomorphisms are compatible with
restriction along continuous homomorphism.The Grassmannian \mathbf {Gr} is the disjoint union of the homogeneous pieces \mathbf {Gr}^{[n]},
and the latter is isomorphic to the semifree orthogonal space {\mathbf {L}}_{O(n),{\mathbb {R}}^n}, via{\mathbf {L}}({\mathbb {R}}^... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
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665205dd76f4657790745bc8eda8a1a267ff9ec3 | subsection | 282 | 1,121 | Global forms of | Moreover, A_{(n)} is G-invariant,
so the restriction \xi _{(n)} of the bundle to A_{(n)} is classified
by a G-map f_{(n)}:A_{(n)}\longrightarrow \mathbf {Gr}^{[n]}(V_n) for some
finite-dimensional G-representation V_n.
Since A is compact, almost all A_{(n)} are empty, so by increasing
the representations, if necessary,... | {
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} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
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b23dd9b41233f095e56a938191e1d919ddcf17bf | subsection | 283 | 1,121 | Global forms of | This special case identifies the global power monoid
{\underline{\pi }}_0({\mathbf {BOP}}) with the global power
monoid \mathbf {RO}\mathbf {RO} - orthogonal representation ring global functorrepresentation ring!orthogonal
of orthogonal representation rings.
For every compact Lie group G the abelian monoid \mathbf {RO}... | {
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] | [
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7c6c0004a443cfb56e4483ab5f4b5ac8a0ce9a13 | subsection | 284 | 1,121 | Global forms of | We have to argue that in addition, the maps (REF )
are also compatible with transfers (or equivalently, with power operations).
The compatibility with transfers can either be deduced directly from
the definitions; equivalently it can be formally deduced from the
compatibility of the isomorphisms {\underline{\pi }}_0(\m... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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fd2dd05506cc55b49c4289f190b41b583d0f2518 | subsection | 285 | 1,121 | Global forms of | We make \mathbf {B}^\circ {\mathbf {O}} into an ultra-commutative monoid by endowing
it with multiplication maps\mu _{V,W}\ : \ (\mathbf {B}^\circ {\mathbf {O}})(V) \times (\mathbf {B}^\circ {\mathbf {O}})(W)
\ \longrightarrow \ (\mathbf {B}^\circ {\mathbf {O}})(V\oplus W)defined as the compositeB ( O(V) ) \times B ( O... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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7112f85b771968428ae8a3e9cd33aa65a3edd160 | subsection | 286 | 1,121 | Global forms of | We define an ultra-commutative monoid \mathbf {B}^{\prime }{\mathbf {O}} by combining the
constructions of \mathbf {B}^\circ {\mathbf {O}} (bar construction) and {\mathbf {BO}} (Grassmannians)
into one definition.
The value of \mathbf {B}^{\prime }{\mathbf {O}} at an inner product space V is(\mathbf {B}^{\prime }{\math... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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affbd1a53bb90f1f79c2f265801941c3d4a2228a | subsection | 287 | 1,121 | Global forms of | We define a continuous map\varphi _\sharp \ : \ {\mathbf {L}}(V,V^2)\ \longrightarrow \ {\mathbf {L}}(W,W^2)by(\varphi _\sharp \psi )(\varphi (v)+ w)\ = \ \varphi ^2( \psi (v) + (w,0) )\ ;here v\in V and w\in W-\varphi (V) is orthogonal to \varphi (V).
The map \varphi _\sharp is compatible with the actions of the ortho... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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a2a8abc903bacef27bc235e6f79bd55482d12744 | subsection | 288 | 1,121 | Global forms of | The right map \beta (V) is the canonical map from homotopy orbits to
strict orbits. As V varies, the \alpha and \beta maps
form morphisms of ultra-commutative monoids\mathbf {B}^\circ {\mathbf {O}}\ \xleftarrow{} \ \mathbf {B}^{\prime }{\mathbf {O}}\ \xrightarrow{}\ {\mathbf {BO}}\ ,essentially by construction.
As we s... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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6a18947903ddef8caccf04bd1d8b73365c784c58 | subsection | 289 | 1,121 | Global forms of | Indeed, the proposition shows that the G-fixed points of
\mathbf {B}^{\prime }{\mathbf {O}}(V)=|B_\bullet ({\mathbf {L}}(V,V^2),O(V), O(V))|/O(V)
are a disjoint union, indexed by conjugacy classes of continuous homomorphisms
\gamma :G\longrightarrow O(V) of the spaces|B_\bullet ({\mathbf {L}}(V,V^2),O(V), O(V))| ^{\Gam... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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665af82c7823cbacfd0efa77a90bf32f497c716f | subsection | 290 | 1,121 | Global forms of | So the upper horizontal quotient map also becomes
arbitrarily highly connected as V grows.
