# Anomalous symmetry breaking in Weyl semimetal CeAlGe

H. Hodovanets,1,∗ C. J. Eckberg,1 Y. Eo, 1 D. J. Campbell,1 P. Y. Zavalij,2 P.
Piccoli,3 T. Metz,1 H. Kim,1,∗ J. S. Higgins,1 and J. Paglione1 1 Maryland
Quantum Materials Center, Department of Physics, University of Maryland,
College Park, Maryland, 20742 USA  2 X-ray Crystallographic Center, Department
of Chemistry and Biochemistry, University of Maryland, College Park, Maryland,
2074 USA  3 Department of Geology, University of Maryland, College Park,
Maryland, 20742 USA  Current address: Department of Physics and Astronomy,
Texas Tech University, Lubbock, Texas, 79409 USA

###### Abstract

CeAlGe, a proposed type-II Weyl semimetal, orders antiferromagnetically below
5 K. Both a spin-flop and a spin-flip transitions to less than 1 $\mu_{B}$/Ce
are observed at 2 K below 30 kOe in the $M(H)$ (H$\|$a and b) and 4.3 kOe
(H$\|$ $\langle 110\rangle$) data, respectively, indicating a four-fold
symmetry of the $M(H)$ along the principal directions in the tetragonal ab-
plane with $\langle 110\rangle$ set of easy directions. However, anomalously
robust and complex two-fold symmetry is observed in the angular dependence of
resistivity and magnetic torque data in the magnetically ordered state once
the field is swept in the ab -plane. This two-fold symmetry is independent of
temperature- and field-hystereses and suggests a magnetic phase transition
that separates two different magnetic structures in the ab-plane. The boundary
of this magnetic phase transition can be tuned by different growth conditions.

Weyl semimetals have attracted much attention due to their intricate
properties associated with the topological manifestation of electronic band
structure and their potential application in spintronics, quantum bits,
thermoelectric and photovoltaic devices Wan  _et al._ (2011); Weng  _et al._
(2015); Hasan  _et al._ (2017); Chang  _et al._ (2017); Yan and Felser (2017);
Armitage  _et al._ (2018). Magnetic semimetals that break spatial inversion
and time-reversal symmetries are relatively scarce, and it is especially hard
to confirm the breaking of the time-reversal symmetry in these materials Wan
_et al._ (2011); Witczak-Krempa and Kim (2012); Liu  _et al._ (2014); Neupane
_et al._ (2014); Wang  _et al._ (2016); Manna  _et al._ (2018); Liu  _et al._
(2018). The RAlGe and RAlSi (R = Ce and Pr) families present a new class of
magnetic Weyl semimetals where both inversion and time-reversal symmetries are
broken due to intrinsic magnetic order Chang  _et al._ (2018); Hodovanets  _et
al._ (2018); Puphal  _et al._ (2019); Yang  _et al._ (2020a); Lyu  _et al._
(2020). With observations of a topological magnetic phase Puphal  _et al._
(2020), anomalous Hall effect (AHE) Meng  _et al._ (2019), topological Hall
effect Puphal  _et al._ (2020), and singular angular magnetoresistance (AMR)
Suzuki  _et al._ (2019), as well as a possible route to axial gauge fields
Destraz  _et al._ (2020), RAlGe is particularly promising.

Noncentrosymmetric CeAlGe, a proposed type-II magnetic Weyl semimetal Chang
_et al._ (2018) that orders antiferromagnetically below 5 K in zero magnetic
field and ferrimagnetically in non-zero field Hodovanets  _et al._ (2018),
hosts several incommensurate multi-$\vec{k}$ magnetic phases, including a
topological phase for H$\|$c Puphal  _et al._ (2020). Motivated by the fact
that its magnetic moments lie in the tetragonal ab-plane, together with the
observation of a sharp singular AMR in its Si-substituted variant Suzuki  _et
al._ (2019), we study pure CeAlGe using magnetization, M, angle-dependent
magnetic torque, $\tau(\varphi)$, and magnetoresistance, $R(\varphi)$
measurements. While we find the expected four-fold tetragonal symmetry in M(H)
when field is swept through the ab-plane, we also observe an anomalous two-
fold symmetry in both angle-dependent magnetic torque and AMR in the ordered
state. In contrast to conventional smoothly changing (i.e. sinusoidal) AMR in
magnetic conductors McGuire and Potter (1975), which is dependent on the
orientation of magnetization and current, the two-fold symmetric ab-plane AMR
of CeAlGe is remarkably history independent and unchanged under magnetic field
and temperature hystereses, highlighting possibilities for device
applications. We discuss the idea of two different magnetic structures in the
ordered state as a likely explanation for the observed two-fold symmetry, and
consider other possibilities.

Figure 1: (color online) Electrode configurations used in the different
resistance as a function of angle measurements. Figure 2: (color online) (a)
Field-dependent magnetization of CeAlGe for H$\|$a, H$\|$b, H$\|$[110], and
H$\|$[$\bar{1}\bar{1}$0]. $H_{1}$ denotes the lowest critical field (beginning
of the canted phase of the spin-flop transition) below which the hysteresis in
the M(H) data starts in the H$\|$a and H$\|$b data. $H_{2}$ marks a critical
field of spin-saturated ferromagnetic state. For H$\|\langle 110\rangle$
directions, a spin-flip transition occurs at the critical field marked
$H^{\prime}$. (b) Angular dependence of the resistivity (AMR) of CeAlGe single
crystal measured in the 4-probe configuration at T = 2 K with H = const and
I$\|$[010]. The field is swept in the tetragonal ab-plane. (c) AMR of CeAlGe
single crystal measured in the 4-probe configuration at H = 2.5 kOe and
selected T = const. AMR of CeAlGe single crystal measured in the 4-probe wire
configuration at 2 K and 2.5 kOe (d) different current and (e) different
conditions on approaching 2 K and 2.5 kOe: cool down from 3 K to 2 K in 2.5
kOe; at 2 K, changed the field from 1.5 to 2.5 kOe; at 2 K, swept through 0 Oe
(-30 to 30 kOe); warmed up to 10 K, set H = 50 kOe, cooled down to 2 K; warmed
up to 10 K, demagnetized with 50 kOe, zero-field cooled to 2 K; warmed up to
10 K at position 90∘, set H = 50 kOe, cooled down to 2 K, started off from 0∘;
warmed up to 10 K at position 90∘, set H = 50 kOe, cooled down to 2 K, started
off from 90∘.(f-h) Angle-dependent magnetic torque data at selected H = const
collected at 2 K. The data for H = 3 and 10 kOe are repeated for clarity.

Single crystals of CeAlGe were grown by the high-temperature flux method
Hodovanets  _et al._ (2018); Canfield  _et al._ (2016). Temperature-, field-,
and angle-dependent magnetization, resistivity, and magnetic torque
measurements were performed in a commercial cryostat. All angle-dependent data
were collected on changing the angle from 0∘ to 360∘ unless otherwise noted.
Resistivity measurements were made in a standard four-probe, van der Pauw van
der Pauw (1958), Hall bar or concentric ring (we will call it a 4-terminal
Corbino) geometry ($I$ = 0.5 or 1 mA), Fig. 1. The samples were polished and
shaped with care to not have any Al inclusions. Electrical contacts to the
samples in the four-probe, van der Pauw, and Hall bar geometry were made with
Au wires attached to the samples using EPOTEK silver epoxy and subsequently
cured at 100∘C. The 4-terminal Corbino was patterned using standard
photolithography followed by a standard metal liftoff. The patterns consist of
20-30 Å/1500 Å Ti/Au contacts made by e-beam evaporation. 25 $\mu$m Au wires
were attached to the gold electrodes by wire bonding. To calculate the
resistivity of the 4-terminal Corbino one needs a geometric factor, which is
difficult to estimate when the sample is not a 2D material or a thin film. To
estimate the geometric factor of a single crystal that has a finite thickness,
we used the effective thickness that was found numerically Eo  _et al._
(2018). We note that this is a nominal resistivity since the resistivity ratio
between the ab-plane and the c-axis is not accurately known.

Figure 3: (color online) AMR for various electrode configurations and
measurement techniques, and magnetic torque of CeAlGe measured at T = 2 K with
(a)-(h) H = 2.5 kOe and (i)-(p) H = 5 kOe. For the sample with I$\|$[110], the
rotation started with the a-axis shifted by 45∘, thus the positions of the a\-
and b\- axes are marked in the graph. The data for the van der Pauw
configuration with I$\|$a, b, and [110] were measured on the same sample. The
sample for the 4-terminal Corbino measurement was mounted, due to the size
restrictions, with approximately -13∘ offset with respect to the a-axis. All
samples are from the same batch.

We now discuss the field-dependent magnetization $M(H)$ data at T = 2 K
measured for H$\|$a, b, [110], and [$\bar{1}\bar{1}$0] axes, shown in Fig.
2(a). For H$\|$a and b (circles), a clear sharp spin-flop transition to a less
than 1 $\mu_{B}$ saturation moment is observed below $\sim$26 kOe as was
reported in Ref. Hodovanets  _et al._ , 2018. The critical fields $H_{1}$ and
$H_{2}$ delineate the canted moment phase. On the contrary, the spin-flip
transition to a slightly higher value of saturated magnetization is observed
for H$\|$[110] and [$\bar{1}\bar{1}$0] data (squares) below $H^{{}^{\prime}}$
= 4.3 kOe, indicating that the easy axes are the $\langle 110\rangle$ set of
directions. The data presented in Fig. 2(a) indicate a four-fold symmetry of
M(H) data in the tetragonal ab-plane.

However, as shown in Fig. 2(b), a sharp two-fold symmetry change is observed
in the AMR data when field is swept in the tetragonal ab-plane. This two-fold
symmetry sets in before the critical fields of $H_{1}$ and $H^{{}^{\prime}}$
(defined from the magnetization data) and is less apparent above $H_{2}$ where
the moments are in the field-saturated ferromagnetic state. Keeping the field
constant at H = 2.5 kOe, the AMR was measured at constant temperatures, Fig.
2(c). The two-fold symmetry holds only in the ordered state, below 5 K, thus
suggesting that the origin of this behavior is due to magnetic order. Neither
the magnitude of the current, nor the different conditions at which the 2 K
temperature is reached, nor at what angle the measurement is started have an
effect on the AMR features, at least for the H = 2.5 kOe data as shown in
Figs. 2(d) and (e), respectively. The AMR for CeAlGe cannot be simply scaled
based on the field-induced magnetization along either the a-axis or [110] axis
(see Fig. S1 below).

To further validate whether the two-fold symmetry in the AMR is due to the
magnetic order or due to the current direction, we measured magnetic torque in
the tetragonal ab-plane at H=const at 2 K, Figs. 2(f)-(h). Figure 2(f) shows
$\tau(\varphi)$ data with a clear two-fold symmetry and very complicated
functional dependence that cannot be fit by a series of even sine functions
below 10 kOe Okazaki  _et al._ (2011); Kasahara  _et al._ (2012). The magnetic
torque changes the location of positive and negative maxima (sign change)
between 3 and 5 kOe, Fig. 3(g). The corresponds in the AMR to the appearance
of plateaus at 45∘ (every 45∘) in Figs. 2(b) between H = 3.5 and 4 kOe. This
region separates the data into two different magnetic regimes and is more
evident in the AMR of the sample for which I$\|$[110], Fig. S1(a). Here, H =
3.5 kOe is further confirmed as a transition field. This value of magnetic
field is slightly above H1(H$\|$a) and much lower than H′(H$\|$[110]) in the
M(H) data, Fig. 2(a).

The two-fold anisotropy in the torque data decreases and is barely observable
at 10 kOe, Fig. 2(g), as the magnitude of the torque increases with the
magnetic field. We would like to note that the two-fold symmetry is observed
in the torque data at 5 K as well (SM below), which is above the magnetic
ordering temperature, indicating some moment fluctuation. As opposed to the
AMR, the magnetic torque data display a clear four-fold symmetry at a lower
magnetic field H = 20 kOe and above, Fig. 2(h). These observations point to
the magnetic order being a culprit of the breaking of the four-fold symmetry
in the measurements (according to neutron studies Suzuki  _et al._ (2019);
Puphal  _et al._ (2020), the crystal structure of CeAlGe remains tetragonal
down to 2 K). This two-fold symmetry is more dramatic and enhanced in the
resistivity measurements.

To further test the effect of the current and its direction, we measured the
AMR with different electrode configurations and techniques. The results are
shown in Fig. 3 for H = 2.5 and 5 kOe left and right panels, respectively. The
two-fold symmetry is present in all measurements. The sample with I$\|$c and
4-terminal Corbino sample display similar functional dependence in the AMR for
both H = 2.5 and 5 kOe. The van der Pauw sample, in-plane Hall sample, and
4-probe sample can be grouped together based on the similar behavior as well.
It is interesting to note that the van der Pauw sample shows similar behavior
in the resistance and Hall configurations. The AMR data of the sample with
I$\|$[110] appear to mirror the behavior of the magnetic torque data, i.e. the
45∘ shift of maxima between H = 2.5 and 5 kOe. As the field is increased to 5
kOe, the current chooses a different preferred direction, compared to that for
H = 2.5 kOe, and is a reflection of a change in the magnetic order/spin
structure between H = 2.5 and 5 kOe. This is more clearly seen in Figs. S1 and
S2. At 0.1 K a yet different functional behavior of AMR is observed for H =
2.5 kOe, Fig. S3, indicating an additional magnetic phase transition. No
functional change is observed for H = 5 kOe at 0.1 K compared to that at 2 K.

Neutron diffraction experiments have reported differing results on the
magnetic structure of CeAlGe. While Ref. Suzuki  _et al._ , 2019 reported a
zero-field coplanar $Fd^{\prime}d2^{\prime}$ magnetic structure (i.e. m =
($m_{x},m_{y}$,0) on sublattice A and (-$m_{x},-m_{y}$,0) on sublattice B
related by a diamond glide-operation) and a collinear magnetic structure
(m$\|$[100]) in non-zero field with independent moments on the symmetrically
nonequivalent A and B sublattice sites, Ref. Puphal  _et al._ , 2020 reported
an incommensurate multi-$\vec{k}$ magnetic ground state in zero field. This
magnetic phase changes to a single-$\vec{k}$ state at the metamagnetic
transition $H_{1}$ = 3 kOe (H$\|$a), consistent with our results. The
single-$\vec{k}$ state evolves into the field polarized ferromagnetic state at
$H_{2}\sim$9 kOe (lower than $H_{2}$ for this work) at 2 K. Thus, the two
different regimes seen in the AMR data would correspond to these two different
magnetic phases. Note that the sample studied in Ref. Puphal  _et al._ , 2020
is stoichiometric and the ones in this work have $\sim 5\%$ deficiency in both
Al and Ge (see Table II in SM). Despite slightly different magnetic ordering
temperatures, the critical field $H_{1}$ is the same. As we discussed above,
the field close to $H_{1}$ determines the boundary of the two magnetic phases.
As is discussed in SM, a large Al deficiency, which depends on different
crystal growth conditions, changes the value of $H_{1}$ and $H^{\prime}$,
making them smaller, and perhaps changing the values of multi-$\vec{k}$
vectors (or magnetic structure altogether) since the features in the AMR in
the lower-field state become slightly different. The magnetic phase above
these two fields remains unchanged. Systematic magnetic structure studies are
needed to confirm this hypothesis.

In Ref. Suzuki  _et al._ , 2019, the observed singular AMR was suggested to
arise, under particular conditions, from momentum space mismatch across real
space domains and was confined to a very narrow angles. These domains form a
single domain once the field is increased so that the sample is in the field
saturated ferromagnetic state. One may assume that if the magnetic field is
subsequently lowered, the sample would break into a different set of magnetic
domains and hence upon remeasuring AMR, a different functional dependence
would be observed. Instead, we still observe the same behavior not matter how
many times the sample is warmed up above the ordered temperature and at which
field the sample is cooled down to 2 K, Fig. 2(e). It is plausible that
structural defects (e.g. micro-cracks in the samples after polishing since the
samples are rather brittle) or some arrangements of sub-micron Al inclusions
may “help” the formation of the domains and once domains are formed, they are
pinned and could only be changed if the defects are removed, e.g. by
controlled annealing in the former case. Such studies, together with the
visualization of magnetic domains Yang  _et al._ (2020b) and defects or
pinning centers in the ordered state at constant applied magnetic field, would
be necessary. Alternatively, single crystals of CeAlGe can be grown using a
different flux (we discuss In-flux grown single crystal of CeAlGe in SM) or a
different single crystal growth technique can be utilized.

In conclusion, a clear two-fold symmetry is observed in the robust and sharp
non-sinusoidal AMR data in the ordered state when the magnetic field is swept
in the tetragonal ab-plane, revealing more detailed and complicated underlying
magnetic structures and the phase transitions between them. The current along
the b-axis enhances this two-fold symmetry compared to the current along [110]
and c-axes, although [110] axis is an easy axis. A clear separation of the AMR
data into two regimes based on the two distinct magnetic phases is observed in
the magnetic torque and AMR data at the magnetic field close to $H_{1}$ and
$H^{\prime}$. Al deficiency controls the value of these two critical fields
and changes the critical field of the phase transition between the two
different magnetic phases and most likely alters the magnetic structure of the
low-field phase in the type II Weyl semimetal CeAlGe.

Acknowledgments. The authors would like to thank J. Checkelsky and J. Lynn for
insightful discussions. H.H. would like to thank D. M. Benson and B. L.
Straughn for fruitful discussions. Materials synthesis efforts were supported
by the Gordon and Betty Moore Foundation’s EPiQS Initiative through grant no.
GBMF9071, and experimental investigations were supported by the Department of
Energy, Office of Basic Energy Sciences, under Award No. DE-SC0019154.

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##  Appendix A Supplementary Materials

Figure S1: (color online) (a) AMR of CeAlGe single crystal measured in the
four-probe wire configuration at 2 K, I$\|$[110]. AMR data are clearly split
into two regimes at H = 3.5 kOe - a value of the phase transition between the
two different magnetic states. Field-dependent resistivity of CeAlGe measured
in the 4-probe wire configuration at constant angles at 2 K (b) I$\|$[010] and
(c) I$\|$[110]. Insets show zoom in of low field data. Figure S2: (color
online) Angle-dependent magnetic torque of CeAlGe single crystal measured at
(a) H = 2 kOe and (b) H = 5 kOe. Figure S3: (color online) Angle-dependent
resistivity of CeAlGe single crystal measured in four-probe wire
configuration, I$\|$[010], at $T\leq$2 K for H = 2.5 and 5 kOe.

##  Appendix B Angle- and field-dependent resistivity

Figure S1(a) shows AMR with I$\|$[110] at 2 K. The shape of AMR for
I$\|$[110], is very different compared to that of I$\|$b discussed in the main
text. The two-fold symmetry is still visible here but is not as drastic.
Interestingly, all peaks at every 90∘ for H = 3.25 kOe that pointed up, point
down at H = 3.75 kOe. In between, at H = 3.5 kOe, they alternate, the ones at
90∘ and 270∘ turn down but the one at 180∘ and 0∘/360∘ stay up. These fields
are slightly over H1(H$\|$a) and much lower than H′(H$\|$[110]) in the M(H)
data, Fig. 2(a). Looking at Fig. 2(b), the corresponding feature in the AMR
data for I$\|$b is the appearance of dips (H = 3.5 kOe) that turn into
plateaus (H = 4 kOe) at 45∘ (every 45∘). Perhaps the narrow region between
$H_{1}$ and $H^{\prime}$ critical fields in the M(H) data of spin
reorientation is captured here. This phase transition at H = 3.5 kOe, seems to
divide AMR into two different regimes.

The field-dependent resistivity data collected every 45∘ at 2 K are shown in
Figs. S1(b) I$\|$b and S1(c) I$\|$[110], respectively. The data fall into
three manifolds for I$\|$b and two manifolds for I$\|$[110] in the ordered
state. For I$\|$b, Fig. S1(b), as opposed to the M(H) data, $\rho(H)$ data
show a clear two-fold symmetry for H$\|$[100] and [010] directions and a clear
four-fold symmetry for H$\|\langle 110\rangle$ directions in the ordered
state. This would be consistent with the current breaking the four-fold
symmetry for 90∘ rotations. On the contrary, for I$\|$[110], Fig. S1(c), the
$\rho(H)$ data seem to follow M(H) behavior except below H = 3 kOe, inset to
Fig. S1(c), where the data for 0∘ (180∘ and 360∘) and 90∘(270∘) are not the
same, i.e. the four-fold symmetry is broken. The difference in the data is not
due to the hysteresis since the data were collected with the same approach.
Thus, the two-fold symmetry in the $\rho(H)$ data for I$\|$[110] is more
subtle. Negative magnetoresistance in the ordered state is followed by a
positive magnetoresistance at the field close to $H_{2}$ with almost no
anisotropy in the field-saturated state for I$\|$b and small anisotropy for
I$\|$[110]. The features in the ordered state are consistent with those
observed in the M(H) data shown in Fig. 2(a), except for I$\|$b sample at 0∘
and 180∘ a clear sharp change in the AMR is seen at about 16 kOe. There is no
corresponding sharp feature in the M(H) data Fig. 2(a). The hysteresis in the
data below H = 4 kOe, consistent with that seen in the M(H) data, is evident
in the data shown in the inset to Fig. S1(a). One thus should expect
hysteresis on increasing and decreasing the angle in the AMR data as well.

##  Appendix C Magnetic torque

The torque magnetometry option of Quantum Design Physical Properties
Measurement System was used to collect magnetic torque data. The sample for
this measurement was secured with the help of N grease. The background of the
Si torque chip with the N grease and sample was measured at 90∘ and was
accounted for in the final result. A markedly different evolution, reflecting
different magnetic state, of the angle-dependent magnetic torque with
temperature at H = 2.5 and 5 kOe is shown in Figs. S2(a) and S2(b),
respectively. The absolute value of the torque decreases as the temperature
increases. At 5 K which is just above the ordering temperature, the torque
data for these two magnetic fields become similar and at 6.5 K, the torque is
almost zero reflecting the paramagnetic state. Interestingly, the torque data
at 5 K is still two-fold symmetric, perhaps indicating some moment
fluctuations.

##  Appendix D Lower than 2 K temperature

Upon lowering the temperature to 0.1 K, the shape of the AMR for I$\|$b
changes at H = 2.5 kOe, Fig. S3. However, the shape of AMR at H = 5 kOe
remains unchanged except the peaks become narrower. This indicates another
magnetic phase below 2 K with the critical field less than 2.5 kOe.

##  Appendix E Single crystals grown under different conditions

Additional two batches of CeAlGe single crystals were grown with the following
conditions: (i) Cerium ingot from Ames Laboratory and Al was used as a flux
and (ii) Cerium ingot from Alfa Aesar and In flux was used to grow CeAlGe
single crystals. Cerium from Alfa Aesar was also used to grow the crystals
presented in the main text. Canfield crucible with the fritCanfield  _et al._
(2016) was used in these two growths to prevent Si substitution from the
quartz wool in the catch crucible. The batches of crystals were grown at
different times with the same temperature profile in the same Lindberg/Blue M
1500∘C box furnace.

Lattice parameters determined through single crystal x-ray diffraction
analysis are shown in the Table 1. Sample with Ames Ce shows only slightly
larger c-axis. In-flux grown sample has both lattice parameters smaller than
Al-flux grown samples.