Hence the map \alpha (V)^G becomes an equivalence\operatorname{tel}_i\, \alpha (V_i)^G\ : \ \operatorname{tel}_i\, (\mathbf {B}^{\prime }{\mathbf {O}}(V_i))^G \ \longrightarrow \ \operatorname{tel}_i\, (\mathbf {B}^\circ {\mathbf... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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68faa51615bf717db3ebae809615d5fab7860135 | subsection | 291 | 1,121 | Global forms of | The claim then follows by passing to colimits over V in s({\mathcal {U}}_G).We showed in part (i) that the
inclusions of G-fixed points {\mathbf {L}}^G(V,V^2)\longrightarrow {\mathbf {L}}(V,V^2)
and O^G(V)\longrightarrow O(V) induce a homeomorphism| B_\bullet ({\mathbf {L}}^G(V,V^2),O^G(V),\ast )|\ \xrightarrow{} \ |B_... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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21ef3415d30590b6cdc7e08e113c88d65f3c28bf | subsection | 292 | 1,121 | Global forms of | Now we determine the entire homotopy types of the
G-fixed point spaces of the three ultra-commutative
monoids \mathbf {B}^\circ {\mathbf {O}}, {\mathbf {BO}} and {\mathbf {BOP}}.Corollary 4.16
Let G be a compact Lie group.The G-fixed point space of \mathbf {B}^\circ {\mathbf {O}} is a
classifying space of the group O^... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
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... |
92443e048b736910e722883947618a61a3e4b3b1 | subsection | 293 | 1,121 | Global forms of | Since G-fixed points also commute with the filtered colimit at hand
(see Proposition REF (ii)), we have(\mathbf {B}^\circ {\mathbf {O}}({\mathcal {U}}_G))^G \ &= \ \left( \operatorname{colim}_{V\in s({\mathcal {U}}_G)} \mathbf {B}^\circ {\mathbf {O}}(V) \right)^G
\\
&\cong \ \operatorname{colim}_{V\in s({\mathcal {U}}... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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03313f17c64985afde6c8eac79ec22b6bb1ed98d | subsection | 294 | 1,121 | Global forms of | The classifying space construction commutes with weak products,
which gives a weak equivalenceB (O^G({\mathcal {U}}_G)) \ \simeq \ {\prod }^{\prime } B( O^G({\mathcal {U}}_\lambda )) \ .Moreover, the group O^G({\mathcal {U}}_\lambda ) is isomorphic to an infinite orthogonal, unitary
or symplectic group, depending on th... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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8d457473cc19544b4ad78fd1650ce18c5732f579 | subsection | 295 | 1,121 | Global forms of | So hitting all the previous examples with the bar construction
yields a commutative diagram of globally connected orthogonal spaces:@C=15mm{
\mathbf {B}^\circ {\mathbf {SU}}[r]^-{\mathbf {B}^\circ \text{incl}} @{-->}[d]_{\mathbf {B}^\circ l} &
\mathbf {B}^\circ {\mathbf {U}}@{-->}[d]_{\mathbf {B}^\circ l} @{-->}[dr]^-{... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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17f510e12e3ab529ed38f22e36778310a08623a6 | subsection | 296 | 1,121 | Global forms of | The structure map {\mathbf {bO}}(\varphi ):{\mathbf {bO}}(V)\longrightarrow {\mathbf {bO}}(W) is given by{\mathbf {bO}}(\varphi )(L) \ = \ (\varphi \oplus {\mathbb {R}}^\infty )(L) + ( (W-\varphi (V))\oplus 0)\ ,the internal orthogonal sum of the image of L under
\varphi \oplus {\mathbb {R}}^\infty :V\oplus {\mathbb {R... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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0a52044d4b6c1aa638a0425a8b4103756eb751cb | subsection | 297 | 1,121 | Global forms of | Moreover, for every linear isometric embedding \varphi :V\longrightarrow W
the relation\left({\mathbf {bO}}(\varphi )(L)\right)^\perp \ &= \ \left( (\varphi \oplus {\mathbb {R}}^\infty )(L) +(W-\varphi (V))\oplus 0 \right)^\perp \\
&= \ \left(\varphi (L^\perp ) + (W^\perp -\varphi (V^\perp ))\right)\oplus 0 \ = \ \left... | {
"cite_spans": []
} | 10.1017/9781108349161 | 1802.09382 | Global homotopy theory | [
"Stefan Schwede"
] | [
"math.AT"
] | 2,018 | en | Mathematics | [
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