Table 1:  Lattice parameters data determined through single-crystal x-ray diffraction of CeAlGe single crystals grown with different conditions. Space group I41md (No. 109). All data were collected at 250 K on Bruker APEX-II CCD system equipped with a graphite monochromator and a MoK$\alpha$ sealed tube (wavelength $\lambda$ = 0.71070 $\mathrm{\AA}$). Lattice parameters | frit (main text) | Ames Cerium/frit | In flux/frit
---|---|---|---
$a$($\mathrm{\AA}$) | 4.2920(2) | 4.2930(2) | 4.2875(2)
$b$($\mathrm{\AA}$) | 4.2920(2) | 4.2930(2) | 4.2875(2)
$c$($\mathrm{\AA}$) | 14.7496(4) | 14.7631(7) | 14.7197(7)
Table 2:  Results of WDS with 2standard deviations for CeAlGe single crystals grown under different conditions. The rows represent different samples. Chemical elements | frit (main text) | Ames Ce/frit | In flux/frit
---|---|---|---
Ce | 1 | 1 | 1
Ge | 0.95(1) | 0.96(2) | 0.98(1)
Al | 0.98(2) | 0.83(2) | 0.90(1)
Ce | 1 | 1 |
Ge | 0.95(4) | 0.95(1) |
Al | 0.95(1) | 0.81(2) |

To determine stoichiometry of the samples, single crystals were analyzed using
a JEOL 8900R electron probe microanalyzer at the Advanced Imaging and
Microscopy Laboratory (AIMLab) in the Maryland Nanocenter using standard
wavelength dispersive spectroscopy (WDS) techniques. The following analytical
conditions were utilized: 15 kV accelerating voltage; 50 nA sample current; a
1 micron beam; and synthetic Al and Ge metal and CePO4 standards. Both K-alpha
(Al, Ge) and L-alpha (Ce) x-ray lines were used. Count times ranged from 20-30
seconds on peak, and 5-10 on background. Raw x-ray intensities were corrected
using a standard ZAF algorithm. The standard deviation due to counting
statistics was generally below 0.5$\%$, 0.3$\%$, and 0.25$\%$ for Ge, Al, and
Ce, respectively. Based on the total amounts recorded for Ce, Al, and Ge, any
additional In doping was not recorded. WDS results are listed in Table 2.
Small and nearly the same Ge deficiency is observed among all samples with the
In-flux grown sample having Ge concentration closest to 1. However, Al
deficiency varies largely among different batches. The first batch listed in
the Table 2 shows about 5$\%$ Al deficiency. Crystals grown with Ames Ce show
surprisingly large Al deficiency at nearly 20$\%$. In-flux grown crystal is
$\sim$10$\%$ Al deficient. There appears to be no correlation between lattice
parameters and either Ge or Al deficiency. Both In and Al inclusions were
observed in In-grown single crystals.

Zero-field cooled (ZFC) and field-cooled (FC) temperature- and field-dependent
magnetization for the batch with Ames Ce/Al flux and In-flux grown single
crystals of CeAlGe are shown in Fig. S4 (a) and (c), respectively. The
ordering temperature of 5 K is the same. The effective moments calculated from
the Curie-Weiss law fits of the polycrystalline average (not shown here) are
consistent with the WDS data. Large difference is seen in the ZFC and FC data
below 4.5 K. Temperature-dependent magnetization is also larger at 100 Oe for
In-grown sample. This is consistent with the M(H) data shown in Fig. S4(d)
where $H_{1}$ = 1 kOe (it may even be lower, the data were collected with 1
kOe step) as opposed to $H_{1}$ = 2 kOe of Ames Ce sample, Fig. S4(b). $H_{1}$
critical fields for both samples and $H^{\prime}$ for Ames Ce sample are lower
than those of the sample presented in the main text. On the other hand,
$H_{2}$ critical fields are somewhat similar. It appears that Al deficiency
affects the values of critical fields of spin reorientation $H_{1}$ and
$H^{\prime}$.

M(H) data for H$\|$c show different behavior. The magnetic moment along the
c-axis for In-flux sample is larger than that of Ames Ce sample and the field
at which $M_{H\|c}>M_{H\|a}$ is lower. Two metamagnetic transitions with
hysteresis are seen below H = 20 kOe for the In-grown sample as well.

The temperature-dependent resistivity of the samples from these batches
measured in the standard 4-probe configuration are shown together with those
from the batch (measured with different techniques) described in the main text
in Fig. 1. The $\rho(T)$ data for Al-flux grown samples show a good agreement.
The $\rho(T)$ data for I$\|$c are 1.6 times larger at 300 K and 1.8 times
larger at 2 K than that of I$\|$b indicating that b-axis is more conductive.
In-flux grown samples show larger resistivity values. The feature around 7 K
is more pronounced and the feature associated with the magnetic order is less
pronounced compared to those for the Al-flux grown samples.

The AMR data measured in the 4-probe configuration with I$\|$b at 2 K at
different constant magnetic fields (the field was rotated in the tetragonal
ab-plane) for the Ames Ce/Al-flux and In-flux grown samples are shown in Fig.
S6(a) and (b), respectively. The AMR shows similar features for H$\geq$ 2 kOe.
The same behavior was observed for H$\geq$ 5 kOe in the sample discussed in
the main text, Fig. 2(b). This corresponds to the region above $H^{\prime}$
(which is different for these samples), however, the behavior of the AMR data
is the same, reflecting the same magnetic state for all three samples above
$H^{\prime}$. On the contrary, the features in the AMR are different below
$H^{\prime}$ for all three samples. The onset of the large and clear two-fold
symmetry occurs at a much lower field of 1 kOe (perhaps even at a smaller
field) as opposed to 2.5 kOe for the sample discussed in the main text, due to
$H_{1}$ for these two sample being much lower than the one for the sample
discussed in the main text.

Al deficiency appears to affect critical fields $H_{1}$ and $H^{\prime}$,
making them smaller than that of a closer to Al stoichiometric sample. This
leads to the second regime, common among all samples, of AMR to appear above
fields as small as 2 kOe. In addition, Al deficiency seems to affect the spin
orientation/structure in the first regime below $H^{\prime}$ as is evident by
a different functional dependence of the AMR.

Figure S4: (color online) (a) ZFC and FC temperature-dependent magnetization
of CeAlGe single crystals grown using Ames cerium. (b) M(H) data of
CeAlGe/Ames cerium and Al flux, (c) ZFC and FC temperature-dependent
magnetization of CeAlGe single crystals grown using In flux, and (d) M(H) data
of CeAlGe/In flux. Figure S5: (color online) (a) Temperature-dependent
resistivity of CeAlGe single crystals grown under different conditions and
measured in different electrode configurations. (b), (c), and (d) Low-
temperature part of the temperature-dependent resistivity showing the features
associated with the magnetic order. Figure S6: (color online) Angle-dependent
resistivity of CeAlGe single crystal measured in four-probe wire configuration
at 2 K and H = 2.5 kOe. (a) Ce from Ames Laboratory was used, Al-flux was used
to grow CeAlGe single crystals. (b) In-flux was used to grow CeAlGe single
crystals.


The homotopy types of Sp(n)-gauge groups over S^4m
Sajjad Mohammadi

Let $m$ and $n$ be two positive integers such that $m < n$. Denote by $P_{n,k}$ the principal $Sp(n)$-bundle over $S^{4m}$ and $\mathcal{G}_{k,m}(Sp(n))$ be the gauge group of $P_{n,k}$ classified by $k\varepsilon'$, where $\varepsilon'$ is a generator of $\pi_{4m}(B(Sp(n)))\cong\mathbb{Z}$. In this article we partially classify the homotopy types of $\mathcal{G}_{k,m}(Sp(n))$. Also we will study the order of the
Samelson product $S^{4m-1}\wedge Q_{n-m+1}\rightarrow Sp(n)$, where $Q_{n-m+1}$ be the symplectic quasi-projective space of rank $n-m+1$.

Keywords: gauge group; homotopy type; Samelson product; Symplectic group.

$2010$ Mathematics Subject Classification: Primary $54C35$; Secondary $55P15$.

§ INTRODUCTION

Let $G$ be a topological group and let $P\rightarrow B$ be a principal $G$-bundle over connected finite complex $B$. The gauge group $\mathcal{G}(P)$ of $P$ is the group of $G$-equivariant automorphisms of $P$ which fix $B$. The topology of gauge groups plays an important role in the physics and identification of four-dimensional manifolds. Crabb and Sutherland [3] have shown that if $B$ and $G$ are as above, then the number of homotopy types amongst all the gauge groups of principal $G$-bundles over $B$ is finite. This is in spite of the fact that the number of isomorphism classes of principal $G$-bundles over $B$ is often infinite.Let $G$ be a simply-connected, simple compact Lie group. For integers $a$ and $b$ let $(a, b)$ be the greatest common divisor of $|a|$ and $|b|$. There is a one-to-one correspondence between the number of equivalence classes of principal $G$-bundles over $B=S^4$ and homotopy classes of maps $[S^4, BG]\cong  \mathbb{Z}$.

The first classification was done by Kono [9] for $G = SU(2)$. He showed that there is a homotopy equivalence $\mathcal{G}_k(SU(2))\simeq \mathcal{G}_{k'}(SU(2))$ if and only if $(12, k) = (12, k')$. Results formally similar to that of Kono have been obtained for principal bundles over $S^4$ with different structure groups. In the following, we mention some results

* $\mathcal{G}_k(SU(3))\simeq\mathcal{G}_{k'}(SU(3))$ if and only if $(24,k)=(24,k')$ [5];

* $\mathcal{G}_k(SU(5))\simeq\mathcal{G}_{k'}(SU(5))$ if and only if $(120,k)=(120,k')$, when localized at any prime $p$ or rationally [18];

* $\mathcal{G}_k(Sp(2))\simeq\mathcal{G}_{k'}(Sp(2))$ if and only if $(40,k)=(40,k')$, when localized at any prime $p$ or rationally [19];

* $\mathcal{G}_k(Sp(3))\simeq\mathcal{G}_{k'}(Sp(3))$ then $(84,k)=(84,k')$ and if $(336,k)=(336,k')$ then $\mathcal{G}_k(Sp(3))\simeq\mathcal{G}_{k'}(Sp(3))$  when localized at any prime $p$ or rationally [2];

* $\mathcal{G}_k(G_2)\simeq\mathcal{G}_{k'}(G_2)$ then $(84,k)=(84,k')$ and if $(168,k)=(168,k')$ then $\mathcal{G}_k(G_2)\simeq\mathcal{G}_{k'}(G_2)$  when localized at any prime $p$ or rationally [12];

* in a non-simply-connected case, $\mathcal{G}_k(SO(3))\simeq\mathcal{G}_{k'}(SO(3))$ if and only if $(12,k)=(12,k')$ [11];

* $\mathcal{G}_k(PU(3))\simeq\mathcal{G}_{k'}(PU(3))$ if and only if $(24,k)=(24,k')$
and $\mathcal{G}_k(PSp(2))\simeq\mathcal{G}_{k'}(PSp(2))$ for any prime $p$ if and only if $(40,k)=(40,k')$[8].

There are also several classification results for gauge groups of principal bundles with base spaces other than
$S^4$ as follow

* for $B=S^6$, $\mathcal{G}_k(SU(3))\simeq\mathcal{G}_{k'}(SU(3))$ if and only if $(120,k)=(120,k')$ [6];

* for $B=S^6$, if $\mathcal{G}_k(SU(n))\simeq\mathcal{G}_{k'}(SU(n))$ then we have $((n-1)n(n+1)(n+2),k)=((n-1)n(n+1)(n+2),k')$ [16];

* for $B=S^8$, $\mathcal{G}_k(Sp(2))\simeq\mathcal{G}_{k'}(Sp(2))$ if and only if $(140,k)=(140,k')$ [7];

* for $B=S^8$, $\mathcal{G}_k(SU(4))\simeq\mathcal{G}_{k'}(SU(4))$ then $(420,k)=(420,k')$ and if $(3360,k)=(3360,k')$ then  $\mathcal{G}_k(SU(4))\simeq\mathcal{G}_{k'}(SU(4))$ [15].

Let $\mathcal{G}_k(SU(n))$ be the gauge group of the principal $SU(n)$-bundle with second Chern class $k$. Stephen Theriault in [20] classify the $p$-local homotopy types of $\mathcal{G}_k(SU(n))$, where $p$ is an odd prime and $n\leq (p-1)^2+1$. Also, in [14] we have studied the homotopy types of $SU(n)$-gauge groups over $S^{2m}$.In [10] Kishimoto and Kono classify the homotopy types of $Sp(n)$ gauge groups over $S^4$, by showing that if there is a homotopy equivalence $\mathcal{G}_{k,1}(Sp(n))\simeq \mathcal{G}_{k',1}(Sp(n))$ then $(4n(2n+1),k)=(4n(2n+1),k')$. In this paper, in general case, we study the classification of the homotopy types of the gauge groups of principal $Sp(n)$-bundles over $S^{4m}$ and we give only lower bounds for the number of homotopy types and does not prove the converse, which is very hard and realistically out of reach. Put $t=(2n+1)2n(2n-1)\cdots (2n-2m+2)$. Let $P_{n,k}$ be the principal $Sp(n)$-bundles over $S^{4m}$ and $\mathcal{G}_{k,m}(Sp(n))$ be the gauge group of $P_{n,k}$ classified by $k\varepsilon'$, where $\varepsilon'$ is a generator of $\pi_{4m}(B(Sp(n)))\cong\mathbb{Z}$. We will prove the following theorems.

Let $\varepsilon_{m,n}\colon S^{4m-1} \rightarrow Sp(n)$ be the represents the generator of $\pi_{4m-1}(Sp(n))$ and $\epsilon_{m,n}\colon Q_{n-m+1}\rightarrow Sp(n)$ be the inclusion, where $Q_{n-m+1}$ be the symplectic quasi projective space of rank $n-m+1$. The order of Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$ is equal to

(2n+1)!/(2n-2m+1)!     if $m$ is even ,

2(2n+1)!/(2n-2m+1)!     if $m$ is odd .

If $\mathcal{G}_{k,m}(Sp(n))$ is homotopy equivalent to $\mathcal{G}_{k',m}(Sp(n))$ then the following hold:

Case one: if $m$ is even integer, then

(k,4t)=(k',4t)     if $n$ is even ,

(k,t)=(k',t)     if $n$ is odd .

Case two: if $m$ is odd integer, then $(k,2t)=(k',2t)$,

The theorems recovers the known case in [10] when $m=1$.

§ PRILIMINARIES

Let $Map_k(S^{4m}, BSp(n))$ be the component of the space of continuous unbased maps from $S^{4m}$ to $BSp(n)$ which contains the map inducing $P$, similarly let $Map_k^\ast(S^{4m}, BSp(n))$  be the spaces of  pointed continuous maps from $S^{4m}$ and  $BSp(n)$ which contains the map inducing $P$. We know that there is a fibration

$$Map_k^*(S^{4m}, BSp(n))\rightarrow Map_k(S^{4m}, BSp(n)) \overset{ev}\longrightarrow  BSp(n),$$

where the map $ev$ is evaluation map at the basepoint of $S^{4m}$. Let $B\mathcal{G}_{k,m}(Sp(n))$ be the classifying spaces of  $\mathcal{G}_{k,m}(Sp(n))$. By Atiyah-Bott [1], there is a homotopy equivalence

$$B\mathcal{G}_{k,m}(Sp(n))\simeq Map_k(S^{4m}, BSp(n)).$$

The evaluation fibration therefore determines a homotopy fibration sequence

\begin{eqnarray}
Sp(n)\overset{\alpha_k}\longrightarrow Map_k^*(S^{4m}, BSp(n))\rightarrow  B\mathcal{G}_{k,m}(Sp(n)) \overset{ev}\longrightarrow BSp(n),
\end{eqnarray}

it is well known that there is a homotopy equivalence
$$Map_k^*(S^{4m}, BSp(n))\simeq Map_0^*(S^{4m}, BSp(n)),$$

we writing $\Omega_{0}^{4m-1}Sp(n)$ for $Map_0^*(S^{4m}, BSp(n))$.Let $G$ be the topological group and let $c\colon G\wedge G \rightarrow G$ denote the commutator of $G$.
The Samelson product of two maps $\alpha\colon X \rightarrow G$ and $\beta\colon Y\rightarrow G$ denoted by $\langle \alpha, \beta \rangle$ and defined to be the composition $\langle \alpha, \beta \rangle\colon X\wedge Y \overset{\alpha\wedge\beta}\longrightarrow  G\wedge G \overset{c}\longrightarrow G$. Let $\varepsilon_{m,n}\colon S^{4m-1}\rightarrow Sp(n)$ be the represents the generator of $\pi_{4m-1}(Sp(n))$.  For an $H$-space $X$, let $k\colon X\rightarrow X$ be the $k^{th}$-power map. By [13] we have the following lemma.

The adjoint of the connecting map $Sp(n)\overset{\alpha_k}\longrightarrow \Omega_{0}^{4m-1}Sp(n)$ is homotopic to the Samelson product $S^{4m-1} \wedge Sp(n) \overset{\langle k\varepsilon_{m,n},1\rangle}\longrightarrow Sp(n)$, where $1: Sp(n) \rightarrow Sp(n)$ is the identity map on $Sp(n)$. $\quad \Box$

The linearity of the Samelson product implies that $\langle k\varepsilon_{m,n},1\rangle\simeq k\langle \varepsilon_{m,n},1\rangle$. Thaking adjoints therefore implies the following.

The connecting map $\alpha_k$ satisfies $\alpha_k\simeq k\circ \alpha_1$. $\quad \Box$

The organization of this article is as follows. In Section <ref>, in cases where $m$ is an even integer and $m$ is an odd integer, we first calculate $[\Sigma^{4m-1}Q_{n-m+1}, Sp(n)]$ and then, regarding the Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$  as an element of $[\Sigma^{4m-1}Q_{n-m+1}, Sp(n)]$, we will study the order of the Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$. In
Section <ref>, we will study the group  $[\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k,m}(Sp(n))]$ and will prove Theorem <ref>.

§ THE GROUP $[\SIGMA^{4M-1}Q_{N-M+1}, SP(N)]$

Our main goal in this section is to study the Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$, where the map $\epsilon_{m,n}\colon Q_{n-m+1}\rightarrow Sp(n)$ be the inclusion map. We denote $Sp(\infty)/Sp(n)$ by $X_n$ and $[X,Sp(n)]$ by $Sp_n(X)$. For $n\leq 3$, we recall that the symplectic quasi projective spaces $Q_n$ have the cellular structures

$$Q_1=S^3,\qquad Q_2=S^3\cup_{v_1}e^7,\qquad Q_3=S^3\cup_{v_1}e^7\cup_{v_2}e^{11}$$

where $v_1\in \pi_6(S^3)\cong \mathbb{Z}_{12}$ and $v_2:S^{10}\rightarrow Q_2$ is some map.Put $X=S^{4m-1}\wedge Q_{n-m+1}$. Note that $X$ has a following cellular structure

$$X\simeq S^{4m+2}\cup e^{4m+6}\cup e^{4m+10}\cup \cdots \cup e^{4n+2}.$$

Recall that as an algebra

\begin{align*}
&H^*(Sp(n);\mathbb{Z})= \bigwedge (y_3,y_7,\cdots, y_{4n-1}),\\
& H^*(Sp(\infty);\mathbb{Z})= \bigwedge (y_3,y_7,\cdots),\\
&H^*(BSp(\infty);\mathbb{Z})= \mathbb{Z}[q_1,q_2,\cdots],
\end{align*}

where $y_{4i-1}=\sigma q_i$, $\sigma$ is the cohomology suspension and $q_i$ is the $i-$th universal symplectic Pontrjagin class. Consider the projection map $\pi:Sp(\infty)\rightarrow X_n$, also an algebra we have

\begin{align*}
& H^*(X_n;\mathbb{Z})= \bigwedge (\bar{y}_{4n+3}, \bar{y}_{4n+7}, \cdots), \\
& H^*(\Omega X_n;\mathbb{Z})= \mathbb{Z}\{b_{4n+2}, b_{4n+6}, \cdots, b_{8n+2} \} \quad (*\leq 8n+2 ),
\end{align*}

where $\pi^*(\bar{y}_{4i+3})=y_{4i+3}$ and $b_{4n+4j-2}=\sigma (\bar{y}_{4n+4j-1})$. Consider the following fibre sequence

\begin{eqnarray}
\Omega Sp(\infty)\overset{\Omega \pi}\longrightarrow \Omega X_n\overset{\delta}\longrightarrow Sp(n) \overset{j}\longrightarrow Sp(\infty)\overset{\pi}\longrightarrow X_n,
\end{eqnarray}

applying the functor $[X,\quad]$ to $(3.1)$, there is an exact sequence

$$[X,\Omega Sp(\infty)]\overset{(\Omega \pi)_*}\longrightarrow [X,\Omega X_n]\overset{\delta_*}\longrightarrow Sp_n(X) \overset{j_*}\longrightarrow [X,Sp(\infty)]\overset{\pi_*}\longrightarrow [X,X_n].$$

Note that $X_n$ has a cellular structure as following

$$X_n\simeq S^{4n+3}\cup_{\eta'} e^{4n+7}\cup e^{4n+11}\cup \cdots,$$
where $\eta'$ is the generator of $\pi_{4n+6}(S^{4n+3})$. Also, we have

$$\Omega X_n\simeq S^{4n+2}\cup e^{4n+6}\cup e^{4n+10}\cup \cdots.$$

According to the $CW$-structure of $X_n$ we have the following isomorphisms

\begin{equation*}
\pi_i(X_n)=0 \quad (for\quad i\leq 4n+2), \qquad \pi_{4n+3}(X_n)\cong \mathbb{Z}.
\end{equation*}

Observe that

$$[X, Sp(\infty)]\cong [\Sigma X, BSp(\infty)]\cong {\widetilde{KSp}}^{-1}(X).$$

Applying ${\widetilde{KSp}}^{-1}$ to the cofibration sequence $S^{4m-1}\wedge Q_{n-m}\rightarrow X\rightarrow S^{4n+2}$. We know that ${\widetilde{KSp}}^{-1}(S^{4n+2})=0$, for every $n\geq 1$, so by use of induction, we can conclude that ${\widetilde{KSp}}^{-1}(X)=0$.On the other hand we know that $\Omega X_n$ is $(4n+1)$-connected and $H^{4n+2}(\Omega X_n)\cong \mathbb{Z}$ which is generated by $b_{4n+2}=\sigma(\bar{y}_{4n+3})$. The map $b_{4n+2}\colon\Omega X_n \rightarrow K(\mathbb{Z},4n+2)$ is a loop map and is a $(4n+3)$-equivalence. Since dim$X\leq 4n+2$, the map $(b_{4n+2})_*\colon[X,\Omega X_n] \rightarrow H^{4n+2}(X)$ is an isomorphism of groups. Thus we get the following exact sequence

\begin{eqnarray}
{\widetilde{KSp}}^{-2}(X)\overset{\psi}\longrightarrow H^{4n+2}(X)\rightarrow Sp_n(X)\rightarrow 0,
\end{eqnarray}

where we use the isomorphism
$${\widetilde{KSp}}^{-i}(X)\cong [\Sigma^iX,BSp(\infty)].$$

So we have the exact sequence

\begin{equation*}
0\rightarrow Coker \psi \overset{\iota}\longrightarrow Sp_n(X)\rightarrow 0.
\end{equation*}

Therefore we get the following lemma.

$Sp_n(X)\cong Coker \psi$. $\quad \Box$

In the following we calculate the image of $\psi$.Let $Y$ be a $CW$-complex with dim $Y \leq 4n+ 2$, we denote $[Y, U(2n+1)]$ by $U_{2n+1}(Y)$. By [4] there is an exact sequence

\begin{equation*}
\tilde{K}^{-2}(Y)\overset{\varphi}\longrightarrow H^{4n+2}(Y)\rightarrow U_{2n+1}(Y)\rightarrow \tilde{K}^{-1}(Y)\rightarrow 0,
\end{equation*}

for any $f\in \tilde{K}^{-2}(Y)$ the map $\varphi$ is define as following

where $ch_{4n+2}(f)$ is the $4n+2$-th part of $ch(f)$. Also, we use the isomorphism

$$\tilde{K}^{-i}(Y)\cong [\Sigma^iY,BU(\infty)].$$

In this paper, we denote both the canonical inclusion $Sp(n)\hookrightarrow U(2n)$ and the induced
map ${\widetilde{KSp}}^{*}(X) \rightarrow {\tilde{K}}^{*}(X)$ by $c'$. By [17] there is a following commutative diagram

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
{{\widetilde{KSp}}^{-2}(X)& H^{4n+2}(X)\\ {\tilde{K}}^{-2}(X)& H^{4n+2}(X).\\}; \path[->,font=\scriptsize]
(m-1-1) edge node[above] {$\psi$}(m-1-2)
(m-1-1) edge node[left] {$c'$} (m-2-1)
(m-1-2) edge node[right] {$(-1)^{n+1}$}(m-2-2)
(m-2-1) edge node[above] {$\varphi$} (m-2-2);
\end{tikzpicture}
\end{equation}

Therefore to calculate the image of $\psi$ we first calculate the image of $\varphi$. We denote the free abelian group with a basis $e_1,e_2,\cdots$, by $\mathbb{Z}\{ e_1,e_2,\cdots \}$. We have the following lemma.

The following hold:

$(a)\colon{\widetilde{KSp}}^{-2}(X)=\mathbb{Z}\{\xi_1,\xi_2,\cdots ,\xi_{n-m+1}\}$, where $\xi_1\in {\widetilde{KSp}}^{-2}(S^{4m+2})$, $\xi_2\in {\widetilde{KSp}}^{-2}(S^{4m+6})$, $\cdots$ and $\xi_{n-m+1}\in {\widetilde{KSp}}^{-2}(S^{4n+2})$,

$(b)\colon {\tilde{K}}^{-2}(X)=\mathbb{Z}\{\xi'_1,\xi'_2,\cdots ,\xi'_{n-m+1}\}$, where $\xi'_1\in {\tilde{K}}^{-2}(S^{4m+2})$, $\xi'_2\in {\tilde{K}}^{-2}(S^{4m+6})$, $\cdots$ and $\xi'_{n-m+1}\in {\tilde{K}}^{-2}(S^{4n+2})$,

$(c):$ if $m$ is even integer then

c'(ξ_1)=ξ'_1,  c'(ξ_2)=2ξ'_2,  ⋯,   c'(ξ_n-m+1) = {
ξ'_n-m+1     $n$ is even ,

2ξ'_n-m+1     $n$ is odd ,

and if $m$ is odd integer then

c'(ξ_1)=2ξ'_1,  c'(ξ_2)=ξ'_2,  ⋯,   c'(ξ_n-m+1) = {
ξ'_n-m+1     $n$ is even ,

2ξ'_n-m+1     $n$ is odd .

First, we consider the case $Q_2$. Put $L_1={\widetilde{KSp}}^{-2}(S^{4m+3})$ and $L_2={\widetilde{KSp}}^{-2}(S^{4m+5})$. The cofibration sequence

$$S^{4m+2}\rightarrow S^{4m-1}\wedge Q_2\rightarrow S^{4m+6}$$

induce the following commutative diagram of exact sequences

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
{L_1&{\widetilde{KSp}}^{-2}(S^{4m+6})&{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge Q_2)&{\widetilde{KSp}}^{-2}(S^{{4m+2}})&L_2\\0&{\tilde{K}}^{-2}(S^{4m+6})&{\tilde{K}}^{-2}(S^{4m-1}\wedge Q_2)&{\tilde{K}}^{-2}(S^{{4m+2}})&0,\\}; \path[->,font=\scriptsize]
(m-1-1) edge (m-1-2)
(m-1-2) edge (m-1-3)
(m-1-3) edge (m-1-4)
(m-1-4) edge (m-1-5)
(m-1-2) edge node[left] {$c'=\sigma_j$}(m-2-2)
(m-1-3) edge node[left] {$c'$}(m-2-3)
(m-1-4) edge node[left] {$c'=\sigma_i$}(m-2-4)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4)
(m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{equation}

where $\sigma_i=1, \sigma_j=2 $ when $m$ is even and $\sigma_i=2, \sigma_j=1$ when $m$ is odd. Since ${\tilde{K}}^{-2}(S^{2i})\cong \mathbb{Z}$ and also

\begin{align*}
{\widetilde{KSp}}^{-2}(S^{4i+2})\cong \mathbb{Z}, \quad {\widetilde{KSp}}^{-2}(S^{4n+1})\cong 0,\quad {\widetilde{KSp}}^{-2}(S^{4n-1})\cong \left\{ \begin{array}{ll}
0 & \textrm{if $n$ is even },\\\\
\mathbb{Z}/2\mathbb{Z} & \textrm{if $n$ is odd},\\
\end{array} \right.
\end{align*}

we have

$${\widetilde{KSp}}^{-2}(S^{4m-1}\wedge Q_2)=\mathbb{Z}\{\xi_1,\xi_2\}, \quad {\tilde{K}}^{-2}(S^{4m-1}\wedge Q_2)=\mathbb{Z}\{\xi'_1,\xi'_2\},$$

where $\xi_1\in {\widetilde{KSp}}^{-2}(S^{4m+2})$, $\xi_2\in {\widetilde{KSp}}^{-2}(S^{4m+6})$, $\xi'_1\in {\tilde{K}}^{-2}(S^{4m+2})$ and $\xi'_2\in {\tilde{K}}^{-2}(S^{4m+6})$.Put $L_3={\widetilde{KSp}}^{-2}(S^{4m}\wedge Q_{n-m})$. There is a cofibration sequence

$$S^{4m-1}\wedge Q_{n-m} \rightarrow X\rightarrow S^{4n+2},$$

this sequence induce the following commutative diagram of exact sequences

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
{L_3&{\widetilde{KSp}}^{-2}(S^{4n+2})&{\widetilde{KSp}}^{-2}(X)&{\widetilde{KSp}}^{-2}(S^{4m-1}\wedge Q_{n-m})&0\\0&{\tilde{K}}^{-2}(S^{4n+2})&{\tilde{K}}^{-2}(X)&{\tilde{K}}^{-2}(S^{4m-1}\wedge Q_{n-m})&0,\\}; \path[->,font=\scriptsize]
(m-1-1) edge (m-1-2)
(m-1-2) edge (m-1-3)
(m-1-3) edge (m-1-4)
(m-1-4) edge (m-1-5)
(m-1-2) edge node[left] {$c'=\sigma_k$}(m-2-2)
(m-1-3) edge node[left] {$c'$}(m-2-3)
(m-1-4) edge node[left] {$c'$}(m-2-4)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4)
(m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{equation}

where $\sigma_k=1$ when $n$ is even and $\sigma_k=2$ when $n$ is odd. By induction, we can conclude that ${\widetilde{KSp}}^{-2}(X)$ and ${\tilde{K}}^{-2}(X)$ are free abelian groups with their basis $\xi_1,\xi_2,\cdots ,\xi_{n-m+1}$ and $\xi'_1,\xi'_2,\cdots ,\xi'_{n-m+1}$, respectively, where $\xi_{n-m+1}\in {\widetilde{KSp}}^{-2}(S^{4n+2})$ and $\xi'_{n-m+1}\in {\tilde{K}}^{-2}(S^{4n+2})$. According to the definition the maps of $c'$ we can choose $\xi_1$, $\xi'_1$, $\xi_2$, $\xi'_2$, $\cdots$, $\xi_{n-m+1}$ and $\xi'_{n-m+1}$ such that if $m$ is even integer then we have

c'(ξ_1)=ξ'_1,  c'(ξ_2)=2ξ'_2,  ⋯,   c'(ξ_n-m+1) = {
ξ'_n-m+1     $n$ is even ,

2ξ'_n-m+1     $n$ is odd ,

and if $m$ is odd integer then we have

c'(ξ_1)=2ξ'_1,  c'(ξ_2)=ξ'_2,  ⋯,   c'(ξ_n-m+1) = {
ξ'_n-m+1     $n$ is even ,

2ξ'_n-m+1     $n$ is odd .

Consider the map $c'\colon Sp(n-m+1)\rightarrow SU(2n-2m+2)$, restriction the map $c'$ to their quasi-projective spaces we get to map ${\bar{c}}'\colon Q_{n-m+1}\rightarrow \Sigma\mathbb{C}P^{2n-2m+1}$. The cohomology of $Q_{n-m+1}$ and $\Sigma\mathbb{C}P^{2n-2m+1}$ are given by

\begin{align*}
&H^* (Q_{n-m+1})=\mathbb{Z}\{{\bar{y}}_3,{\bar{y}}_7, \cdots ,{\bar{y}}_{4n-4m+3}\},\\\\
&H^* (\Sigma\mathbb{C}P^{2n-2m+1})=\mathbb{Z}\{{\bar{x}}_3,{\bar{x}}_5, {\bar{x}}_7,\cdots, {\bar{x}}_{4n-4m+3}\},
\end{align*}

such that

$${\bar{c}}'({\bar{x}}_3)={\bar{y}}_3,\quad{\bar{c}}'({\bar{x}}_5)=0,\quad {\bar{c}}'({\bar{x}}_7)={\bar{y}}_7,\quad \cdots, \quad, {\bar{c}}'({\bar{x}}_{4n-4m+3})={\bar{y}}_{4n-4m+3}.$$

Denote by $\zeta_n$ a generator of $\tilde{K}(S^{2n})$. Recall that

$$H^*(\mathbb{C}P^{2n-2m+1})=\mathbb{Z}[t]/(t^{2n-2m+2}),\qquad K(\mathbb{C}P^{2n-2m+1})=\mathbb{Z}[x]/(x^{2n-2m+2}),$$

where $|t|=2$. Note that ${\tilde{K}}^{-2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})\cong {\tilde{K}}^{0}(\Sigma^{4m+2}\mathbb{C}P^{2n-2m+1})$ is a free abelian group generated by $\zeta_{2m+1}\otimes x$, $\zeta_{2m+1}\otimes x^2$, $\cdots$, $\zeta_{2m+1}\otimes x^{2n-2m+1}$. we have the following

\begin{align*}
&ch_{2n+1}(\zeta_{2m}\otimes x)=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x)=\frac{1}{(2n-2m+1)!}\sigma^{4m}t^{2n-2m+1},\\\\
&ch_{2n+1}(\zeta_{2m}\otimes x^2)=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x^2)=A\sigma^{4m},\\\\
&ch_{2n+1}(\zeta_{2m}\otimes x^3)=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x^3)=B\sigma^{4m},\\
\vdots\\
&ch_{2n+1}(\zeta_{2m}\otimes x^{2n-2m+1})=ch_{2m}(\zeta_{2m})ch_{2n-2m+1}(x^{2n-2m+1})=C\sigma^{4m},
\end{align*}

where $A$, $B$ and $C$ are equal to

$$A= ch_{2n-2m+1}(x^2)= \sum_{\substack {i_1+i_2=2n-2m+1,\\ 1\leq i_1\leq n-m}}ch_{i_1}x ch_{i_2}x =\sum_{k=1}^{n-m}\frac{1}{k!(2n-2m+1-k)!}t^{2n-2m+1},$$

\begin{align*}
B= ch_{2n-2m+1}(x^3)=&ch_1x\sum_{\substack {i_1+i_2=2n-2m,\\ 1\leq i_1\leq i_2}}ch_{i_1}x ch_{i_2}x+ch_2 x\sum_{\substack {i_1+i_2=2n-2m-1,\\ 2\leq i_1\leq i_2}}ch_{i_1}x ch_{i_2}x+\cdots \\
&+ ch_{n-m-1}x\sum_{\substack {i_1+i_2=n-m+2,\\ n-m-1\leq i_1\leq i_2}}ch_{i_1}x ch_{i_2}x  + \sum_{\substack {i_1+i_2=2n-2m+1, \\ i_1, i_2\geq 1}}ch_{i_1}x ch_{i_2}x^2,
\end{align*}

\begin{align*}
C= ch_{2n-2m+1}(x^{2n-2m+1})=&ch_1x\sum_{\substack {i_1+\cdots + i_{2n-2m}=2n-2m,\\ 1\leq i_1\leq i_2 \leq \cdots \leq i_{2n-2m}}} ch_{i_1}x^{i_1}\cdots ch_{i_{2n-2m}}x^{i_{2n-2m}} \\&+ch_2x^2\sum_{\substack {i_1+\cdots + i_{n-m-1}=2n-2m-1,\\ 2\leq i_1\leq i_2\leq \cdots \leq i_{n-m-1}}}ch_{i_1}x^{i_1} \cdots ch_{i_{n-m-1}}x^{i_{n-m-1}} \\
&+ch_3x^3 \sum_{\substack {i_1+\cdots +i_{n-m-2}=2n-2m-2,\\ 3\leq i_1\leq i_2\leq \cdots \leq i_{n-m-2}}}ch_{i_1}x^{i_1}\cdots  ch_{i_{n-m-2}}x^{i_{n-m-2}} +\cdots \\&+ ch_{n-m}x^{n-m}\sum_{\substack {i_1=n-m+1,\\ i_1\geq n-m}}ch_{i_1}x^{i_1}.
\end{align*}

We can put $\xi'_1$, $\xi'_2$, $\cdots$ and  $\xi'_{n-m+1}$ as

$$\xi'_1={\bar{c}}'(\zeta_{2m}\otimes x),\quad \xi'_2={\bar{c}}'(\zeta_{2m}\otimes x^3),\quad \cdots, \quad \xi'_{n-m+1}={\bar{c}}'(\zeta_{2m}\otimes x^{2n-2m+1}).$$

The Image of $\psi$ is generated by

(2n+1)!/(2n-2m+1)! σ^4m-1⊗y̅_4n-4m+3      if $m$ is even ,

2(2n+1)!/(2n-2m+1)! σ^4m-1⊗y̅_4n-4m+3      if $m$ is odd .

Consider the following commutative diagram

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
{{\tilde{K}}^{-2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})& H^{4n+2}(\Sigma^{4m}\mathbb{C}P^{2n-2m+1})\\ {\tilde{K}}^{-2}(S^{4m-1}\wedge Q_{n-m+1})& H^{4n+2}(S^{4m-1}\wedge Q_{n-m+1}),\\}; \path[->,font=\scriptsize]
(m-1-1) edge node[above] {$\varphi'$}(m-1-2)
(m-1-1) edge node[left] {${\bar{c}}'$} (m-2-1)
(m-1-2) edge node[right] {$\cong$}(m-2-2)
(m-2-1) edge node[above] {$\varphi$} (m-2-2);
\end{tikzpicture}
\end{equation}

where the map $\varphi'$ is defined as the map $\varphi$. By definition of the map of $\varphi'$ we have

\begin{align*}
&\varphi'(\zeta_{2m}\otimes x)=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes x)=
\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m}t^{2n-2m+1},\\\\
&\varphi'(\zeta_{2m}\otimes x^2)=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes x^2)=
&\varphi'(\zeta_{2m}\otimes x^3)=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes x^3)=
\vdots\\
&\varphi'(\zeta_{2m}\otimes x^{2n-2m+1})=(2n+1)!ch_{2n+1}(\zeta_{2m}\otimes x^{2n-2m+1})=
\end{align*}

Therefore according to the commutativity of diagram $(3.6)$ we have

\begin{align*}
&\varphi(\xi'_1)=\varphi'(\zeta_{2m}\otimes x)=
\frac{(2n+1)!}{(2n-2m+1)!}\sigma^{4m-1}\otimes \bar{y}_{4n-4m+3},\\\\
&\varphi(\xi'_2)=\varphi'(\zeta_{2m}\otimes x^3)=B'(2n+1)!\sigma^{4m-1}\otimes \bar{y}_{4n-4m+3},\\
\vdots\\
&\varphi(\xi'_{n-m+1})=\varphi'(\zeta_{2m}\otimes x^{2n-2m+1})=C'(2n+1)!\sigma^{4m-1}\otimes \bar{y}_{4n-4m+3},
\end{align*}

where $B=B't^{2n-2m+1}$ and $C=C't^{2n-2m+1}$. Thus by the commutativity of diagram $(3.3)$ we get

ψ(ξ_1)= φ(c'(ξ_1))=±{
(2n+1)!/(2n-2m+1)! σ^4m-1⊗y̅_4n-4m+3     if $m$ is even ,

2(2n+1)!/(2n-2m+1)! σ^4m-1⊗y̅_4n-4m+3     if $m$ is odd .

ψ(ξ_2)= φ(c'(ξ_2))=±{
2B'(2n+1)! σ^4m-1⊗y̅_4n-4m+3     if $m$ is even ,

B'(2n+1)! σ^4m-1⊗y̅_4n-4m+3     if $m$ is odd .

ψ(ξ_n-m+1)= φ(c'(ξ_n-m+1))= {
-(2n+1)! C' σ^4m-1⊗y̅_4n-4m+3     if $n$ is even ,

2(2n+1)! C' σ^4m-1⊗y̅_4n-4m+3     if $n$ is odd .

Thus we can conclude that

Im ψ≅{
ℤ{(2n+1)!/(2n-2m+1)! σ^4m-1⊗y̅_4n-4m+3}     if $m$ is even ,

ℤ{2(2n+1)!/(2n-2m+1)! σ^4m-1⊗y̅_4n-4m+3}     if $m$ is odd .

Therefore by Lemma <ref> and Proposition <ref> we get the following theorem.

There is an isomorphism

[X, Sp(n)]≅{
ℤ_(2n+1)!/(2n-2m+1)!     if $m$ is even ,

ℤ_2(2n+1)!/(2n-2m+1)!     if $m$ is odd .

Here we will prove Theorem <ref>.

$\textbf{Proof of Theorem \ref{a1}}$

The following theorem was proved in [17].

Let $X$ be a $CW$-complex with dim $X \leq 4n+2$. For $\alpha,\beta\in Sp_n(X)$, the commutator $[\alpha, \beta]$ in $Sp_n(X)$ is given as

$$[\alpha, \beta]=\iota([\sum_{i+j=n+1}\alpha^*(y_{4i-1})\beta^*(y_{4j-1})]). \quad \Box$$

Consider the following composition

$$S^{4m-1}\wedge Q_{n-m+1}\overset{\bar{\Lambda}}\longrightarrow S^{4m-1}\wedge Q_{n-m+1}\wedge S^{4m-1}\wedge Q_{n-m+1} \overset{p_1\wedge p_2}\longrightarrow S^{4m-1}\wedge Q_{n-m+1}, $$

where $\bar{\Lambda}$ is the reduced diagonal map and $p_1$ and $p_2$ are
the first and the second projections respectively. Since $(p_1\wedge p_2 )\circ \bar{\Lambda}=1$, the Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$ is equal to the commutator $[\varepsilon_{m,n}\circ p_1,\epsilon_{m,n}\circ p_2]$ in the group $[S^{4m-1}\wedge Q_{n-m+1}, Sp(n)]$. So by Theorem <ref> we have

$$[\varepsilon_{m,n}\circ p_1,\epsilon_{m,n}\circ p_2 ]=\iota ([{\varepsilon_{m,n}}^*(y_{4m-1})\otimes {\epsilon_{m,n}}^*(y_{4n-4m+3})])= \iota([\sigma^{4m-1}\otimes \bar{y}_{4n-4m+3}]).$$

Therefore by Theorem <ref> when $m$ is even then the order of Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$ is equal to $\frac{(2n+1)!}{(2n-2m+1)!}$ and when $m$ is odd then the order of Samelson product $\langle \varepsilon_{m,n},\epsilon_{m,n} \rangle$ is equal to $\frac{2(2n+1)!}{(2n-2m+1)!}$.

§ PROOF OF THEOREM <REF>

In this section we will prove Theorem <ref>.Since the dimension of $\Sigma^{4n-4(m+1)}Q_2$ is equal to $4(n-m)+3$, we have

$$[\Sigma^{4n-4(m+1)}Q_2, BSp(n)]\cong [\Sigma^{4n-4(m+1)}Q_2, BSp(\infty)] \cong \widetilde{KSp}(\Sigma^{4n-4(m+1)}Q_2).$$

The cofibration sequence $ S^{4(n-m)-1}\rightarrow \Sigma^{4n-4(m+1)}Q_2\rightarrow S^{4(n-m)+3}$ induces the following exact sequence

$$\cdots \rightarrow \widetilde{KSp}(S^{4(n-m)+3})\rightarrow \widetilde{KSp}(\Sigma^{4n-4(m+1)}Q_2)\rightarrow \widetilde{KSp}(S^{4(n-m)-1})\rightarrow \cdots.$$

For all $i \geq 1$, we have $\widetilde{KSp}(S^{4i-1})=0$, this implies that $\widetilde{KSp}(\Sigma^{4n-4(m+1)}Q_2)=0$. Thus we get the following lemma.

There is an isomorphism $[\Sigma^{4n-4(m+1)}Q_2, BSp(n)]\cong 0$. $\quad \Box$

Apply the functor $[\Sigma^{4n-4(m+1)}Q_2,\quad ]$ to fibration $(2.1)$, we get the following exact sequence

\begin{align*}
[\Sigma^{4n-4(m+1)}Q_2,Sp(n)]\overset{(\alpha_k)_*}\longrightarrow [\Sigma^{4n-5}Q_2,Sp(n)]
&\longrightarrow [\Sigma^{4n-4(m+1)}Q_2, BSp(n)], \hspace{10mm}(4-1)
\end{align*}

where by Lemma <ref>, we have $[\Sigma^{4n-4(m+1)}Q_2, BSp(n)]\cong 0$. Therefore we get the following lemma.

There is an isomorphism $[\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k,m}(Sp(n))]\cong Coker(\alpha_k)_*$. $\quad \Box$

In what follows we will calculate $[\Sigma^{4n-5}Q_2,Sp(n)]$ and Im $(\alpha_k)_*$.

Consider the following commutative diagram

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
{{\widetilde{KSp}}^{-2}(\Sigma^{4n-4m-5}Q_2)& H^{4(n-m)+2}(\Sigma^{4n-4m-5}Q_2)\\ {\tilde{K}}^{-2}(\Sigma^{4n-4m-5}Q_2)& H^{4(n-m)+2}(\Sigma^{4n-4m-5}Q_2)\\}; \path[->,font=\scriptsize]
(m-1-1) edge node[above] {$\vartheta$}(m-1-2)
(m-1-1) edge node[left] {$c'$} (m-2-1)
(m-1-2) edge node[right] {$(-1)^{n-m+1}$}(m-2-2)
(m-2-1) edge node[above] {$\phi$} (m-2-2);
\end{tikzpicture}
\end{equation}

we first calculate the image of $\vartheta$. We have the following lemma.

The following hold:

$(a)\colon {\widetilde{KSp}}^{-2}(\Sigma^{4n-4m-5} Q_{2})=\mathbb{Z}\{\alpha,\beta\}$, where $\alpha\in {\widetilde{KSp}}^{-2}(S^{4n-4m-2})$ and
$\beta\in {\widetilde{KSp}}^{-2}(S^{4n-4m+2})$,

$(b)\colon {\tilde{K}}^{-2}(\Sigma^{4n-4m-5} Q_{2})=\mathbb{Z}\{\alpha' ,\beta'\}$, where $\alpha'\in {\tilde{K}}^{-2}(S^{4n-4m-2})$ and $\beta'\in {\tilde{K}}^{-2}(S^{4n-4m+2})$,

(c): if $m$ is even integer then

\begin{align*}
c'(\alpha) = \left\{ \begin{array}{ll}
2\alpha' & \textrm{$n$ is even },\\\\
\alpha' & \textrm{$n$ is odd },
\end{array} \right. &  c'(\beta) = \left\{ \begin{array}{ll}
\beta' & \textrm{$n$ is even },\\\\
2\beta' & \textrm{$n$ is odd },
\end{array} \right.
\end{align*}

and if $m$ is odd integer then

\begin{align*}
c'(\alpha) = \left\{ \begin{array}{ll}
\alpha' & \textrm{$n$ is even },\\\\
2\alpha' & \textrm{$n$ is odd },
\end{array} \right. & c'(\beta) = \left\{ \begin{array}{ll}
2\beta' & \textrm{$n$ is even },\\\\
\beta' & \textrm{$n$ is odd }.
\end{array} \right.
\end{align*}

Put $L_4={\widetilde{KSp}}^{-2}(S^{4(n-m)-1})$ and $L_5={\widetilde{KSp}}^{-2}(S^{4(n-m)+1})$. There is a cofibration sequence
$$S^{4n-4m-2} \rightarrow \Sigma^{4n-4m-5}Q_2\rightarrow S^{4n-4m+2},$$
this sequence induce the following commutative diagram of exact sequences

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
&L_5\\0&{\tilde{K}}^{-2}(S^{4n-4m+2})&{\tilde{K}}^{-2}(\Sigma^{4n-4m-5}Q_2)&{\tilde{K}}^{-2}(S^{4n-4m-2})&0,\\}; \path[->,font=\scriptsize]
(m-1-1) edge (m-1-2)
(m-1-2) edge (m-1-3)
(m-1-3) edge (m-1-4)
(m-1-4) edge (m-1-5)
(m-1-2) edge node[left] {$c'$}(m-2-2)
(m-1-3) edge node[left] {$c'$}(m-2-3)
(m-1-4) edge node[left] {$c'$}(m-2-4)
(m-2-1) edge (m-2-2)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4)
(m-2-4) edge (m-2-5);
\end{tikzpicture}
\end{equation}

where $L_5$ is zero and $L_4$ is zero when $n-m$ is even and is $\mathbb{Z}/2\mathbb{Z}$ when $n-m$ is odd. Thus we have

$${\widetilde{KSp}}^{-2}(\Sigma^{4n-4m-5}Q_2)=\mathbb{Z}\{\alpha,\beta\}, \quad {\tilde{K}}^{-2}(\Sigma^{4n-4m-5}Q_2)=\mathbb{Z}\{\alpha',\beta'\},$$

where $\alpha\in {\widetilde{KSp}}^{-2}(S^{4n-4m-2})$, $\beta\in {\widetilde{KSp}}^{-2}(S^{4n-4m+2})$, also $\alpha'\in {\tilde{K}}^{-2}(S^{4n-4m-2})$ and $\beta'\in {\tilde{K}}^{-2}(S^{4n-4m+2})$. We can choose $\alpha, \alpha', \beta, \beta'$ such that

Case one: $m$ is even integer

when $n$ is even then $c'(\alpha)=2\alpha', c'(\beta)=\beta'$ and when $n$ is odd then $c'(\alpha)=\alpha', c'(\beta)=2\beta'$.

Case two: $m$ is odd integer

when $n$ is even then $c'(\alpha)=\alpha', c'(\beta)=2\beta'$ and when $n$ is odd then $c'(\alpha)=2\alpha', c'(\beta)=\beta'$.

Consider the map $c'\colon Sp(n)\rightarrow SU(2n)$, restriction the map $c'$ to their quasi-projective spaces we get to map ${\bar{c}}'\colon Q_2\rightarrow \Sigma\mathbb{C}P^3$. The cohomology of $Q_2$ and $\Sigma\mathbb{C}P^3$ are given by

$$H^* (Q_2)=\mathbb{Z}\{{\bar{y}}_3,{\bar{y}}_7\},\qquad H^* (\Sigma\mathbb{C}P^3)=\mathbb{Z}\{{\bar{x}}_3,{\bar{x}}_5, {\bar{x}}_7\},$$

such that

$${\bar{c}}'({\bar{x}}_3)={\bar{y}}_3,\qquad {\bar{c}}'({\bar{x}}_5)=0,\qquad {\bar{c}}'({\bar{x}}_7)={\bar{y}}_7.$$

Note that ${\tilde{K}}^{-2}(\Sigma^{4(n-m)-4}\mathbb{C}P^3)\cong {\tilde{K}}^{0}(\Sigma^{4(n-m)-2}\mathbb{C}P^3)$ is a free abelian group generated by $\zeta_{2(n-m)-1}\otimes x$, $\zeta_{2(n-m)-1}\otimes x^2$ and $\zeta_{2(n-m)-1}\otimes x^3$. We can put $\alpha', \beta'$ as

$$\alpha'={\bar{c}}'(\zeta_{2(n-m)-2}\otimes x),\quad \beta'={\bar{c}}'(\zeta_{2(n-m)-2}\otimes x^3).$$

The image of map ${\widetilde{KSp}}^{-2}(\Sigma^{4n-4m-5}Q_2)\overset{\vartheta}\longrightarrow H^{4(n-m)+2}(\Sigma^{4n-4m-5}Q_2)$ is isomorphic to

Case one: $m$ is even integer

Im ϑ≅{
ℤ{1/3(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is even ,

ℤ{1/6(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is odd .

Case two: $m$ is odd integer

Im ϑ≅{
ℤ{1/6(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is even ,

ℤ{1/3(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is odd .

Consider the following commutative diagram

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.20ex]
{{\tilde{K}}^{-2}(\Sigma^{4(n-m)-4}\mathbb{C}P^3)& H^{4(n-m)+2}(\Sigma^{4(n-m)-4}\mathbb{C}P^3)\\ {\tilde{K}}^{-2}(\Sigma^{4(n-m)-5} Q_2)& H^{4(n-m)+2}(\Sigma^{4(n-m)-5} Q_2),\\}; \path[->,font=\scriptsize]
(m-1-1) edge node[above] {$\phi'$}(m-1-2)
(m-1-1) edge node[left] {${\bar{c}}'$} (m-2-1)
(m-1-2) edge node[right] {$\cong$}(m-2-2)
(m-2-1) edge node[above] {$\phi$} (m-2-2);
\end{tikzpicture}
\end{equation}

where the map $\phi'$ is defined similarly to the map $\varphi$. By definition of the map of $\phi'$ we have

\begin{align*}
\phi'(\zeta_{2(n-m)-2}\otimes x)&=(2(n-m)+1)!ch_{2(n-m)+1}(\zeta_{2(n-m)-2}\otimes x)\\&=
\frac{1}{3!}(2(n-m)+1)!\sigma^{4(n-m)-4}t^3,\\\\
\phi'(\zeta_{2(n-m)-2}\otimes x^2)&=(2(n-m)+1)!ch_{2(n-m)+1}(\zeta_{2(n-m)-2}\otimes x^2)\\&=
\frac{1}{2}(2(n-m)+1)!\sigma^{4(n-m)-4}t^3,\\\\
\phi'(\zeta_{2(n-m)-2}\otimes x^3)&=(2(n-m)+1)!ch_{2(n-m)+1}(\zeta_{2(n-m)-2}\otimes x^3)\\&=
\end{align*}

Now according to the commutativity of diagram $(4.3)$ we have

\begin{align*}
\phi(\alpha')=\phi({\bar{c}}'(\zeta_{2(n-m)-2}\otimes x))&=\phi'(\zeta_{2(n-m)-2}\otimes x)\\&=
\frac{1}{3!}(2(n-m)+1)!\sigma^{4(n-m)-5}\otimes \bar{y}_7,\\\\
\phi(\beta')=\phi({\bar{c}}'(\zeta_{2(n-m)-2}\otimes x^3))&=\phi'(\zeta_{2(n-m)-2}\otimes x^3)\\&=
2(2(n-m)+1)!\sigma^{4(n-m)-5}\otimes \bar{y}_7.
\end{align*}

Thus by the commutativity of diagram $(4.1)$ we get two following cases.

Case one:  $m$ is even integer

ϑ(α)= ϕ(c'(α))= {
-1/3(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is even,

1/6(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is odd ,

ϑ(β)= ϕ(c'(β))= {
-2(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is even,

4(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is odd.

Case two: $m$ is odd integer

ϑ(α)= ϕ(c'(α))= {
1/6(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is even ,

-1/3(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is odd.

ϑ(β)= ϕ(c'(β))= {
4(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is even,

-2(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7     if $n$ is odd.

Therefore when $m$ is even integer then

Im ϑ≅{
ℤ{1/3(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is even,

ℤ{1/6(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is odd .

and when $m$ is odd integer then

Im ϑ≅{
ℤ{1/6(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is even ,

ℤ{1/3(2(n-m)+1)!σ^4(n-m)-5⊗y̅_7}     if $n$ is odd.

There is an isomorphism

ℤ_2/3!(2n+1)!     if $n$ is even,

ℤ_1/3!(2n+1)!     if $n$ is odd.

Applying the functor $[\Sigma^{4n-5}Q_2,\quad]$ to the fibre sequence $(3.1)$, there is an exact sequence of groups

\begin{align*}
[\Sigma^{4n-5}Q_2,\Omega Sp(\infty)]\overset{(\Omega \pi)_*}\longrightarrow [\Sigma^{4n-5}Q_2,\Omega X_n]&\overset{\delta_*}\longrightarrow [\Sigma^{4n-5}Q_2,Sp(n)]\\& \overset{j_*}\longrightarrow [\Sigma^{4n-5}Q_2,Sp(\infty)]\overset{\pi_*}\longrightarrow [\Sigma^{4n-5}Q_2,X_n].
\end{align*}

We have the following isomorphism

$$[\Sigma^{4n-5}Q_2, Sp(\infty)]\cong [\Sigma^{4n-4}Q_2, BSp(\infty)]\cong {\widetilde{KSp}}^{-1}(\Sigma^{4n-5}Q_2).$$

Now applying the ${\widetilde{KSp}}^{-1}$ to the cofibration sequence $S^{4n-2}\rightarrow \Sigma^{4n-5}Q_2\rightarrow S^{4n+2}$. Since ${\widetilde{KSp}}^{-1}(S^{4n+2})=0$, we have ${\widetilde{KSp}}^{-1}(\Sigma^{4n-5}Q_2)=0$. Thus we get the following exact sequence

\begin{equation*}
{\widetilde{KSp}}^{-2}(\Sigma^{4n-5}Q_2)\overset{\psi'}\longrightarrow H^{4n+2}(\Sigma^{4n-5}Q_2)\rightarrow [\Sigma^{4n-5}Q_2,Sp(n)]\rightarrow 0.
\end{equation*}

Therefore there is an isomorphism $[\Sigma^{4n-5}Q_2,Sp(n)]\cong Coker \psi'$. On the other hand similar to the proof of Proposition <ref>, for $m=0$ we can conclude that

Im ψ'≅{
ℤ{2/3!(2n+1)!σ^4n-5⊗y̅_7}     if $n$ is even ,

ℤ{1/3!(2n+1)!σ^4n-5⊗y̅_7}     if $n$ is odd .

Thus the proof is completed.

In continuation of the article, we compute the Im $(\alpha_k)_*$. For this, we need the following lemmas. Put $A=\Sigma^{4n-4m-5}Q_2$ and $X'=S^{4m}\wedge A$.

Let $\mu\colon\Sigma A\rightarrow Sp(n)$ be an element of $\widetilde{KSp}(\Sigma^2 A)$ and $\mu'\colon S^{4m}\wedge A\rightarrow Sp(n)$ be the adjoint of the following composition

$$ S^{4m}\wedge \Sigma A \overset {\Sigma \varepsilon_{m,n}\wedge \mu}\longrightarrow \Sigma Sp(n)\wedge Sp(n)\overset {[ev,ev]}\longrightarrow BSp(n).$$

Then there is a lift $\tilde{\mu}$ of $\mu'$

\begin{equation*}
\begin{tikzpicture}
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.25ex]
{ & \Omega X_n\\ S^{4m}\wedge A &Sp(n)\\}; \path[->,font=\scriptsize]
(m-2-1) edge node [above] {$\mu'$} (m-2-2)
edge node [above] {$\tilde{\mu}$}(m-1-2)
(m-1-2) edge  (m-2-2);
\end{tikzpicture}
\end{equation*}

such that ${\tilde{\mu}}^*(a_{4n+2})=\sigma^{4m}\otimes \Sigma^{-1} {\mu}^*(y_{4(n-m+1)-1})$.

By [17], we know that there is a lift $\lambda\colon\Sigma Sp(n)\wedge Sp(n)\longrightarrow X_{n}$ of $[ev,ev]$ such that

$$\lambda^*(\bar{y}_{4n+3})=\sum\limits_{i+j=n+1}\Sigma y_{4i-1}\otimes y_{4j-1}.$$

Let $\tilde{\lambda}$ be the following composition

$$\tilde{\lambda}\colon  S^{4m}\wedge \Sigma A \overset {\Sigma \varepsilon_{m,n}\wedge \mu}\longrightarrow \Sigma Sp(n)\wedge Sp(n)\overset {\lambda}\longrightarrow X_{n}.$$

We have

\begin{align*}
{\tilde{\lambda}}^*(\bar{y}_{4n+3})&=(\Sigma \varepsilon_{m,n}\wedge \mu)^*{\lambda}^*(\bar{y}_{4n+3})\\&= (\Sigma \varepsilon_{m,n}\wedge \mu)^*(\sum\limits_{i+j=n+1}\Sigma y_{4i-1}\otimes y_{4j-1})\\&=
\sigma^{4m}\otimes {\mu}^* (y_{4(n-m+1)-1}).
\end{align*}

Now we take $\tilde{\mu}\colon S^{4m}\wedge A \longrightarrow \Omega X_{n}$ to be the adjoint of the following composition

$$\Sigma S^{4m}\wedge A \overset {S}\longrightarrow S^{4m}\wedge \Sigma A\overset {\tilde{\lambda}}\longrightarrow X_{n},$$

that is $\tilde{\mu}\colon ad (\tilde{\lambda}\circ S)$, where the map $S\colon\Sigma S^{4m}\wedge A \longrightarrow S^{4m}\wedge \Sigma A $ is the swapping map and the map $ad\colon [\Sigma S^{4m}\wedge A, X_{n} ]\longrightarrow [S^{4m}\wedge A, \Omega X_{n}] $ is the adjunction. Then $\tilde{\mu}$ is a lift of $\mu$. Note that

\begin{align*}
(\tilde{\lambda}\circ S)^*(\bar{y}_{4n+3})=S^* \circ \tilde{\lambda}^*(\bar{y}_{4n+3})&=
S^*(\sigma^{4m}\otimes {\mu}^*(y_{4(n-m+1)-1}))\\&= \sigma^{4m+1} \otimes \Sigma^{-1} {\mu}^*(y_{4(n-m+1)-1}),
\end{align*}

thus we get

$${\tilde{\mu}}^*(a_{4n+2})=\sigma^{4m}\otimes \Sigma ^{-1} {\mu}^* (y_{4(n-m+1)-1}).$$

Consider the map $\theta:\Sigma A\rightarrow Sp(n)$, then by Lemma <ref> there is a lift $\tilde{\theta}$ of $(\alpha_k)_*(\theta)$ such that
$${\tilde{\theta}}^*(a_{4n+2})=\sigma^{4m}\otimes \Sigma ^{-1} {\theta}^* (y_{4(n-m+1)-1}).$$

We define the map $\beta_k\colon \widetilde{KSp}(\Sigma^2A) \rightarrow H^{4n+2}(X')$ by $\beta_k(\theta)={\tilde{\theta}}^*(a_{4n+2})=a_{4n+2}\circ \tilde{\theta}$. Now Consider the following commutative diagram

\begin{equation}
\begin{tikzpicture}[baseline=(current bounding box.center)]
\matrix(m)[matrix of math nodes,
row sep=2em, column sep=2em,
text height=1.5ex, text depth=0.15ex]
{{[\Sigma^{4n-4(m+1)}Q_2,\mathcal{G}_k]}& {[\Sigma^{4n-4(m+1)}Q_2,Sp(n)]}&{[\Sigma^{4n-5}Q_2,Sp(n)]}&\quad\\ {\widetilde{KSp}}^{-2}(\Sigma^{4n-5} Q_2)& H^{4n+2}(\Sigma^{4n-5} Q_2)&{[\Sigma^{4n-5}Q_2,Sp(n)]}&0\\}; \path[->,font=\scriptsize]
(m-1-2) edge node[above] {$(\alpha_k)_*$}(m-1-3)
(m-1-2) edge node[left] {$\beta_k$}(m-2-2)
(m-2-1) edge node[above] {$\psi'$} (m-2-2)
(m-1-1) edge (m-1-2)
(m-1-3) edge (m-1-4)
(m-2-2) edge (m-2-3)
(m-2-3) edge (m-2-4);
\draw[double distance =2pt,font=\scriptsize]
(m-1-3) -- (m-2-3);
\end{tikzpicture}
\end{equation}

where $[\Sigma^{4n-4(m+1)}Q_2,Sp(n)]\cong\widetilde{KSp}(\Sigma^2A)$. We have $Im(\alpha_k)_*=Im\beta_k/Im\psi'$. Note that in the proof of Theorem <ref> we calculate the $Im\psi'$, thus we need to calculate $Im\beta_k$.Let $a: \Sigma Q_2\rightarrow BSp(\infty)$ be
the adjoint of the composition of the inclusions $Q_2 \rightarrow Sp(2) \rightarrow Sp(\infty)$ and also $b: \Sigma Q_2\rightarrow BSp(\infty)$ be the pinch map of the bottom cell $q: \Sigma Q_2 \rightarrow S^8$
followed by a generator of $\pi_8(BSp(\infty))\cong \mathbb{Z}$. Note that $\widetilde{KSp} (\Sigma Q_2)$ is a free abelian group with a basis $a , b$. Then we have

$$ch(c'(a))=\Sigma y_3-\frac{1}{6} \Sigma y_7, \qquad ch(c'(b))=-2\Sigma y_7.$$

Let $\theta_1= \textbf{q}(\zeta_{2(n-m+1)-4}c'(a))\in \widetilde{KSp}(\Sigma^{4(n-m+1)-7}Q_2)$, where $\textbf{q}:K \rightarrow KSp$ is the quaternionization. Also let $\theta_2:\Sigma^{4(n-m+1)-7}Q_2 \rightarrow BSp(\infty)$ be the composite of the pinch map to the top cell $\Sigma^{4(n-m+1)-7}Q_2 \rightarrow S^{4(n-m+1)}$ and a generator of $\pi_{4(n-m+1)}(BSp(\infty))\cong \mathbb{Z}$. Then $\widetilde {KSp} (\Sigma^2A)$ is a free abelian group with a basis $\theta_1, \theta_2$. We have

\begin{align*}
ch(c'(\theta_1))= \left\{ \begin{array}{ll}
\Sigma^{4(n-m+1)-7} y_3-\frac{1}{6} \Sigma^{4(n-m+1)-7} y_7  & \textrm{if $n-m+1$ is even},\\\\
2\Sigma^{4(n-m+1)-7} y_3+\frac{1}{3} \Sigma^{4(n-m+1)-7} y_7& \textrm{if $n-m+1$ is odd},\\
\end{array} \right.
\end{align*}

\begin{align*}
ch(c'(\theta_2))= \left\{ \begin{array}{ll}
-2\Sigma^{4(n-m+1)-7} y_7 & \textrm{if $n-m+1$ is even},\\\\
\Sigma^{4(n-m+1)-7} y_7 & \textrm{if $n-m+1$ is odd}.\\
\end{array} \right.
\end{align*}

$Im\beta_k$ is isomorphic to

ℤ{1/6(2(n-m+1)-1)! kσ^4n-5⊗y_7}     if $n-m+1$ is even,

ℤ{1/12(2(n-m+1)-1)! kσ^4n-5⊗y_7}     if $n-m+1$ is odd .

For $i=1,2$, by Lemma <ref>, $(\beta_k)_*(\theta_i)$ has a lift ${\tilde{\theta}}_{i,k}\colon S^{4m}\wedge A\rightarrow \Omega X_n$ such that

$${{\tilde{\theta}}_{i,k}}^*(a_{4n+2})=k\sigma^{4m}\otimes \Sigma ^{-1} {\theta_i}^* (y_{4(n-m+1)-1}).$$

Since $y_{4i-1}=\sigma (q_i)$ and $q_i={c'}^*(c_{2i})$, so we get

$${{\tilde{\theta}}_{i,k}}^*(a_{4n+2})=k\sigma^{4m}\otimes \Sigma ^{-2} q_{(n-m+1)}(\theta_i)=k\sigma^{4m}\otimes \Sigma ^{-2} c_{2(n-m+1)}(c'(\theta_i)). $$

On the other hand we have

\begin{align*}
c_{2(n-m+1)}(c'(\theta_1))= \left\{ \begin{array}{ll}
\frac{1}{6}(2(n-m+1)-1)! \Sigma^{4(n-m+1)-7} y_7  & \textrm{if $n-m+1$ is even},\\\\
\frac{1}{3}(2(n-m+1)-1)! \Sigma^{4(n-m+1)-7} y_7& \textrm{if $n-m+1$ is odd},\\
\end{array} \right.
\end{align*}

\begin{align*}
c_{2(n-m+1)}(c'(\theta_2))= \left\{ \begin{array}{ll}
2(2(n-m+1)-1)!\Sigma^{4(n-m+1)-7} y_7 & \textrm{if $n-m+1$ is even},\\\\
(2(n-m+1)-1)!\Sigma^{4(n-m+1)-7} y_7 & \textrm{if $n-m+1$ is odd}.\\
\end{array} \right.
\end{align*}

Therefore we get

\begin{align*}
{{\tilde{\theta}}_{1,k}}^*(a_{4n+2})= \left\{ \begin{array}{ll}
k\sigma^{4m}\otimes \frac{1}{6}(2(n-m+1)-1)! \Sigma^{4(n-m+1)-9} y_7  & \textrm{if $n-m+1$ is even},\\\\
k\sigma^{4m}\otimes\frac{1}{3}(2(n-m+1)-1)! \Sigma^{4(n-m+1)-9} y_7& \textrm{if $n-m+1$ is odd},\\
\end{array} \right.
\end{align*}

\begin{align*}
{{\tilde{\theta}}_{2,k}}^*(a_{4n+2})= \left\{ \begin{array}{ll}
k\sigma^{4m}\otimes 2(2(n-m+1)-1)! \Sigma^{4(n-m+1)-9} y_7  & \textrm{if $n-m+1$ is even},\\\\
k\sigma^{4m}\otimes(2(n-m+1)-1)! \Sigma^{4(n-m+1)-9} y_7& \textrm{if $n-m+1$ is odd}.\\
\end{array} \right.
\end{align*}

If $n-m+1$ is even, then $Im\beta_k \cong \mathbb{Z}\{\alpha_1, \beta_1\}$, where

\begin{align*}
&\alpha_1=\frac{1}{6}(2(n-m+1)-1)! k\sigma^{4n-5}\otimes y_7, \\
& \beta_1 = 2(2(n-m+1)-1)! k\sigma^{4n-5}\otimes y_7.
\end{align*}

Also $Im\beta_k$ is generated by $2\alpha_1 - \frac{1}{12}\beta_1=\frac{1}{6}(2(n-m+1)-1)! k\sigma^{4n-5}\otimes y_7$. If $n-m+1$ is odd, then $Im\beta_k \cong \mathbb{Z}\{\alpha_2, \beta_2\}$, where

\begin{align*}
&\alpha_2=\frac{1}{3}(2(n-m+1)-1)! k\sigma^{4n-5}\otimes  y_7, \\
& \beta_2 = (2(n-m+1)-1)! k\sigma^{4n-5}\otimes y_7.
\end{align*}

Also $Im\beta_k$ is generated by $\alpha_2 - \frac{1}{4}\beta_2=\frac{1}{12}(2(n-m+1)-1)! k\sigma^{4n-5}\otimes y_7$. Therefore we can conclude

Imβ_k ≅{
ℤ{1/6(2(n-m+1)-1)! kσ^4n-5⊗y_7}     if $n-m+1$ is even,

ℤ{1/12(2(n-m+1)-1)! kσ^4n-5⊗y_7}     if $n-m+1$ is odd .

Therefore we obtain the following proposition.

The following hold:

Case one:  $m$ is even integer then

Im (α_k)_*≅{
ℤ/((2n+1)!/(3(k,4t)))     if $n$ is even ,

ℤ/((2n+1)!/(6(k,t)))     if $n$ is odd .

Case two:  $m$ is odd integer then

Im (α_k)_*≅{
ℤ/((2n+1)!/(3(k,2t)))     if $n$ is even ,

ℤ/((2n+1)!/(6(k,2t)))     if $n$ is odd .

Therefore by Theorem <ref> and Proposition <ref> we get the following theorem.

There is an isomorphism

Case one: $m$ is even integer then

[Σ^4n-4(m+1)Q_2, B𝒢_k,m(Sp(n))]≅{
ℤ_(k,4t)     if $n$ is even ,

ℤ_(k,t)     if $n$ is odd ,

Case two:  $m$ is odd integer then

$$[\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k,m}(Sp(n))]\cong \mathbb{Z}_{(k,2t)}. \quad \Box$$

Here we prove Theorem <ref>.

$\textbf{Proof of Theorem \ref{a2}}$

We write $|H|$ for the order of a group $H$. Consider the exact sequence $(4-1)$. Suppose that $\mathcal{G}_{k,m}(Sp(n))\simeq \mathcal{G}_{k',m}(Sp(n))$, then there is an isomorphism of groups

$$[\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k,m}(Sp(n))]\cong [\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k',m}(Sp(n))].$$

Thus we have

$$|[\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k,m}(Sp(n))]|=|[\Sigma^{4n-4(m+1)}Q_2, B\mathcal{G}_{k',m}(Sp(n))]|.$$

Therefore by Theorem <ref> we can conclude the Theorem <ref>.

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Department of Mathematics, Faculty of Sciences,  Urmia University, P.O. Box $5756151818$, Urmia, Iran
E-mail<EMAIL_ADDRESS>